Dynamical Systems Seminar

Suppose we have a collection of objects such as such as the group of homeomorphisms or diffeomorphisms. We choose an object from this collection at “random”. What dynamical properties does this object have? The usual notion of a “random” or a “large” collection is that of generic or comeager, i.e., we show that the collection of all objects with certain dynamical properties contain a countable intersection of dense open sets. There is an alternate natural notion of bigness called prevalence and its complement called shy sets or Haar null sets. This notion of smallness often highlights different properties than the usual notion of generic. In this introductory talk, we give a background on prevalence, discuss a variety of results involving prevalence and analysis, prevalence and automorphism groups and state some open problems concerning prevalence and dynamics.

The variational principle states that the topological entropy of a compact dynamical system is a supremum of measure-theoretic entropies of invariant measures supported on this system. Therefore, one may ask whether we can get a similar formula for the topological entropy of a dynamical system restricted to the level sets, which are usually not compact. In several cases it was then possible to prove the so-called restricted variational principle formula: For every possible value $\alpha$ of the Lyapunov exponent, the topological entropy of the set of points with the Lyapunov exponent $\alpha$ is equal to the supremum of measure-theoretic entropies of invariant measures with Lyapunov exponent $\alpha$. In this talk, I will investigate the structure of the level sets of all Lyapunov exponents for typical cocycles. I will show that the restricted variational principle formula for a vector of Lyapunov exponents holds for typical cocycles. This generalizes the works of Barreira-Gelfert and Feng-Huang.

In 1950, Utz introduced expansive homeomorphisms, a concept widely acknowledged in current literature. Recently, there has been notable progress in exploring this idea from a pointwise dynamics perspective. This not only enhances existing theories but also reveals insights not accessible through a global dynamics approach. This talk focuses on two such notions: uniformly expansive points and minimally expansive points. We will explore the presence of uniformly expansive points in homeomorphisms on bounded intervals and their implications for identifying expansive homeomorphisms on the unit circle. Additionally, we will discuss how minimally expansive points contribute to topological stability of a point by establishing a pointwise form of Walters' stability theorem. We will discuss how this result can be applied for homeomorphisms on intervals which is not the case with other existing forms of Walters' stability theorem. This talk is based on joint work with Tarun Das.

Bounded density shifts are examples of hereditary subshifts. Bounded density shifts are defined by disallowing words whose sum of entries exceeds a value depending on the length of the word. After presenting some examples and reviewing the concepts of topological entropy and measure-theoretic entropy, we will provide sufficient conditions for bounded density shifts to have a unique measure of maximal entropy. This is joint work with Felipe García-Ramos and Ronnie Pavlov.

Let $G$ be an infinitely countable amenable group and $K$ be a finite subset of $G$, containing the unit $e$ and at least one more element. A set $V\subset G$ is called $K$-separated if the sets $Kv$, where $v$ ranges over $V$, are pairwise disjoint. A $K$-separated set $V$ is called maximal if for every $g\notin V$, $V\cup\{g\}$ is not $K$-separated. The collection of indicator functions of all maximal $K$-separated sets, treated as symbolic elements over $G$, is a closed and shift-invariant subset of $\{0,1\}^G$, hence it forms a topological dynamical system. We call such system the $K$-shift and denote it by $\Omega_K$. We show that if a $K$-shift supports at least one free measure, then it is universal in the sense that for every free ergodic measure-theoretic system $(Y,\nu,G)$ of entropy smaller than (topological) entropy of $\Omega_K$, there exists a measure $\mu$ supported by $\Omega_K$, which is isomorphic to $\nu$. We also show some partial generalizations of this result for general $G$-subshifts with specifications (we say that a symbolic system $(X,G)$ has specification with margin $M$ if for any two subsets $F_1,F_2$ of $G$, such that $MF_1\cap MF_2=\varnothing$, any $X$-admissible pattern over $F_1$ and any $X$-admissible pattern over $F_2$ coexist in some element of $X$). The talk is based on the joint work with Tomasz Downarowicz, Benjamin Weiss and Guohua Zhang.

Various notions of dimension capture how efficiently a set can be covered from different aspects. These can lead to different values for a set which is “inhomogeneous” in some sense. A recent trend in dimension theory has been to define one parameter families of dimensions which interpolate between two well-known dimensions in order to uncover finer geometric scaling properties of such sets. The talk will present how this can be done in a meaningful way and present results and applications for the class of planar self-affine carpets. Based on joint works with A. Banaji, J. M. Fraser and A. Rutar.

Topological dynamical systems with measure theoretically discrete spectrum can be organized into a hierarchy. This is done by the invertibility properties of their continuous factor maps to their maximal equicontinuous factor (MEF). Many such invertibility properties correspond to equivalent intrinsic dynamical rigidity properties. One of those is mean equicontinuity, which can be described in various ways. Its corresponding invertibility property is that the continuous factor map to the MEF is measure theoretically an isomorphism. Outside of the hierarchy lie systems like the Thue-Morse subshift whose continuous factor map to the MEF is measure theoretically 2:1. Other examples of such a form are given by Iwanik and Lacroix or by Williams. They exhibit rich spectral properties such as a singularly continuous part of the spectrum and thus do not fall into the aforementioned hierarchy. Our aim is to extend the hierarchy to also handle m:1 extensions. In the past few years those classes of systems have been studied under the viewpoint of multivariate forms of mean sensitivity, as for example by Li, Ye, Yu (2021). We propose a notion of multivariate mean equicontinuity called mean m-equicontinuity, which satisfies a Auslander-Yorke type dichotomy to the aforementioned multivariate mean sensitivities. We call a continuous extension of another system that is measure theoretically m:1 an m:1-topomorphic extension and conjecture an equivalence between mean m-equicontinuity and being an m:1-topomorphic extension of the MEF. We prove that m:1-topomorphic extensions of equicontinuous systems are mean m-equicontinuous and provide some ideas for the converse direction. This is joint work with J. Breitenbücher and T. Jäger.

Both Roman and I grew up scientifically on Ważewski Retract Theorem. Our research interests sometimes diverged, sometimes converged, but were always somehow intertwined. Recently, they merged again in a joint paper touching combinatorial topological dynamics. In the talk I will present a short story of half a century of our friendship and collaboration.

Lorenz maps are one-dimensional maps with a single discontinuity, which appear in a natural way as Poincaré maps in geometric models of the well-known Lorenz attractor. In this talk we will discuss connections between periodic points, completely invariant sets, transitivity and renormalizations of expanding Lorenz maps. This talk will be based in part on joint works with P. Porotski and P. Raith and with Ł. Cholewa.

One of the central discoveries in the theory of dynamical systems was that differentiable (or smooth) systems can display strongly chaotic behavior and in many ways behave like a sequence of random coin tosses. In this talk we will describe appearance and interactions of chaotic properties in smooth dynamics. We will highlight main developments, describe the state of the art and discuss some open problems in the field.

We introduce a continuous-time version of the Feldman-Katok pseudometric (FK-pseudometric) following some ideas of Marina Ratner. For discrete-time dynamical systems FK-pseudometric is a useful tool in the study of zero entropy loosely Bernoulli transformations (following Ratner’s suggestion we call them loosely Kronecker). García-Ramos and Kwietniak characterized loosely Kronecker transformations using the FK-pseudometric. With our version of the FK-pseudometric for flows (continuous-time dynamical systems) we have a characterization of loosely Kronecker flows analogous to the one by García-Ramos and Kwietniak. Using FK-pseudometric we also provide a purely topological characterization of every topological model of a loosely Kronecker flow.

A topological dynamical system is a pair (X,T) consisting of a compact metric space X and a continuous self-map T of X. Recently Downarowicz and Weiss [Ergodic Theory and Dynamical Systems, to appear] asked what is the relation between the vague specification property (introduced by Teturo Kamae in the 1970s) and the weak specification property (a variant of the specification property introduced by Bowen). We prove that the vague specification property is equivalent to the asymptotic average shadowing property, a variant of the classical shadowing property introduced by Gu in 2007. Since the weak specification property implies the asymptotic average shadowing property but the converse is not true, we have a complete answer to Downarowicz and Weiss' question. This is joint work with Alexandre Trilles.

One way to describe Toeplitz subshifts is via so-called semicocycles. The semicocycle's discontinuity points and their orbits are related to non-periodic positions and proximal components. In this talk we consider the question whether we can, in the case of a Toeplitz subshift with infinitely many discontinuity points, find a factor subshift with only finitely many of them. For this we discuss a sufficient and a necessary condition. The results are part of ongoing joint work with Franziska Sieron (FSU Jena).

The backward dynamics of a system could be far richer than its forward counterpart. The talk will consist of an example illustrating such behavior, demonstrating how one can define the backward limit sets (so called alpha limit sets) of mappings on compact metric spaces in different ways. Then it will proceed to the investigation of the connection between these limit sets and the Birkhoff center, how the alpha limit sets are related to omega limit sets and whether their relation is somehow restricted.

A "single orbit" approach to dynamics is the study of the interplay between the global properties of a dynamical system and the behaviour of its orbits. The talk will investigate how much information about the system can be deduced from the existence of an orbit satisfying the Wiener-Wintner theorem. We will use this information to provide a simple criterion for discrete spectrum based on properties of a generic point of an ergodic invariant measure. The talk is based on joint work with Melih Emin Can, Dominik Kwietniak and Piotr Oprocha.

Multiple ergodic averages for sparse sequences along primes Abstract: We discuss convergence results (in L^2) for multiple ergodic averages along sequences of polynomial growth evaluated at primes. Building on the work of Frantzikinakis, Host, and Kra who showed that polynomial ergodic averages along primes converge, we generalize their results to other sequences with polynomial growth. Combining our results with Furstenberg's correspondence principle, we derive several applications in combinatorics. The most interesting application is that positive density subsets of natural numbers contain arbitrarily long arithmetic progressions with common difference of the form [p^c], where c is a positive non-integer and p is a prime number. The main tools in the proof are a recent result of Matomäki, Shao, Tao, and Teräväinen on the uniformity of the von Mangoldt function in short intervals, a polynomial approximation of our sequences with good equidistribution properties, and a lifting trick that allows us to replace Z-actions on a probability space by R-actions on an extension of the original system. Joint work with A. Koutsogiannis.

Multiple ergodic averages for polynomial sequences along primes Abstract: After Furstenberg's celebrated extension of Szemeredi's theorem (every subset of natural numbers of positive upper density contains arbitrarily long arithmetic progressions) and its (integer) polynomial extension by Bergelson-Leibman, one of the main objects of study in ergodic theory is the understanding of the limiting behavior of multiple ergodic averages. In this talk, we will deal with the case of polynomial iterates and present how the problem evolved during the past few years. Also, because of the uniformity, in terms of Gowers norms, of the von Mangoldt function (a recent result by Green-Tao), we will see how one can extend the averaging along natural numbers results to the corresponding ones, averaging along primes.

The rank of a group is the minimum size of a generating set. The classical Nielsen-Schreier theorem implies that the rank of subgroups is at worst sublinear in the index. In many cases however there exist significantly more efficient generating sets. Rank gradient is a notion that quantifies this. In this talk, I will introduce rank gradient and explain its connection to another invariant from ergodic theory (the cost). I will also explain some interesting examples where we are able to compute the cost and thus show vanishing of rank gradient. No prior knowledge of rank gradient or cost will be assumed.

In this talk, I will introduce the classical example of P. Walters of the pseudo-Anosov diffeomorphism of the two-dimensional sphere, note several properties that are not very clear at first glance, and discuss a generalization of hyperbolicity called continuum-wise hyperbolicity that is present in this example. Then I will discuss some consequences of cw-hyperbolicity and recent advances in the classification of cw-hyperbolic surface homeomorphisms. If there is still time, I will prove the density of periodic points in the non-wandering set of cw-hyperbolic homeomorphisms with jointly continuous stable/unstable holonomies. These are joint works with Alfonso Artigue, José Vieitez, Welington Cordeiro, Rodrigo Arruda, Alberto Sarmiento, and Elias Rego.

We explore the notion of Gibbsiannes in several frameworks including symbolic dynamics, probability, mathematical physics, and operator algebras. We show the equivalence among several different notions/definitions of Gibbs state in different settings and possible consequences. Mainly based on a joint work with Bruno Fukushima-Kimura (Hokkaido University, Japan), Rafael Pereira Lima (Victoria University of Wellington, New Zealand) and Thiago Raszeja (AGH University of Krakow, Poland)

In this talk, I will present results obtained in a joint work with Katrin Gelfert. I will provide historical context motivating our study, define iterated functions systems (IFS) and associated contracting on average conditions, and present illustrative examples on the interval. I will also explore cases involving IFSs of circle diffeomorphisms, highlighting instances where contraction is not uniform. Our main result establishes the existence of a metric that makes an IFS of circle diffeomorphisms contracting on average. To finish I will define the Lyapunov exponent in this context and I will present a result establishing large deviations for this exponent.

We consider billiards on rational polygonal tables on the hyperbolic plane, and look into the dynamical properties that they display.

I will talk about recent results on topological dynamics of the Hénon maps obtained in joint work with Jan Boroński. For a parameter set generalizing the Benedicks-Carleson parameters (the Wang-Young parameter set) we obtain the following: The pruning front conjecture (due to Cvitanović, Gunaratne, and Procacci); A kneading theory (realizing a conjecture by Benedicks and Carleson); A classification: two Hénon maps are conjugate on their strange attractors if and only if their sets of kneading sequences coincide, if and only if their folding patterns coincide. The classification result relies on a further development of the authors’ recent inverse limit description of the Hénon attractors in terms of densely branching trees. (Joint work with Jan P. Boroński)

By Ornstein's isomorphism theorem, two Bernoulli shifts are isomorphic if and only if they have the same entropy. It is natural to ask whether a classification up to isomorphism is possible for other families of transformations, such as Kolmogorov automorphisms (or K-automorphisms for brevity). It is known that K-automorphisms cannot be classified by a complete numerical Borel invariant. This left open the possibility of classifying K-automorphisms with a more complex type of Borel invariant. We show that this is impossible, by proving that the collection of isomorphic pairs of K-automorphisms is not a Borel set. Our results also hold for K-automorphisms that are smooth diffeomorphisms of the five-dimensional torus. I will include a brief introduction to Kakutani equivalence and indicate how to simultaneously obtain non-classifiability for the isomorphism and Kakutani equivalence relations. This is joint work with Philipp Kunde.

We discuss the combinatorial construction in Ergodic Theory. We briefly discuss a very special technique “Approximation by conjugation,” that allows the construction of exciting maps on the manifolds with pre- scribed interesting topological and measure-theoretic properties. We present an example of an Invariant measure for the smooth category, which is a generic but non-ergodic measure satisfying other topological, mixing and ergodic properties on the 2-Torus. Also, present an explicit collection of the set containing the generic points of the system with interesting values of its Hausdorff dimension. As such, this talk should interest a broad readership, including those interested in Smooth Ergodic Theory, the Existence of Invariant measure, the Anosov Katok Method, Generic measures of the smooth category and the Hausdorff Dimension of the set containing only generic points. Keywords: Ergodic theory, Invariant measure, Generic and non-generic points, Hausdorff dimension, Anosov Katok method

Sequence entropy was introduced by Kushnirenko and it provided the first link between the functional analytic ergodic theory of von Neumann and the entropy-related ergodic theory of Kolmogorov's school, by proving that a system has pure point spectrum if and only if it has zero sequence entropy. Sensitivity (to initial conditions) is a classical notion for topological dynamics that is helpful to define chaos. For measure preserving systems sensitivity is a very weak notion. Nonetheless, a notion called called μ-mean sensitive is much natural in this context and García-Ramos proved that every system with an ergodic measure is either μ−mean sensitive or has discrete spectrum. In consequence, every system with an ergodic measure is μ-mean sensitivity if and only if it has positive sequence entropy. In this talk I will present a local version of the last result, which answers a question of Li and Yu. This is joint work with Felipe García-Ramos.

In recent years, there has been a flurry of activity in what is known as “Hyperbolic Linear Dynamics”. We introduce the basic notions such as hyperbolicity, shadowing and structural stability in the context of linear operators on separable Banach spaces. We discuss some recent developments and open problems.

A compact space $Y$ is called homeo-product-minimal if, given any minimal system $(X, T)$, it admits a homeomorphism $S : Y \rightarrow Y$ such that the product system $(X × Y, S × T)$ is minimal. The idea of homeo-product-minimality is motivated by the fact, that there exist minimal spaces, whose Cartesian powers do not admit minimal homeomorphisms. In [1, Theorem B] the authors listed the homeo-product-minimal spaces and raised the question whether the R. H. Bing’s pseudo-circle is homeo-product-minimal. In the talk we will show how to modify Handel’s construction [2] of the pseudo-circle to obtain the following result, which answers the question from [1]:
Theorem 1. The pseudo-circle is homeo-product-minimal. The results presented during the talk are obtained as a joint work with Jan Boroński and Piotr Oprocha.
References:
[1] M. Dirbák, L. Snoha, V. Špitalsky, ˇ Minimal direct products, Trans. Amer. Math. Soc. 375 (2022), 6453–6505.
[2] M. Handel, A pathological area preserving C∞ diffeomorphism of the plane, Proc. Amer. Math. Soc. 86 (1982), 163–168.

Many examples illustrating the limitations of the dynamical systems theory are defined on quite exotic spaces. We also know that the geometry of one-dimensional manifolds severely restricts available dynamics. On the other hand, physically relevant models of real-life phenomena result in smooth dynamical systems acting on manifolds. These observations are behind one of the most intriguing questions in the dynamical systems theory: given a property P that a dynamical system may or may not have, and a family C of dynamical systems decide whether a system exhibiting P exists in C. A very famous instance of that question is the smooth realisation problem. It asks if every measure preserving system isomorphic to a system given by a smooth diffeomorphism of a compact manifold preserving a measure equivalent to the volume element. The question is inspired by a brief remark made in 1932 by Johnny von Neumann in his foundational paper on ergodic theory and remains unresolved. During my talk, I will discuss the recent progress on the following variant of the realisation problem: which nonempty Choquet simplices can be realised as simplices of invariant measures for minimal homeomorphisms on manifolds. (joint work with Sejal Babel, Jernej Cinc, Till Hauser, Piotr Oprocha)

This is joint work with Tobias Jäger. Due to a result by Glasner and Downarowicz, it is known that a minimal system is mean equicontinuous if and only if it is an isomorphic extension of its maximal equicontinuous factor (MEF). The majority of well-known examples of this type are in addition almost automorphic, that is, the factor map to the MEF is almost one-to-one. The only cases of isomorphic extensions which are not almost automorphic are again due to Glasner and Downarowich, who in the same article construct such systems in a rather general and abstract setting. In this talk I present a construction using the Anosov-Katok method which provides an alternative way to produce such examples, and to obtain as a byproduct that these can be realised as smooth torus diffeomorphisms.

Generalised polynomials are polynomial-like expressions that additionally allow the use of the floor function. They can be studied using ergodic theory---in fact, they arise by evaluating a semialgebraic function along an orbit on a nilmanifold. In previous work with Jakub Konieczny we have noticed that you can find a generalised polynomial expression that vanishes precisely at the Fibonacci numbers. Which other linear recurrence sequences can you realise that way? We show that this is the case for linear recurrences with characteristic polynomial the minimal polynomial of a: i) Pisot unit of degree 2; ii) Pisot unit of degree 3 that is not totally real; and iii) Salem number. We expect that the above list is essentially complete (except for some cheap tricks). However, proving that you cannot realise something is much more difficult. Jakub Konieczny did that for powers of an integer k>=2, and in this paper we introduce the notion of a generalised polynomial on a number field and we use methods from diophantine geometry to prove a result that hopefully is a first step in extending this further. The talk is based on joint work with Jakub Konieczny.

Finding a ground state of a given Hamiltonian of an Ising model on a graph G=(V,E) is an important but hard problem. The standard approach for this kind of problem is the application of algorithms that rely on single-spin-flip Markov chain Monte Carlo methods, such as the simulated annealing based on Glauber or Metropolis dynamics. In this work, we investigate new algorithms, the so-called Digital Annealer's algorithm, and some particular kinds of stochastic cellular automata, the SCA and the $\epsilon$-SCA. We prove that if the temperature drops in time n as $\frac{1}{\log(n)}$, then the Markov chain asymptotically converges to the ground states. We also provide some simulations of these algorithms and show their superior performance compared to the conventional simulated annealing.

The pseudoarc is an intriguing continuum that was discovered by Knaster 100 years ago. While its structure is very complicated, Bing proved that from topological perspective, pseudoarc is the typical continuum to encounter in the plane. In this talk I will present selected results connecting pseudoarc as topological object with selected properties of dynamical systems.

For a topological dynamical systems (T, X) and a fixed x ∈ X we are interested in the distribution of prime and semi-prime orbits, i.e. {T^p(x)} where p runs over the primes and {T^{p1p2} x} where p1,p2 are primes. We are interested in systems for which the related sequences of empirical measures have a limit for every x in X (we then say that (T, X) satisfies a PNT or SPNT respectively). It turns out that there are not so many examples of such systems. Our knowledge is especially limited for systems which have some non-trivial chaotic behavior, for example weak-mixing or mixing systems. I will discuss some known results and some recent progress. I will focus on examples of smooth and weakly mixing systems which satisfy a PNT and also on some mixing systems which satisfy a PNT. Finally I will discuss the case of SPNT for horocycle flows.

The notion of generic/mean points goes back to the seminal work of Krylov and Bogolyubov. The first to investigate the question of what happens when all points of a dynamical system are generic for some invariant measure seem to be Dowker and Lederer in 1964. As it turns out, combining this property with other topological regularity criteria yields measure-theoretic rigidity results of the dynamical system. For example, minimality of the system implies its unique ergodicity in this setting. Another natural topological criterion in place of minimality is to assume that the map, which assigns each point its invariant measure to which it is generic, is continuous. By several recent works by different authors, the following picture emerges for abelian group actions in this setting: each point is generic for some ergodic measure and even stronger, each orbit closure is uniquely ergodic. In my talk, I will show that this is no longer the case for general actions by topological amenable groups, providing concrete counter-examples involving the group of all orientation preserving homeomorphisms on the unit interval as well as the Lamplighter group. Moreover, in the course of the talk I will elaborate on the recently introduced notion of weak mean equicontinuity. This is joint work with G. Fuhrmann and T. Hauser.

The aim of this talk is to present some aspects of the problem of the existence of distorted elements in groups of transformations. In particular we focus on our recent paper joint with Tomasz Szarek, constructively proving that rotations are distorted in the group of piecewise–affine circle homeomorphisms.

Dating back to the foundational paper by John von Neumann, a fundamental theme in ergodic theory is the isomorphism problem to classify invertible measure-preserving transformations (MPT's) up to isomorphism. Starting from the late 1990s, so-called anti-classification results use concepts from descriptive set theory to show precisely that classification for ergodic MPT's is impossible in terms of computable invariants, even with a very inclusive understanding of "computability''. %In a series of papers, Matthew Foreman, Daniel Rudolph and Benjamin Weiss have shown in a rigorous way that such a classification is impossible.In a landmark paper, M. Foreman, D. Rudolph and B. Weiss showed that the measure-isomorphism relation for ergodic MPT's is not a Borel set. Informally speaking, this result says that determining isomorphism between ergodic transformations is inaccessible to countable methods that use countable amount of information. Recently, this anti-classification result has been generalized to ergodic diffeomorphisms of compact surfaces and other equivalence relations like Kakutani equivalence. In the search for classification or anti-classification results, we place increasingly more stringent conditions on the family of transformations being considered for classification. Bernoulli shifts, which can be viewed as the most random type of transformations, have the strongest type of classification, a complete Borel invariant. But what about classes of transformations that are more random than ergodic transformations, but less so than Bernoulli shifts, such as weakly mixing transformations, mixing transformations, and K-automorphisms? In this talk we show how to extend anti-classification results in ergodic theory to the collection of weakly mixing systems.

The dynamical zeta function of a dynamical system is defined as the exponential power series of the sequence (a_n), where a_n is the number of periodic points of period n (not necessarily minimal). We consider a particular class of dynamical systems, namely, toral endomorphisms, induced by integer matrices. We characterise pairs of matrices inducing systems with the same dynamical zeta function. We also study dynamical systems such that the number of periodic points is equal to b_n, where b_n is a qth root of the number of periodic points of a given toral endomorphism. Assuming that the characteristic polynomial of a matrix inducing the endomorphism is irreducible, we show that there exist such systems only if the degree of the root q is equal to 2 and the polynomial is reciprocal. For tori of dimension at most 4, we provide a full characterization of endomorphisms such that a dynamical system like the above does exist.

A two-dimensional flat torus is often presented as a quotient of the plane by a lattice of translations, or as the result of gluing opposite sides of a parallelogram by translations. But whether one can perform such gluings in Euclidean space to transform a paper parallelogram into a polyhedral flat torus is not obvious. Diplotori are a family of such polyhedral flat tori. Though they were introduced in the late 1970s, many mathematicians are still unaware of their existence, including dynamicists working on tori, and a systematic study did not happen until the 2020s. In this talk, you will fold your own diplotorus and we will show that diplotori realize all flat tori. This is joint work with Pierre Arnoux and Samuel Lelievre.

It has been a general belief that entropy relates to the complexity of the behavior of a dynamical system. However, it has been shown in 2009 by Harańczyk and Kwietniak that exact maps on an interval attain lower entropy than pure mixing maps. This phenomenon has been extended to graph maps by Harańczyk, Kwietniak and Oprocha in 2014. We studied an analogous question whether the same assertion holds for dendrites, and specifically, for Gehman dendrite. It is known by works of Špitalský that for exact maps on Gehman dendrite the infimum of entropies is zero and is not attainable. We will present the construction of a family of pure mixing maps on Gehman dendrite satisfying the same property.

Furstenberg’s dynamical proof of the Szemerédi theorem initiated a thorough examination of multiple ergodic averages, laying the grounds for a new subfield within ergodic theory. Of the many families of multiple ergodic averages, special attention has been paid to averages of commuting transformations with polynomial iterates since they play a central role in Bergelson and Leibman’s proof of the polynomial Szemerédi theorem. Their norm convergence has been established in a celebrated paper of Walsh, but for a long time, little more has been known due to insurmountable obstacles encountered by existing methods. Recently, there has been an outburst of research activity which sheds new light on their limiting behaviour. I will discuss a number of novel results, including new seminorm estimates and limit formulas for these averages. Additionally, I will talk about new criteria for joint ergodicity of general families of integer sequences whose potential utility reaches far beyond polynomial sequences. The talk will be based on two recent papers written jointly with Nikos Frantzikinakis.

We study the descriptive complexity of sets of points de fined by placing restrictions on statistical behaviour of their orbits in dynamical systems. An example is the set of generic points of a given invariant measure. We obtain general upper bounds on the complexity, and lower bounds under a certain kind of specification property. The talk is based on joint work with Steve Jackson, Dominik Kwietniak and Bill Mance.

Given a measure-preserving group action $(\mu, X, G)$ one can associate to it a character (positive-definite conjugacy invariant function) on $G$ by \begin{align*} \chi(g)=\mu(\{x \in X: g x=x\}), g \in G . \end{align*} Anatoly Vershik suggested that for a "sufficiently rich" simple group $G$ every indecomposable character (extreme point in the space of characters) on $G$ can be obtained by formula (1) from some ergodic measure-preserving action of $G$. Given a Cantor minimal system one can associate to it two important groups of actions: the topological full group of the system and the approximately finite group of the related Bratteli diagram. In my talk I will explain the correspondence (1) for these groups and outline the proof of a generalization of Vershik's conjecture for them. The talk is based on an ongoing joint work with Konstantin Medynets.

The spectral structure of aperiodic tilings and point sets is reviewed, with some emphasis on pure point spectra and the projection method. In contrast, singular continuous spectra can be detected via exact renormalisation techniques, as will be demonstrated with a planar example. Finally, we will discuss the visible lattice points related dynamical systems of number-theoretic origin, with focus on examples with a two-dimensional shift action.

The projections of large circles in R^2 onto the standard torus T^2 become equidistributed as the radius of the circles goes to infinity. In this talk, we consider the analogous problem in the hyperbolic setting; more precisely, for any compact hyperbolic surface, we provide a precise asymptotic expansion of the equidistribution rate of arbitrary circle arcs of large radius. The method we use is inspired by the works of Ratner on quantitative mixing properties of the geodesic flow and of Burger. Furthermore, we discuss related distributional limit theorems and we give an explicit bound on the error term in the corresponding hyperbolic lattice counting problem (albeit weaker than the known estimates, which have been proved by Selberg and others using number theoretical methods). This is a joint work with E. Corso.

Following Hochman, say that a countable group G has the strong topological Rohlin property (STRP) if it admits a continuous action on the Cantor space whose conjugacy class is comeager (in an appropriate topology). It is a well-known result, originally proved by Kechris and Rosendal and then re-proved by different methods by Akin, Glasner and Weiss, that there is a generic homeomorphism on the Cantor space - or in other words, that the group of integers has the STRP. It was later proved by Hochman that Z^d, for d>1, does not have the STRP, and by Kwiatkowska that free groups on finitely many generators do have the STRP. We show that a countable group G has the STRP if and only if for any finite set A with at least two elements, in the space of all closed subshifts of A^G equipped with the Vietoris topology the isolated points are dense. The latter can be more conveniently restated in the language of SFTs, which will be done during the talk. Some applications will be presented, including a new proof of the STRP for Z.

We give a general introduction to C*-algebras with a particular emphasis on those arising from étale groupoids. These include C*-algebras arising from topological dynamical systems and various other combinatorial objects like directed graphs. Indeed, étale groupoids provide a natural lens through which to examine various combinatorial/dynamical properties and their C*-algebraic counterparts (e.g. minimality vs simplicity). Time permitting we will also outline our recent noncommutative extension of the classic Gelfand duality which unifies classic work of Dauns and Hofmann on C*-bundles with the more recent Weyl groupoid construction of Kumjian and Renault.

We are interested in the limit behaviour of Birkhoff sums over an infinite sigma-finite measure space. If the observable is integrable then – by a classical theorem by Aaronson – there exists no sequence of real numbers such that the Birkhoff sum normed by this sequence converges almost surely to 1. Under strong mixing conditions on the underlying system we prove a generalized strong law of large numbers for integrable observables using a truncated sum. We will see how this truncation works, how it is related to trimming in a finite system. For f not integrable we give conditions on f such that the Birkhoff sum normed by a sequence of real numbers converges almost surely to 1. This is joint work with Claudio Bonanno.

From the beginning of the algebraic topology, then also called the combinatorial topology, i.e. from the beginning of the 20th century, its basic object is the simplicial complex. The representation of a given topological space $X$ as a simplicial complex (i.e. a homeomorphism with it) is called triangulation. The name comes from the fact that when space X is a two-dimensional surface, triangulation means representing it as the union of adjacent triangles with edges meeting at the vertices. In the case of higher dimensions, the basic cells are the "i" - dimensional simplices. One of the natural questions is to find a triangulation with the minimum number of vertices, respectively of all simplexes (or estimate these numbers). This lecture will be devoted to this problem. We will present a new method based on the notion of covering type estimating from below the number of vertices by the weighted length of the elements in the cohomology ring $ H^*(X) $, or the weighted Lusternik-Schirelmann category theory. As a consequence, we got not only a unified method of proof of estimates of the number of vertices of the minimal triangulations derived originally by ad hock combinatorial methods, but also sharper estimates, or estimates for the families of manifolds not studied earlier.

Dendrites and measures with discrete spectrum Möbius disjointness conjecture proposed by Sarnak in 2009 ([2]) states that Möbius function is linearly disjoint from any dynamical system with zero dynamical entropy. The conjecture was confirmed on various one-dimensional spaces, such as interval, circle, topological graphs and some dendrites. It is also known that if all invariant measures have discrete spectrum, then the conjecture holds. In the talk we focus on the following question stated in [1]: Which one-dimensional continua $X$ have the property that every invariant measure of $(X, f)$ has discrete spectrum, assuming $f$ is a zero-entropy map? In particular we focus on dendrites with countable closure of the set of endpoints. We show that for such dendrites every recurrent point is minimal, which is the generalization of the property of zero-entropy interval maps. Then we prove that all invariant measures of zero-entropy maps of such dendrites have discrete spectrum, which confirms the Möbius disjointness conjecture on those spaces. Those results almost completely characterize dendrites for which all invariant measures of zero-entropy maps have discrete spectrum, leaving the case of dendrites with endpoint set countable with uncountable closure unsolved. All results presented during the talk are obtained as a joint work with J. Hantáková, J. Kupka, P. Oprocha and S. Roth. REFERENCES [1] J. Li, P. Oprocha, G. Zhang, Quasi-graphs, zero entropy and measures with discrete spectrum Nonlinear. (2022) 35 no. 3, 1360-1379, [2] P. Sarnak, Three lectures on the Mobius function randomness and dynamics, https://www.math.ias.edu/files/wam/2011/PSMobius.pdf

We consider a continuous-time dynamical system with a section. The associated Poincare map is defined as the first-return map to the section. That map provides an important information on the system, for example its periodic and fixed points correspond to periodic trajectories of the system. The existence of such points can detected by the homological Conley index, a topological invariant of the map. In the talk we present a theorem on determining the homological Conley index even if there is no information on the values of the Poincare map or its approximations, but a suitable information on some singular cycles for an index pair with respect to a small-step discretization of the system is provided. The talk is based on the preprint arXiv2106.14293

I will define two new constructions of finitely generated simple left orderable groups (in recent joint work with Hyde and Rivas). Among these examples are the first examples of finitely generated simple left orderable groups that admit a minimal action by homeomorphisms on the torus, and the first family that admits such an action on the circle. I shall also present examples of finitely generated simple left orderable groups that are uniformly simple (these were constructed by me with Hyde in 2019). And present new examples that, somewhat surprisingly, have infinite commutator width. Finally, I will present some new results around the second bounded cohomology of these groups (joint with Fournier-Facio).

We will examine three commonly used variants of van der Corput's Difference Theorem (vdCDT) in Hilbert spaces and show that they are associated with the notions of weak mixing, strong mixing, and Bernoullicity. We will then use this association to derive 2 new vdCDTs corresponding to ergodicity and mild mixing. We remark that our methods naturally yield vdCDTs for a class of unbounded sequences of vectors. We will then obtain an application to recurrence in measure preserving systems by giving a partial answer to a question of Frantzikinakis. If time permits, we will also discuss analogues of these vdCDTs in the context of uniform distribution and the classes of "mixing distributions" that they produce.

We will discuss some classical ergodic (Bernoulli, K-property, positive entropy...) and statistical (limit theorems, quantitative mixing...) properties of smooth dynamical systems. We will discuss their flexibility (i.e. non-trivial examples of systems which satisfy some but not all of them) and rigidity (i.e. some properties imply other). We will mostly focus on two results: 1) exponential mixing implies Bernoulli 2) existence of zero entropy systems satisfying a central limit theorem.

In their recent paper, S. Kiriki, Y. Nakano, and T. Soma introduced a concept of pointwise emergence to measure the complexity of irregular orbits. In our joint work with Yushi Nakano we consider the set of points with high pointwise emergence for topologically mixing subshifts of finite type. We show that this set has full topological entropy, full Hausdorff dimension, and full topological pressure for any H\"older continuous potential. Furthermore, we show that this set belongs to a certain class of sets with large intersection property. In my most recent paper I consider the set of points with high pointwise emergence for $C^{1+\alpha}$ diffeomorphisms preserving a hyperbolic measure. I find a lower bound on the Hausdorff dimension of this set in terms of unstable Hausdorff dimension of the hyperbolic measure. If the measure is an SRB, I prove that the set of points with high emergence has full Hausdorff dimension.

Generalised polynomials are expressions constructed from polynomials with the use of the floor function, addition and multiplication, such as [√2n [√3n2] + √6n + 1/2]. Despite superficial similarity, generalised polynomials exhibit many phenomena which are impossible for ordinary polynomials. In particular, there exist generalised polynomial sequences which take only finitely many values but are not constant. This is the case, for instance, for Sturmian sequences. In my talk, I will discuss finitely-valued generalised polynomial sequences from the perspective of combinatorics on words. The talk will include a survey of existing results on generalised polynomials, adapted to the current context, as well as several new results obtained in joint works with Boris Adamczewski and with Jakub Byszewski.

It has been observed in several contexts that certain substitutive sequences defined using substitutions of non-constant length could in fact also be obtained from substitutions of constant length. While it is easy to construct such examples artificially, they also occur naturally, and the corresponding constant-length substitution is often by no means obvious. A particularly striking example is the Lysënok morphism related to the presentation of the first Grigorchuk group. In the opposite direction, a problem of showing that a given substitutive sequence is not automatic has also appeared in several contexts, e.g. in the study of gaps between factors in the famous Thue–Morse sequence or in the mathematical description of the drawing of the classical Indian kolam. The problem of how to recognize that a substitutive sequence is automatic has been raised recently by Allouche, Dekking and Queffélec. In this talk we will give a simple complete characterisation of automaticity of uniformly recurrent (minimal) substitutive sequences. This is joint work with Clemens Müllner.

A system is said to have historic behaviour if there is a positive Lebesgue measure set of points having non convergent Birkhoff averages. The question of knowing whether systems with historic behaviour are abundant in some families of dynamics has recently regained attention, with the recent works of Kiriki and Soma, and Berger's definition of (local) emergence, which measures how big is the set of accumulation points of Birkhoff averages. In this talk, I will present two examples of systems with historic behaviour. The first one, obtained with Guarino and Santiago, is a modification of Bowen's eye example in which the set of points with historic behaviour is of positive Lebesgue measure but nowhere dense. The second one, in collaboration with Andersson, is the study of reparametrized linear flows of the two torus with two fixed points; we obtain some Diophantine conditions on the flow's parameters under which the system has/has not historic behaviour.

I will introduce the notion of a convolution system and show formulas for the topological pressure in this setting. As an application, I will give a plain formula for the topological pressure for the hereditary closure of any B-free system (for an arbitrary continuous potential). The talk will be based on joint paper with Joanna Kułaga-Przymus.

Normally hyperbolic invariant manifold theory deals with persistence results of invariant structures under perturbations. In the talk we will discuss a topological reformulation of the theory, which allows us to extend the results to an earlier unattainable setting. We consider perturbations of normally hyperbolic invariant manifolds, under which they can lose their hyperbolic properties. We show that if the perturbed map which drives the dynamical system preserves the properties of topological expansion and contraction, then the manifold is perturbed to an invariant set. The main feature is that our results do not require the `rate conditions' to hold after the perturbation. In this case the manifold can be perturbed to an invariant set, which is not a topological manifold.

Tree-shifts were introduced by Aubrun and Béal as a step-in between one-dimensional shifts and multi-dimensional shifts defined over the d-dimensional lattice. They are less complicated than multi-dimensional shifts but still enrich the dynamics of the classical one-dimensional case. At the same time they preserve a lot of important properties of one-dimensional shifts, which often fail for multi-dimensional case. During the talk I will introduce the notions of tree-shifts and generalise definitions from symbolic dynamics, focusing on tree-shifts of finite type. I will present attempts of generalisations of some theorems concerning one-dimensional shifts. In particular I will show relations between being of finite type, shadowing, openness and stability.

During the talk we will remind the definition and basic properties of the Feldman-Katok topology defined on the space of probability invariant (for a given map of a compact metric space) measures. Further, we will show that the rank function is lower semi-continuous with respect to this topology. As a corollary we will get that all royal measures, which were introduced by Gorodetsky, Ilyashenko, Kleptsyn and Nalski, have rank one. The talk is based on the joint work with Dominik Kwietniak.

Dating back to the foundational paper by John von Neumann, a fundamental theme in ergodic theory is the isomorphism problem to classify invertible measure-preserving transforma- tions (MPT's) up to isomorphism. In a series of papers, Matthew Foreman, Daniel Rudolph and Benjamin Weiss have shown in a rigorous way that such a classication is impossible. Besides isomorphism, Kakutani equivalence is the best known and most natural equivalence relation on ergodic MPT's for which the classication problem can be considered. In joint work with Mar- lies Gerber we prove that the Kakutani equivalence relation of ergodic MPT's is not a Borel set. This shows in a precise way that the problem of classifying such transformations up to Kakutani equivalence is also intractable.

The basic form of Stone duality tells us that a certain subcategory of topological spaces (zero-dimensional compact Hausdorff spaces) is opposite to the category of Boolean rings. We sketch this correspondence and explain how it can be used to provide algebraic duals to categories of symbolic dynamical systems. In the remainder of the talk, we show what some objects commonly studied in symbolic dynamics look like after this dualization.

I will survey what we know, and what we do not know about possible Borel complexities of sets of generic points in topological dynamical systems.

The pseudo-arc is besides the arc the only planar continuum (i.e. compact connected metric space) so that every of its proper subcontinua is homeomorphic to itself (Hoehn and Oversteegen 2020). Its first description appeared in the literature about a hundred years ago and due to many of its remarkable properties it is an object of interest in several branches of mathematics. There are results indicating that the pseudo-arc appears as a generic continuum in very general settings. For instance, Bing has proven that in any manifold M of dimension at least 2, the set of subcontinua homeomorphic to the pseudo-arc is a dense residual subset of the set of all subcontinua of M (equipped with the Vietoris topology). In this talk I will present a result which reveals that pseudo-arc is a generic object also in a certain measure theoretical setting; namely, I will show that the inverse limit of the generic Lebesgue measure preserving interval map is the pseudo-arc. Building on this result I will construct a family of attractors with attracting sets being the pseudo-arc with several interesting topological and measure-theoretical properties.

Non-stationary shift spaces are basic models of sequential dynamical system who were intensively studied in order to construct symbolic models for ergodic automorphism (Vershik) or in the context of the isomorphism problem of shift spaces (Krieger). Recently, the focus moved towards thermodynamic formalism and related questions. A fundamental tool of thermodynamic formalism, Ruelle's operator theorem, has no immediate generalization to the non-stationary setting as invariant functions intrinsically may not exist. However, it is possible to establish geometric ergodicity for a family of ratios of operators. This approach has applications to a classical problem in metric number theory. For a sequence $(\alpha_n)$ converging to $\infty$, set \[X_\alpha := \left\{ x = \frac{1}{x_1 + \frac{1}{x_2 + \cdots} } : x_n \in \mathbb N, x_n \geq \alpha_n \hbox{ for all } n \right\}.\] That is, $X$ is the subset of $[0,1]$ such that the $n$-th entry of the continued fraction expansion of each element is bigger than or equal to $\alpha_n$. In this setting, for $\alpha_n \gg n^{1+\epsilon}$, the geometric ergodicity implies a law of the iterated logarithm for square integrable functions from geometric ergodicity. If, in addition, $(\alpha_n)$ does not behave too wildly, the reference measure is absolutely contiunous with respect to the Hausdorff measure (and the Hausdorff dimension is $1/2$). \\

We study Iterated Function Systems (IFS) with compact parameter space. We show that the compactness of the phase space permits us to obtain a natural metric on the space of IFS which extends $C^0$-topology to the space of IFS. We then use this metric to define topological stability and to prove that in this context the classical results saying that shadowing property is a necessary condition for topological stability and that shadowing property together with expansiveness are sufficient conditions. For a proof of these statements, in fact we use a stronger type of shadowing property which we show to be different than the standard one. This is joint work with Alexander Arbieto.

A sequence is called automatic if it can be obtained as a coding of a fixed point of a substitution of constant length. We study the class of automatic systems, that is systems which arise as orbit closures of automatic sequences. Since there are only countably many automatic sequences, and since automatic systems usually have uncountably many points, it is interesting to study the combinatorial structure of the subset of an automatic system which comprises all of its points which are automatic. We give a dynamical description of this set, which is analogous to the one obtained by Holton and Zamboni for minimal substitutive systems. In particular, we show that automatic sequences in an infinite minimal automatic system correspond to the rationals in the ring of k-adic integers, the maximal connected equicontinuous factor of the system. As an application, we show that any minimal substitutive system which factors onto an infinite k-automatic system is itself k-automatic. We also state several conjectures which generalise our results to arbitrary substitutive systems, and explain their relation to Cobham-type results (connected with the ones obtained by Durand in 2011).

Given a 0-1 infinite matrix $A$, R. Exel and M. Laca have introduced a kind of \textit{generalized countable Markov shift} (GCMS) $X_A=\Sigma_A \cup Y_A$, which is a locally compact (in many important cases compact) version of $\Sigma_A$, the standard countable Markov shift. The elements of $Y_A$ are finite words, possibly including multiplicities. We develop the thermodynamic formalism for GCMS, where we introduced the notion of conformal measure on $X_A$, and we explored its connections with the usual formalism on $\Sigma_A$. Among the results, we highlight the finding of new conformal measures that are not detected by the thermodynamic formalism on $\Sigma_A$ and new phase transition phenomena: for a wide class of GCMS and potentials, we determined regions for the inverse of the temperature $\beta$, where we absence\existence of these new conformal probabilities, living on $Y_A$. The Gurevich entropy $h_G$ plays a fundamental role in determining these regions since the critical value for gauge potentials is $h_G$ when finite. We also have phase transition results for $h_G = \infty$, including the full shift. In addition, for the eigenmeasures of Ruelle's transformation, we discovered a length-type phase transition in the renewal shift: the existence of a critical value for $\beta$ where the measure passes from living on $\Sigma_A$ to live on $Y_A$. We showed that the notion of pressure introduced by M. Denker and M. Yuri for Iterated Function Systems (IFS) is a natural definition of pressure for $X_A$, and it coincides with the Gurevich pressure for GCMS basically for the same generality on which the thermodynamic formalism is developed for the standard countable Markov shifts and potentials. Joint work with R. Bissacot (University of São Paulo (USP), Brazil), R. Exel (Federal University of Santa Catarina (UFSC), Brazil), and R. Frausino (University of Wollongong (UOW), Australia).

In a joint work with Piotr Zgliczyński, we show that the methods used by Burns and Hasselblatt to prove Sharkovskii's Theorem can be generalized to the case of higher-dimensional maps with an attracting periodic orbit. As an application, we prove the existence of $n$-periodic orbits for almost all $n\in\mathbb{N}$ in the R\"ossler system with an attracting periodic orbit, for two sets of parameters. A part of the proof is computer-assisted.

We study properties and applications of the mean orbital pseudo-metric $\bar{\rho}$ on a topological dynamical system $(X,T)$ defined by \[ \bar{\rho}(x,y)= \limsup_{n\to \infty} \min_{\sigma \in S_n} \frac{1}{n}\sum_{k=0}^{n-1} d(T^k(x), T^{\sigma(k)}(y)), \] where $x,y\in X$, $d$ is a metric for $X$, and $S_n$ is the permutation group of the set $\{0,1,\ldots,n-1\}$. Writing $\hat{\omega}(x)$ for the set of $T$-invariant measure generated by the orbit of a point $x\in X$, we prove that the function $x\mapsto \hat{\omega}(x)$ is $\bar{\rho}$ uniformly continuous. This allows us to characterise equicontinuity with respect to the mean orbital pseudo-metric ($\bar{\rho}$-equicontinuity) and connect it to such notions as uniform or continuously pointwise ergodic systems studied recently by Downarowicz and Weiss. This is joint work with F. Cai, D. Kwietniak, and J. Li.

Given a compact subset $K$ of $R^3$, one may ask whether it can be realized as a (Lyapunov stable) attractor of a homeomorphism of $R^3$. In this generality the problem seems extremely complicated. We focus on a class of compacta $K$ called toroidal, which are those that have a neighbourhood basis comprised of solid tori. For instance, generalized solenoids and some wild knots are toroidal. Using techniques from geometric topology we give obstructions for a toroidal set to be realized as an attractor and show different methods to construct plenty of examples of toroidal sets that cannot be attractors in $R^3$. These results were obtained in collaboration with J.J. Sánchez-Gabites.

For $\mathscr{B}\subset \mathbb{N}$, let $\mathcal{M}_{\mathscr{B}}:=\bigcup_{b\in\mathscr{B}}b\mathbb{Z}$ and $\mathcal{F}_{\mathscr{B}}:=\mathbb{Z}\setminus\mathcal{M}_{\mathscr{B}}$ be the corresponding set of multiples and the $\mathscr{B}$-free set. The subshift $(X_\eta,\sigma)$, where $X_\eta\subset \{0,1\}^{\mathbb{Z}}$ is the orbit closure of $\eta=\mathbf{1}_{\mathcal{F}_{\mathscr{B}}}$ is called the $\mathscr{B}$-free system. Motivated by the special case of the square-free system (interesting from the number-theoretic viewpoint), where $\eta=\boldsymbol{\mu}^2$ (and $\boldsymbol{\mu}$ is the M\"obius function), Sarnak suggested to study the properties of $(X_\eta,\sigma)$. In particular, he showed that the square-free subshift is proximal, with the unique minimal subset being a singleton. There are two possible natural counterparts of $\mathcal{M}_{\mathscr{B}}$ in the multidimansional setting: countable unions of lattices in $\mathbb{Z}^m$ or countable unions of ideals in the ring of integers of algebraic number fields (see, e.g., works by Cellarosi, Vinogradov, Baake and Huck). I will concentrate on the latter approach and show that each mutlidimensional $\mathscr{B}$-free system has a~unique minimal subset and describe its structure. I will also discuss minimality, proximality and describe the maximal equicontinuos factor. The talk is based on a part of my doctoral dissertation.

Amorphic complexity, originally introduced for integer actions, is a topological invariant which measures the complexity of dynamical systems in the regime of zero entropy. We will explain its definition for actions by locally compact sigma-compact amenable groups on compact metric spaces. Afterwards, we will illustrate some of its basic properties and show why it is tailor-made to study strictly ergodic group actions with discrete spectrum and continuous eigenfunctions. This class of actions includes, in particular, Delone dynamical systems related to regular model sets obtained via cut and project schemes (CPS). Finally, for this family of Delone dynamical systems we present sharp upper bounds on amorphic complexity utilizing basic properties of the corresponding CPS. This is joint work with G. Fuhrmann, T. Jäger and D. Kwietniak.

We show that some results in additive combinatorics can be translated into corresponding results that are relevant to the mathematical theory of quasicrystals. Specifically, we will use the Freiman-Ruzsa theorem, characterising finite sets with bounded doubling, to obtain an alternative proof of a characterisation of Meyer sets, that is, relatively dense subsets of Euclidean spaces whose difference sets are uniformly discrete.

Article
https://arxiv.org/abs/2103.02289

The continuum random tree and Brownian map are important metric spaces in probability theory and represent the "typical" tree and metric on the sphere, respectively. The Brownian map in particular is linked to Liouville Quantum Gravity but the exact nature of the correspondence is unknown. In this talk I will explain a fairly dynamical construction of these spaces and show how recent advances in the dimension theory of self-similar sets can be used to shed light on general embedding problems. In particular, I will show that the Assouad dimension of these metric spaces is infinite and show how this restricts the nature of embeddings. Time permitting, I will also indicate how the construction of continuum trees may be used to analyse highly singular functions such as the Weierstrass-type functions.

In my talk I shall discuss some of my recent work on parametric families of maps and their strange attractors on surfaces, which employed inverse limit approach. They were focusing on computing accessible rotation numbers (e.g. for reduced Arnold Standard Family [1]), and building 1-dimensional models that are reductions of 2-dimensional dynamics in the presence of strong (mild) dissipation [2]. The latter was inspired by recent results of Crovisier and Pujals [3] (see also [4]).

References
[1] Boroński, J. P.; Činč, J.; Liu, X-C "Prime ends dynamics in parametrised families of rotational attractors". J. Lond. Math. Soc. (2) 102 (2020), no. 2, 557–579.
[2] Topological and Smooth Dynamics on Surfaces, Mathematisches Forschungsinstitut Oberwolfach Report No. 27/2020, DOI: 10.4171/OWR/2020/27
[3] S. Crovisier, E. Pujals, "Strongly dissipative surface diffeomorphisms", Commentarii Mathematici Helvetici 93 (2018), 377–400.
[4] S. Crovisier, E. Pujals, C, Tresser, "Mild dissipative diffeomorphisms of the disk with zero entropy", arXiv 2020.

Given two sequences over a finite alphabet, one can measure the distance between them by looking how their asymptotic behaviours differ. This gives rise to dynamically generated Besicovitch pseudometric. I will talk about the generalisation of this concept to actions of countable amenable groups. I will show that it induces a metric on the space of ergodic measures invariant under the action, and that in the case of the shift space, entropy function is continuous with respect to this metric. As an application of these results, I will show that if the considered group is in addition residually finite, then uniquely ergodic measures are entropy dense in the set of totally ergodic measures. This is a joint work with Martha Łącka.

Thermodynamic formalism has a lot to say in the context of sufficiently regular dynamical systems in compact spaces, for example about the existence and uniqueness properties of equilibrium states, and their characterisation as some derivative of the pressure function. This talk considers non-compact settings, particularly the case of countable Markov shifts. A first natural approach is to take the completion of the space and hope that the boundary created doesn’t interfere with too many thermodynamic properties. I’ll look at how one might do this, some drawbacks, and how they can, in some cases, be overcome.

In this talk we study different notions of entropy for Delone sets of finite local complexity in the setting of (metrizable and sigma-compact) locally compact Abelian groups (LCA groups). For Delone sets of finite local complexity (FLC) in the euclidean space it is well known that the patch counting entropy equals the topological entropy of an associated shift system. We present an example of a FLC Delone set in a LCA group for which the topological entropy and the patch counting entropy are not equal. It was suggested by J. Lagarias for FLC Delone sets in the euclidean space that the patch counting entropy can always be computed as a limit. We discuss why the Ornstein-Weiss lemma can not directly be used in order to see this claim and present that the correspondence between the topological and the patch counting entropy can be used in order to show that the limit in the patch counting entropy formula exists for compactly generated LCA groups. We present counterexamples where the limit does not exist in the context of general LCA groups.

The thermodynamic formalism for transitive uniformly hyperbolic dynamics is nowadays well understood and, among other aspects, it is worth mentioning that regular potentials (meaning Holder continuous) are so that the pressure function is differentiable and admit unique equilibrium states. The situation changes drastically in simple examples beyond uniform hyperbolicity, as the case of the Manneville-Pomeau maps, where different kinds of phase transitions appear due to the phenomenon of intermittency of an indifferent fixed point. In this talk I will focus on this family and discuss a new aspect of the phase transitions, namely the appearance of finitely additive absolutely continuous invariant measures.In particular, the second-order phase transition can be detected as a first-order phase transition for an extended pressure function. This is part of an ongoing work with A. Castro (UFBA) and L. Cioletti (UnB).

I will present some examples of skew-products with concave interval fiber maps over a certain subshift. Here the subshift occurs as the projection of those orbits that stay in a given neighborhood and gives rise to a new type of symbolic space which is (essentially) coded. The fiber maps have expanding and contracting regions. As a consequence, the skew-product dynamics has pairs of horseshoes of different type of hyperbolicity. In some cases, they dynamically interact due to the superimposed effects of the (fiber) contraction and expansion, leading to nonhyperbolic dynamics that is reflected on the ergodic level (existence of nonhyperbolic ergodic measures). The space of ergodic measures of the shift space is shown to be an entropy-dense Poulsen simplex, ergodic measures lift canonically to ergodic measures for the skew-product. Such skew-products can be embedded in increasing entropy one-parameter family of diffeomorphisms which stretch from a heterodimensional cycle to a collision of homoclinic classes. I will discuss some ingredients of associated bifurcation phenomena that involve a jump of the space of ergodic measures and, in some cases, also of entropy. (Joint work with L.J.Díaz and M.Rams)

I will present the notion of a Multiorder of a countable group, a particular case of an Invariant Random Order introduced by John Kieffer in 1975. I will discuss how multiorder is related to orbit equivalence to Z-actions and I will prove that if the group is amenable then each multiorder has the Følner property. If time permits, I will also show how to construct a uniformly Følner multiorder of entropy zero, using a tiling system.

Quantum dynamical entropy quantifies the irreducible randomness of the sequences of outcomes generated by a repetitively measured quantum system that between each two consecutive measurements is subject to unitary evolution. For several classes of quantum measurements, we derive an efficient formula for dynamical entropy by establishing the limiting measure of the Markov chain generated by the system and evaluating the Blackwell integral entropy formula. We also discuss the class of chaotic unitaries, i.e., those with potential to generate maximally random sequences of outcomes. Employing the notion of complex Hadamard matrices, we give a necessary condition for chaoticity (expressed in terms of the operator’s trace and determinant), which in dimensions 2 and 3 is sufficient as well.

The entropy of the product of stationary processes is related to Furstenberg’s filtering problem. In its classical version one deals with the sum $\mathbf{X}+\mathbf{Y}$, where $\mathbf{X}$ corresponds to the signal and $\mathbf{Y}$ to the noise. In his seminal paper from 1967, Furstenberg showed that under the natural assumption of the disjointness of underlying dynamical systems, the information about $\mathbf{X}$ can be retrieved from $\mathbf{X}+\mathbf{Y}$. Instead of the sum, we study the product $\mathbf{X}\cdot\mathbf{Y}$. We give a formula for the entropy rate of $\mathbf{X}\cdot\mathbf{Y}$ (relative to that of $\mathbf{Y}$, for $\mathbf{X}$ and $\mathbf{Y}$ being independent). As a consequence, $\mathbf{X}$ cannot be recovered from $\mathbf{X}\cdot\mathbf{Y}$ for a wide class of positive entropy processes, including exchangeable processes, Markov chains and weakly Bernoulli processes. Moreover, we answer some open problems on the dynamics of $\mathscr{B}$-free systems (including the square-free system given by the square of the Moebius function). The talk is based on joint work with Michał Lemańczyk, see https://arxiv.org/pdf/2004.07648.pdf

We study which properties of shift spaces transfer to their Hausdorff metric dbar-limits. In particular, we study shift spaces we call dbar-approachable, which are Hausdorff metric dbar-limits of their own k-step Markov approximations. We provide a topological characterisation of chain mixing dbar-approachable shift spaces using the dbar-shadowing property. This can be considered as an analogue for Friedman and Ornstein's characterisation of Bernoulli processes. We prove that many classical specification properties imply chain mixing and dbar-approachability. It follows that there are tons of interesting dbar-approachable shift spaces (mixing shifts of finite type, or more generally mixing sofic shifts, or even more generally, shift spaces with the specification or beta-shifts). In addition, we construct minimal and proximal examples of dbar-approachable shift spaces, thus proving dbar-approachability is a more general phenomenon than specification. We also show that dbar-approachability and chain-mixing imply dbar-stability, a property recently introduced by Tim Austin in his study of Bernoulliness of equilibrium states. This allows us to provide first examples of minimal or proximal dbar-stable shift spaces, thus answering a question posed by Austin. Finally, we show that the set of shift spaces with entropy-dense ergodic measures is closed wrt dbar Hausdorff metric. Note that entropy-density of ergodic measures is known to hold for many classes of shift spaces with variants of the specification property, but our result show that in these cases the entropy-density is a mere consequence of entropy-density of mixing shifts of finite type and dbar-approachability. Since we know there are examples of minimal or proximal dbar-approachable shifts, we see that our technique yields entropy-density for examples which were beyond the reach of methods based on specification properties. Finally, we apply our technique to hereditary closures of B-free shifts (a class including many interesting B-free shifts). These shift spaces are not chain-mixing, hence they are not dbar-approachable, but they are easily seen to be approximated by naturally defined sequences of transitive sofic shifts, and this implies entropy-density. This is a joint work with Jakub Konieczny and Michal Kupsa.

Given a set K of probability measures on a Cantor set X, one can ask whether there exists a minimal homeomorphism (= all orbits are dense) whose invariant probability measures are exactly the elements of K. We say that K is a dynamical simplex if such a homeomorphism exists; I will present a characterization of dynamical simplices, which is based in large part on work of T. Ibarlucia and myself; and try to explain the proof strategy, based on the notion of Kakutani-Rokhlin partitions. The talk will be introductory in nature and not assume prior knowledge of Cantor dynamics.

Automatic sequences - that is, sequences computable by finite automata - give rise to one of the most basic models of computation. As such, for any class of sequences it is natural to ask which sequences in it are automatic. In particular, the question of classifying automatic multiplicative sequences has attracted considerable attention in recent years. In the completely multiplicative case, such classification was obtained independently by S. Li and O. Klurman and P. Kurlberg. The main topic of my talk will be the resolution of the general case, obtained in a recent preprint with M. Lemańczyk and C. Müllner.

Tameness is a notion which - very roughly speaking - refers to the absence of topological complexity of a dynamical system. The last decades saw an increased interest in tame systems revealing their connections to other areas of mathematics like Banach spaces, substitutions and tilings or even model theory and logic. In this talk, we will assume a dynamical systems perspective.
Huang showed that, given a minimal system, tameness implies almost automorphy [1]. That is, after discarding a meagre set of points, the factor map of a tame minimal system to its maximal equicontinuous factor is one-to-one. This structural theorem got recently extended to actions of general groups by Glasner [2].
In a collaboration with Glasner, Jäger and Oertel, we could further improve this result by showing that tame minimal systems are actually regularly almost automorphic [3]. In this talk, we will show a closely related statement which, however, is way easier to prove: every symbolic almost automorphic extension of an irrational rotation whose non-invertible fibres form a Cantor set is non-tame. We will further discuss some related results from a collaboration with Kwietniak [4]. Finally, if time allows, we will come to discuss tameness in substitutive subshifts and more general classes of Toeplitz flows [5].
All (non-standard) notions will be introduced in the talk. In other words: we prioritise accessibility over the number of results to be discussed.
[1] W. Huang, Tame systems and scrambled pairs under an abelian group action, Ergodic Theory Dynam. Systems 26 (2006), 1549-1567.
[2] E. Glasner, The structure of tame minimal dynamical systems for general groups, Invent. Math. 211 (2018), 213-244.
[3] G. Fuhrmann, E. Glasner, T. Jäger, C. Oertel, Irregular model sets and tame dynamics, arXiv:1811.06283, (2018), 1-22.
[4] G. Fuhrmann, D. Kwietniak, On tameness of almost automorphic dynamical systems for general groups, Bull. Lon. Math. Soc. 52 (2020), 24-42.
[5] G. Fuhrmann, J. Kellendonk, R. Yassawi, work in progress.

Model sets have been introduced by Yves Meyer in 1972. As the underlying cut and project schemes present a quite general method to construct aperiodic point-sets with long-range order, they are often studied in the theory of mathematical quasicrystals. At the same time, they present an interesting class of examples in the context of topological dynamics. In this talk, we will concentrate on the dynamics of so-called irregular model sets, whose dynamics are generally more complicated and less understood than that of regular models (like the Fibonacci quasicrystal). We show that the Delone dynamical systems associated to irregular model sets often show positive entropy, but the construction also allows for uniquely ergodic zero entropy examples. However, irregular models sets cannot be tame, which provides a lower bound for the complexity of their dynamics.

We investigate the flexibility of the entropy (topological and metric) for the class of piecewise expanding unimodal maps. We show that the only restrictions for the values of the topological and metric entropies in this class are that both are positive, the topological entropy is at most log 2, and by the Variational Principle, the metric entropy is not larger than the topological entropy. In order to have a better control on the metric entropy, we work mainly with topologically mixing piecewise expanding skew tent maps, for which there are only two different slopes. For those maps, there is an additional restriction that the topological entropy is larger than ½ log 2. We also make the interesting observation that for skew tent maps the sum of reciprocals of derivatives of all iterates of the map at the critical value is zero. It is a generalization and a different interpretation of the Milnor-Thurston formula connecting the topological entropy and the kneading determinant for unimodal maps.

For a noninvertible dynamical system $(X,f)$ a point $x$ can have many possible “pasts.” Special alpha-limit sets were defined to contain all the limit points of all those backward orbits, and it turns out that for interval maps they have many good properties. For example, a point belongs to its own special alpha-limit set (this is like “backward recurrence”) if and only if it is in the attracting center of the interval map [Hero, 1992].

One of the last papers by Sergei Kolyada proposes several conjectures and open problems about topological properties of special-alpha limit sets. This talk will address those problems. The project is joint work with Jana Hantáková.

Recall that an infinite sequence over a finite alphabet $A$ is quasi-regular, if it is a generic point for a (non-necessarily ergodic) shift-invariant measure. Given a quasi-regular point $x$ in the full shift over $A$ we write $h(x)$ for the Kolmogorov-Sinai entropy of the shift invariant Borel probability measure generated by $x$. We prove that $h$ is uniformly continuous on the set of all quasi-regular points endowed with the $\overline{f}$ (pseudo)distance. We also give an alternative proof of Abramov's formula for the entropy of induced transformations. This is a joint work with Tomasz Downarowicz and Martha Łącka.

Given a countable group $G$ acting on a Cantor set $X$ by transformations preserving a probability measure, the action is essentially free if the set of points with trivial stabilizers has full measure. On the other hand, there are many examples of group actions, where every point has a non-trivial stabilizer. In this talk, we generalize the notion of an essentially free action to such actions, using the notion of holonomy. For equicontinuous actions of countable groups on Cantor sets, we answer the following question: under what conditions there exists a subgroup $H$ of $G$, such that the stabilizers of almost all points in $X$ are conjugate to $H$? These conditions are that the action is locally quasi-analytic and that the action is uniformly non-constant. The notion of a locally quasi-analytic action was previously used by the speaker in the joint works with Hurder to classify equicontinuous actions on Cantor sets. The notion of a uniformly non-constant action is new and was introduced in this work. This is joint work with Maik Groeger.

A subset of the integers $P$ is called predictive if for all zero-entropy processes $X_i$; $i\in\mathbb{Z}$, $X_0$ can be determined by $X_i$; $i\in P$. The classical formula for entropy shows that the set of natural numbers forms a predictive set. In joint work with Benjamin Weiss, we will explore some necessary and some sufficient conditions for a set to be predictive. These sets are related to Riesz sets (as defined by Y. Meyer) which arise in the study of singular measures. This and several other questions will be discussed during the talk.

We provide a purely topological description of minimal and uniquely ergodic systems whose unique invariant measure is loosely Bernoulli and has zero entropy (we call such measure preserving systems loosely Kronecker). At the heart of our result lies Feldman-Katok continuity, that is, continuity with respect to the Feldman-Katok pseudometric, which is a topological counterpart of the pseudometric f-bar on a symbolic space. The talk is based on a joint work with Felipe García-Ramos (CONACyT & UASLP, Mexico).

In my talk I will present a result on the Conley Index of rest points of a gradient-like flows on manifolds proven using the notion of Lusternik-Shnirelmann cathegory given by Jonathan Pears in a paper "Degenerate Critical Points and the Conley Index".

Let us fix an integer $b>1$. We say that a real number $x\in [0,1)$ with base-$b$ expansion $x=0.a_1 a_2 a_3\ldots$ is normal if for every finite word $s=d_1 d_2\ldots d_k$, where $0\leq d_i< b$, we have that $\lim_{N\to \infty}\frac{v(s,x,N)}{N}=\frac{1}{b^k},$ where $v(s,x,N)$ denotes the number of times that $s$ appears as a subword of the first $N+k-1$ digits of $x$. The aim of the paper is to introduce the notion of deterministic sequence, present examples of deterministic sequences, and show the following result: let $x\in [0,1)$ with base-$b$ expansion $x=0.a_1 a_2 a_3\ldots$ be normal. Let $(n_{i})$ be an increasing sequence of positive integers. Then the number $0.a_{n_1} a_{n_2} a_{n_3} \ldots$ is normal if and only if $(n_i)$ is deterministic. The talk is based on the paper 'A Hot Spot Proof of the Generalized Wall Theorem' by Bergelson and Vendehey.

The talk presents some topological theorems on the existence of periodic orbits of autonomous differential equations

Using Morse decomposition we define Conley category of an isolated, invariant set. We discuss some of its properties and the relation to the discrete Conley index in the invariant subspace and in the entire space.

In this continuation of the previous talks more properties of the Conley index will be discussed, including its appropriation to discrete time dynamical systems.

In this continuation of the previous talk more properties of the Conley index will be discussed, including its appropriation to discrete time dynamical systems.

Let $(X,T)$ be a minimal topological dynamical system, and let $m$ be a probabilistic invariant measure. We say that a point $x$ in $X$ is generic for $m$, if for any continuous real-valued function $f$ on $X$ the sequence of averages $(f(x) + f(Tx) + ... + f(T^n(x))/(n+1)$ converges to the integral of $f$ wrt. $m$. We study the Borel complexity of the set of generic points $\mbox{Gen}(m)$. We prove that $\mbox{Gen}(m)$ is always an $F_{\sigma\delta}$ set (a countable intersection of $F_\sigma$ sets), and produce an example which matches this upper bound, that is, an example of a minimal dynamical system with an invariant measure $m$ such that its set of generic points $\mbox{Gen}(m)$ does not appear at a lower level of Borel hierarchy ($\mbox{Gen}(m)$ is neither $F_\sigma$ nor $G_\delta$).

Toeplitz subshifts are examples of almost periodic but not periodic symbolic dynamical systems. During the talk, we show that under some mild assumptions (a properly generalized version of the regularity of a Topeplitz configuration) the naive entropy of a regular Toeplitz subshift over a residually finite group is zero, which implies the same result for Rokhlin and sofic entropy. We reduce the problem to the computation of entropy of odometers, which are factors of Toeplitz subshifts. The presentation will be based on my Masters Thesis.

The entropy of obstructions to positive expansivity at scale $\varepsilon$ is the supremum of the entropies $h_\mu (f)$ taken over the set of ergodic $f$-invariant Borel probability measures giving a positive measure to the set of points not satisfying the condition for positive expansivity. The entropy of obstructions to specification at scale $\varepsilon$ is the infimum of the entropies of special collections $P\cup S$ of orbit segments. These collections $P$ and $S$ are such that any finite collection of orbit segments can be $\varepsilon$-shadowed (with a fixed gap size) as long as one is allowed to remove elements of $P$ from the beginning of each piece and elements of $S$ from the end of each piece. In my talk, we will see if the entropy of such obstructions is smaller than the topological entropy of the continuous map $f$, then $f$ is intrinsically ergodic, i.e. has a unique measure of maximal entropy. The talk is based on the paper "Intrinsic ergodicity via obstruction entropies" by V. Climenhaga and J. Thompson.

During my talk I will introduce and explain sofic group by Gromov-Weiss and dynamical motivation behind them (Gottschalk's surjunctivity conjecture and dual Gottschalk's conjecture on Bernoulli shifts, Kaplansky direct finiteness conjecture on group rings). In particular, we look at metric approximations of groups and certain limit objects, called metric ultraproducts, which are groups and are useful in describing the general theory of approximations by metric groups. The importance of these limit objects in group theory became apparent recently and they are intensively studied. In the second part of the talk I will concentrate on simplicity and amenability of metric ultraproducts of groups. Examples are: some classes of linear groups over infinite fields, which are uniformly metric simple and Higman-Thompson groups, which are not-uniformly metric amenable and IET group. These generalize, respectively, the previously studied notions of uniform amenability and uniform simplicity.

The entropy of obstructions to positive expansivity at scale $\varepsilon$ is the supremum of the entropies $h_\mu (f)$ taken over the set of ergodic $f$-invariant Borel probability measures giving a positive measure to the set of points not satisfying the condition for positive expansivity. The entropy of obstructions to specification at scale $\varepsilon$ is the infimum of the entropies of special collections $P\cup S$ of orbit segments. These collections $P$ and $S$ are such that any finite collection of orbit segments can be $\varepsilon$-shadowed (with a fixed gap size) as long as one is allowed to remove elements of $P$ from the beginning of each piece and elements of $S$ from the end of each piece. In my talk, we will see if the entropy of such obstructions is smaller than the topological entropy of the continuous map $f$, then $f$ is intrinsically ergodic, i.e. has a unique measure of maximal entropy. The talk is based on the paper "Intrinsic ergodicity via obstruction entropies" by V. Climenhaga and J. Thompson.

I will present a new proof of the Abramov formula for the Kolmogorov-Sinai entropy of the induced measure preserving transformation.

The talk will cover an overview of the Brouwer degree, Morse index, Brock Fuller index, Conley index (in detail), and the Ważewski theorem.

In 1957, Jaroslav Kurzweil introduced in the literature a class of integral equations called generalized ordinary differential equations (GODEs, for short). These equations have been shown to be a powerful tool to investigate measure functional differential equations, measure neutral functional differential equations, among other types of equations, obtaining more general results for these equations. In this talk, we provide a basic overview of the Kurzweil integration and its contribution to the theory of functional differential equations as well as we present some open problems in the area.

The celebrated concept of specification, introduced by R. Bowen, serves as a tool for many results both in ergodic theory and topological dynamics. Regrettably, a smooth map satisfying the specification property is rare even among the class of partially hyperbolic diffeomorphisms. In this talk, I will recall a weakening of the specification property, the gluing orbit property (also known as transitive specification or weak specification), describe its abundance on the space of homeomorphisms and its relation with partially hyperbolic dynamics.

Renyi introduced representations of real numbers with an arbitrary base. Given a real number $\beta > 1$, not necessarily an integer, one considers the map $T_\beta (x) = \beta *x \mbox{ mod } 1$ defined on the interval [0, 1). This map gives rise to a numeration system leading to expansions to the base $\beta$. There exists a unique $T_\beta$-invariant measure on [0, 1) maximizing the entropy of $T_\beta$; it is called the Parry measure. It is absolutely continuous with respect to the Lebesgue measure. A tentative classification of numbers according to the dynamical properties of the associated beta-shifts was introduced by F. Blanchard. During the talk, we pay special attention to two of these classes: $C_3$, where $\beta$ is in $C_3$ when the expansion of {$\beta$} (fractional part of beta) in base $\beta$ is not periodic, i.e., when the associated beta-shift is not a sofic system; $C_5$, where $\beta$ is in $C_5$ when the expansion of {$\beta$} in base $\beta$ is dense, or equivalently, the beta-shift is not a so-called synchronized system. We also provide estimates on the size of the sets of $\beta$ sharing some ergodic properties of their corresponding symbolic spaces which are expressed in terms of the behaviour of the orbit of 1 with respect to $T_\beta$. The talk is based on the paper of Joerg Schmeling (Ergod. Th. & Dynam. Sys. (1997), 17, 675-694).

Ricardo Mañe gave a seminal talk on the ergodic properties of $C^1$-generic diffeomorphisms. He divided his discussion into two parts, one dealing with conservative (i.e., volume preserving) diffeomorphisms, the other with non-conservative ones. The aim of the paper is to realize some of Mañe’s vision of an ergodic theory for non-conservative $C^1$-generic diffeomorphisms. More precisely, first, we see that homoclinic classes of arbitrary diffeomorphisms exhibit ergodic measures whose supports coincide with the homoclinic class. Second, we see that generic (for the weak topology) ergodic measures of $C^1$-generic diffeomorphisms are nonuniformly hyperbolic. Third, we see extension of a theorem by Sigmund on hyperbolic basic sets. The talk is based on the paper "Nonuniform hyperbolicity of $C^1$-generic diffeomorphisms" by Abdenur, Bonatti, and Crovisier.

Bellon and Viallet introduced algebraic entropy in 1998. Algebraic entropy of a rational self-map of a projective space equals the growth rate of degrees of iterates of the map. Bellon and Vaillet proposed two conjectures concerning this topic. The first one claims that the algebraic entropy of a rational map is always an algebraic integer; the second one says that the sequence of degrees of iterates of a rational map satisfies a linear recurrence. The second claim has been proven to be false by Propp and Hasselblatt in 2007. They gave a counterexample to that conjecture involving a specific monomial map. In my master thesis, I gave more general counterexamples with weaker assumptions. In this presentation, we will discuss these counterexamples. The talk will be based on the paper "Degree-growth of monomial maps" by Propp and Hasselblatt from 2007 and my master thesis.

Many of the classic results for estimating the entropy of a dynamical system combine notions of dimension  (Hausdorff dimension, topological dimension, dimension of the manifold, etc.) and expansion  (the Jacobian matrix, Lyapunov exponents, etc.). One of these results says that for a Lipschitz continuous map on a compact metric space, the topological entropy is bounded above by the Hausdorff dimension times the logarithm of the Lipschitz constant. We explore whether or not this bound becomes sharp when we take an infimum over all compatible metrics. We give a positive answer for two classes of systems: linear maps on the torus, and 1-sided expansive mappings. As a corollary, we recover a 1-sided version of an old theorem by Mañé, that a topological space admitting an expansive mapping is necessarily finite-dimensional. This talk is coautored with Zuzana Roth.

We present two characterizations of amenability of finitely generated groups G, in terms of the space of configurations Aᴳ, where A is a finite alphabet: the equivalence between nonwanderingness and surjectivity for cellular automata (shift-equivariant continuous maps of Aᴳ), and the existence of a sequence $(X_n)$ of subsets of G for which the Besicovitch pseudodistance (defined for two colorings x,y∈Aᴳ as the limsup, when n grows, of the proportion of i∈$F_n$ for which xᵢ≠y) is shift-invariant. This is a joint work with Silvio Capobianco.

Let G be a group generated by a finite symmetric set S. In the simple random walk on G we start at identity and at each step we move by a uniform random element of S. Kesten’s criterion says that the group G is amenable if and only if the probability of return at time n decays sub-exponentially fast. After the original work of Kesten in 50’ this theorem was generalised in many ways. In my talk I will present some of these generalisations, including my own work on random walks on topologically minimal G-systems and stationary random graphs.

Let $T$ be a quasihyperbolic (that is, ergodic and not hyperbolic) automorphism of a $d$-dimensional torus. The aim of the talk is to prove a version of Mertens' theorem for it. More precisely, we prove that there exist constants $m$ and $C$ such that $$ \sum_{|\tau |\leq N}\frac{1}{e^{h|\tau |}}=m\log N +C+O(N^{-1}), $$ where the sum is taken over all closed orbits of length not greater than $N$ and $h$ is the topological entropy of $T$. We will show that in the generic situation the number $m$ is equal to $2^{d}$. We will also construct examples of automorphisms $T$ for which $m< 2^{d}$ and $m>2^{d}$. The talk is based on the paper "Mertens' Theorem for Toral Automorphisms" by Jaidee, Stevens and Ward.

We will study transitive sets (typically, homoclinic classes) which are partially hyperbolic with one dimensional center direction. We are specially interested in the case where this direction is genuinely non-hyperbolic (i.e., there are some hyperbolic periodic points which are expanding in the central direction and other periodic points which are contracting). In this setting, the space of ergodic measures splits into three parts according to the exponent corresponding to the central direction: positive (expanding), negative (contracting), and zero (neutral). In many cases, in very rough terms, the expanding and contracting measures are glued throughout the neutral ones. But this is not always the case, and in some case special configurations arise. A key ingredient in those discussions are the so-called exposed pieces of dynamics.

In the uniformly hyperbolic setting it is well known that the measure supported on periodic orbits is dense in the convex space of all invariant measures. We study the reverse question: assuming that some ergodic measure converges to a convex sum of hyperbolic ergodic measures, what can be deduced about the initial measures? To every hyperbolic measure whose stable/unstable Oseledets splitting is dominated we associate canonically a unique class of periodic orbits for the homoclinic relation, that we call its intersection class. In a dominated setting, we prove that a convex sum of finitely many ergodic hyperbolic measures of the same index is accumulated by ergodic measures if, and only if, they share the same intersection class. We provide examples which indicate the importance of the domination assumption. This is joint work with Christian Bonatti and Jairo Bochi.

In 2010 Sarnak formulated the conjecture on Möbius disjointness of topological zero entropy systems, that is: for each zero entropy homeomorphism T of a compact metric space X, we have (*) $\lim_{N\to\infty}\frac1N\sum_{n\leq N}f(T^nx)\mu(n)=0$ for all continuous $f:X\rightarrow\mathbb{R}$ and $x\in X$. Here $\mu$ stands for the classical arithmetic Möbius function. Sarnak's conjecture turns out to be closely related with some open problems in number theory, the most prominent illustration of such an interaction is being "almost equivalent" to the Chowla conjecture (from 1965) on the correlations of the Möbius function. As shown recently by Downarowicz and Serafin, there are however also positive entropy homeomorphisms satisfying (*). I will try to explain that nevertheless the assumption of zero entropy in Sarnak's conjecture is "correct" by exploiting the idea of convergence on so called short intervals and show that it leads to single sequences which can distinuish between zero and positive entropy systems.

Tiling substitution rules are divided into two broad classes: geometric and combinatorial. Geometric substitution tilings include self-similar tilings such as the well-known Penrose tilings; for this class there is a substantial body of research in the literature. Combinatorial substitutions are just beginning to be examined. We give numerous examples, mention selected major results and discuss connections between the two classes of substitutions. The talk is based on the paper "A primer of substitution tilings of the Euclidean plane" by Natalie Priebe Frank.

For a fixed real number $\beta$ > 1 we consider the $\beta$-transformation of the unit interval and the corresponding $\beta$-expansion. We investigate the structure of the set Per($\beta$) of periodic points of such a transformation (this is exactly the set of points with periodic $\beta$-expansion). We show that certain algebraic and topological properties of Per($\beta$) are closely related to $\beta$ being a Pisot or Salem number. In particular, we show that if $\beta$ is a Pisot number, then Per($\beta$) is the intersection of Q($\beta$) and the unit interval. The talk is mainly based on the paper "On periodic expansions of Pisot numbers and Salem numbers" by Klaus Schmidt.

Investigation of dynamics may concern systems for which the exact equation describing their behaviour is unknown. However, by gathering samples of dynamics, we can attempt to retrieve some features of the underlying system. Our strategy is to embed sampled dynamics on simplicial complex, considered later as a finite topological space, and then, to study Morse decomposition of such system. Moreover, we introduce the homological persistence of Morse decomposition as a tool for validating this reconstruction. Our framework can be viewed as a step toward extending the classical persistence theory to the "vector cloud" data.

T. K. Dey, M. Juda, T. Kapela, J. Kubica, M. Lipiński, M.Mrozek. Persistent Homology of Morse Decompositions in Combinatorial Dynamics, arXiv preprint arXiv:1801.06590

Markov field is a generalization of Markov process into a graph, especially a lattice. Hammersley-Clifford theorem says that they are equivalent with Gibbs field, like Ising model. I will introduce to this topic from perspective of computer science application for 2D Fibonacci coding: storing information in a 0/1 lattice without two neighboring '1', for example to improve hard disk capacity. While there are many approximate methods, I will show how to approach in the optimal way. I will also present results about multidimensional Markov types, which extreme points allow for systematic classification of periodic tilings.

Krieger's generator theorem shows that any free invertible ergodic measure preserving action $(Y,\mu, S)$ can be modelled by $A^\mathbb{Z}$ (equipped with the shift action) provided the natural entropy constraint is satisfied; we call such systems (here it is $A^\mathbb{Z}$) universal. Along with Tom Meyerovitch, we establish general specification-like conditions under which $\mathbb{Z}^d$-dynamical systems are universal. These conditions are general enough to prove that:

1) a self-homeomorphism with almost weak specification on a compact metric space (answering a question by Quas and Soo)

2) proper colourings of the $\mathbb{Z}^d$ lattice with more than two colours and the domino tilings of the $\mathbb{Z}^2$ lattice (answering a question by Şahin and Robinson)

are universal. Our results also extend to the almost Borel category giving partial answers to some questions by Gao and Jackson.

Dynamical systems $f$ in $\mathbb{R}^d$ are studied. Let $\Omega\subset\mathbb{R}^d$ be a bounded open set. We will be interested in those periodic orbits such that at least one of its points lies inside $\Omega$ and at least one of its points lies outside $\Omega$; the orbits with this property are called $\Omega$-broken. Information about the structure of the set of $\Omega$-broken orbits is suggested; results are formulated in terms of topological degree theory. The talk is based on the paper by A. V. Pokrowski and O. A. Rasskazow.

The talk will be based on a joint work with Yonatan Gutman. The aim of the project is to study connections between the compression theory for analog signals (modeled as subshifts in $[0,1]^\mathbb{Z}$) and the metric mean dimension - dynamical invariant introduced by B. Weiss and E. Lindenstrauss. Our main results are upper and lower bounds (in terms of the metric mean dimension and its variants) for certain compression rates, where one imposes various regularity conditions on the compressor and decompressor functions (e.g. linearity, Höldericity). An essential tool is the variational principle for the metric mean dimension in terms of the rate-distortion function, obtained recently by E. Lindenstrauss and M. Tsukamoto.

The notion of a Toeplitz sequence can be generalized to any residually finite countable group. As a result for every countable residually finite group we have the class of Toeplitz dynamical systems. All Toeplitz systems are minimal. Since every residually finite group is sofic it is natural to ask about sofic entropy of Toeplitz systems. In particular, it is natural to study sufficient conditions for a Toeplitz system to be zero entropy system. We will present a partial answer to these questions for regular Toeplitz systems on residually finite groups.

Kolmogorov-Sinai entropy can be generalised from $\mathbb Z^d$ lattices to amenable groups and further yet to sofic groups. In this talk I will give some motivation for this generalisation, define sofic groups and prove some of their properties. Then I will introduce sofic entropy with some examples.

A topological dynamical system is substitutive if it arises as the orbit closure of a substitutive sequence. During the talk we will present a classification of subsystems of a substitutive system; in particular we will show that all transitive subsystems of a substitutive system are substitutive. As an application we will obtain a complete characterisation of sets of words that can occur as common factors of two multiplicatively independent automatic sequences. This generalises the classical Cobham's theorem that states that two multiplicatively independent automatic sequences are equal if and only if they are eventually periodic (based on joint work with Jakub Byszewski and Jakub Konieczny).

We study periodic points of endomorphisms of abelian varieties over fields of positive characteristic p, which can be regarded as a positive characteristic analogue of (complex) toral endomorphisms. Periodic points are counted by the dynamical zeta function introduced by Artin and Mazur. In many contexts the dynamical zeta function is rational (this is the case, e.g., for axiom A diffeomorphisms). We prove that in our situation the dynamical zeta function is either rational or transcendental depending on the local behaviour of the endomorphism on the p-torsion. We also study the questions of analytic continuation, asymptotic behaviour of the orbit length distribution, and the analogues of the Prime Number Theorem. This is joint work with Gunther Cornelissen. We will also briefly discuss ongoing work concerning extending these results to general algebraic groups (with Gunther Cornelissen) and dynamically affine maps (with Gunther Cornelissen, Mark Houben, and Lois van der Meijden).

We study periodic points of endomorphisms of abelian varieties over fields of positive characteristic p, which can be regarded as a positive characteristic analogue of (complex) toral endomorphisms. Periodic points are counted by the dynamical zeta function introduced by Artin and Mazur. In many contexts the dynamical zeta function is rational (this is the case, e.g., for axiom A diffeomorphisms). We prove that in our situation the dynamical zeta function is either rational or transcendental depending on the local behaviour of the endomorphism on the p-torsion. We also study the questions of analytic continuation, asymptotic behaviour of the orbit length distribution, and the analogues of the Prime Number Theorem. This is joint work with Gunther Cornelissen. We will also briefly discuss ongoing work concerning extending these results to general algebraic groups (with Gunther Cornelissen) and dynamically affine maps (with Gunther Cornelissen, Mark Houben, and Lois van der Meijden).

The Multiplicative Weights Update method is a ubiquitous meta-algorithm used e.g. in machine learning, optimization, theoretical computer science and game theory. We will analyze convergence of MWU for congestion games to exact Nash equilibria. We will show how aggressive behavior (fast learning) of players may result in lack of convergence - appearance of limit cycles and chaotic behavior.

It is known that two number fields are isomorphic if and only if the Galois groups of their algebraic closures (an analogue of fundamental groups) are isomorphic as topological groups - this is an example of Grothendieck s anabelian philosophy and due to Neukirch and Uchida in the 1970s. However, such Galois groups are not at all understood; their representation theory is the "Langlands programme". We can consider their abelianizations (an analogue of homology groups): these are very well understood through a part of algebraic number theory called class field theory. On the other hand, they fail miserably in determining the number field. In this talk, I will show that there exists a dynamical system, consisting of a topological space modelled on the abelianized Galois groups together with the action of a countably generated monoid such that two number fields are isomorphic if and only if the systems are topologically conjugate, if and only if they are orbit equivalent (joint work with Xin Li, Matilde Marcolli and Harry Smit).

Sarnak’s conjecture is one of the most intensely studied problems in number theory and dynamics in the recent years. During the talk I will discuss the history of this problem, with special emphasis on sequences defined in terms of expansion of a number in a fixed basis. New results concerning $q$-semimultiplicative sequences will be presented.

Arcwise connectedness and entropy-density of extremal points of the simplex of invariant meaures for $\beta$- free shifts will be presented. We touch as well some motivations connected to number theory and possible generalization of our results. This is a joint work with D. Kwietniak.

In the talk, I will discuss a recent result of Gomilko, Kwietniak, and Lemańczyk saying that Sarnak’s conjecture implies Chowla’s conjecture along a subsequence. The proof of this fact is based on properties of logarithmic empirical measure for dynamical systems.

In the talk we will recall basic notions and results related to substitutions acting on finite alphabets. In particular, we will prove that a symbolic dynamical system associated with a primitive substitution is minimal and uniquely ergodic. We will also investigate further properties for specific cases of symbolic dynamical systems, e.g., those associated with Chacon and Fibonacci substitutions. The talk will be based on Chapter 5 of the book ”Substitutions in Dynamics, Arithmetics and Combinatorics” by N. Pytheas Fogg.

In the talk we will recall basic notions and results related to substitutions acting on finite alphabets. In particular, we will prove that a symbolic dynamical system associated with a primitive substitution is minimal and uniquely ergodic. We will also investigate further properties for specific cases of symbolic dynamical systems, e.g., those associated with Chacon and Fibonacci substitutions. The talk will be based on Chapter 5 of the book ”Substitutions in Dynamics, Arithmetics and Combinatorics” by N. Pytheas Fogg.

For projectivizations of rational maps, Bellon and Viallet defined the notion of algebraic entropy using the exponential growth rate of the degrees of iterates. The particular rational maps which I study are monomials. In my presentation I would like to give some ideas on how to compute algebraic entropy and compare it with the topological entropy.

Motivated by the recent results around Zimmer’s program, we study group actions on manifolds, with singular regularity (we require that every element is differentiable at all but countably many points). The groups under considerations have a fixed point property, named FW, which generalizes Kazhdan’s property (T) (in particular we can consider actions of lattices in higher-rank simple Lie groups). The main result is that if a group G has property FW, any singular action of $G$ on a closed manifold 1) either has a finite orbit, 2) or is conjugate to a differentiable action, up to changing the differentiable structure of the manifold. This is a joint work with Yash Lodha and Nicolas Matte Bon.

Over the last few years, anomalous behaviors have been observed for one-dimensional chains of oscillators. The rigorous derivation of such behaviors from deterministic systems of Newtonian particles is very challenging, due to the existence of conservation laws, which impose very poor ergodic properties to the dynamical system. A possible way out of this lack of ergodicity is to introduce stochastic models, in such a way that the qualitative behaviour of the system is not modified. One starts with a chain of oscillators with a Hamiltonian dynamics, and then adds a stochastic which keeps the fundamental conservation laws (energy, momentum and stretch, usually). For the unpinned harmonic chain where the velocities of particles can randomly change sign (and therefore the only conserved quantities of the dynamics are the energy and the stretch), it is known that, under a diffusive space-time scaling, the energy profile evolves following a non-linear diffusive equation involving the stretch. Recently it has been shown that in the case of one-dimensional harmonic oscillators with noise that preserves the momentum, the scaling limit of the energy fluctuations is ruled by the fractional heat equation.

In this talk we will introduce the rotation theory of the two dimensional torus. We will focus on the Franks-Misiurewicz conjecture and the related recent advances.

For a locally compact group $G$, let $C_{b}(G)$ be the space of all complex-valued, continuous and bounded functions on $G$ equipped with the sup-norm, and $LUC(G)$ be the subspace of $C_{b}(G)$ consisting of all functions $f$ such that the map $G\to C_b(G);x\mapsto l_xf$ is continuous, where $l_xf$ is the function defined by $l_xf(y)=f(xy)$, for each $y\in G$. The subspace $LUC(G)$ forms a unital commutative C*-algebra. We can induce a multiplication on the Gelfand spectrum of $LUC(G)$, $G^{LUC}$, with which $G^{LUC}$ forms a semigroup. When $G$ is discrete, $G^{LUC}$ is in fact the Stone-Cech compactification of $G$. In this talk, I study some properties of $G^{LUC}$, the so called right topological semigroup compactification of $G$. I also discuss the question of when the corona, $G^{LUC}\setminus G$, determines the underlying topological group $G$.

The notion of mean equicontinuity, under a different name, was first introduced by Fomin in 1951 to deduce a dynamical property implying pure point spectrum. It was later studied by Auslander in more detail and has gained renewed interest in recent years for several different reasons. The talk will survey parts of the old and new insights with the particular focus on long-range order and actions of general amenable groups.

A classical result by Markley and Paul states that irregular almost automorphic systems over irrational rotations are typically not uniquely ergodic and have positive entropy. By constructing a particular Cantor set, we prove that for each irrational rotation there still are such almost automorphic extensions which are mean-equicontinuous (and hence have zero entropy and are uniquely ergodic). We will shortly review the results by Markley and Paul and then discuss the construction of the Cantor set. This is a joint work with Eli Glasner, Tobias Jager and Christian Oertel.

We study the periodic point free maps and Lefschetz periodic point free maps on connected retract of a finite simplicial complex using the Lefschetz numbers. Special emphasis is placed on the self-maps of products of spheres and wedge sums of spheres (based on an article by J. Llibre and V. F. Sirvent).

The dynamics of expansive homeomorphisms with the shadowing property may be very complicated but it is quite well understood (see Aoki and Hiraide’s monograph, for example). It is known that these systems admit only a finite number of chain recurrent classes (Spectral Decomposition Theorem). In 2012, Morales introduced a generalization of expansivity property, called N-expansive property. For every $N\in\mathbb N$, we will exhibit an $N$-expansive homeomorphism, which is not $(N − 1)$-expansive, has the shadowing property, and admits an infinite number of chain-recurrent classes. We discuss some properties of the local stable (unstable) sets of N-expansive homeomorphisms with the shadowing property and use them to prove that some types of the limit shadowing property are present. We will discuss a Spectral Decomposition Theorem for $N$-expansive homeomorphisms with the shadowing property defined on surfaces.

The Lagarias Wang finiteness conjecture was introduced in 1995 in connection with problems related to spectral radius computation of finite sets of matrices. The conjecture has been proved to be false recently and there are alternative proofs of it. One of the proofs on specific matrices will be presented.

Consider two trajectories, each forming a Jordan curve with a finite length in the two dimensional plane. We show that the integral of the velocity of the portion of one curve that is included in the inside of the second curve is bounded by half the length of the latter curve (based on an article by Zvi Artstein and Ido Bright).

Abraham and Smale proved that hyperbolic diffeomorphisms are not dense among all diffeomorphisms of a given manifold. This inspired Pesin to define a weaker notion of hiperbolicity: he proposed a notion of a hyperbolic ergodic measure. Gorodetsky, Ilyashenko, Kleptsyn, and Nalsky constructed an open set of diffeomorphisms of a three dimensional torus such that every diffeomorphism in this set admits a non-hyperbolic ergodic measure. Their method leads, however, to measures with zero entropy. Bonatti, Bochi, and Diaz in the series of two papers proved that for every $n\geq 3$ there exist a manifold of dimension $n$ and an open set of diffeomorphisms admitting a non-hyperbolic ergodic measure with positive entropy and an open set of diffeomorphisms admitting a nonhyperbolic ergodic measure with full support. We will show that one can strengthen their results by constructing diffeomorphisms with a non-hyperbolic ergodic measure with positive entropy and full support. The talk will be based on the joint work with Ch. Bonnati, L. Diaz, and D. Kwietniak.

Consider an arbitrary automorphism of an Enriques surface with its lift to the covering $K3$ surface. We prove a bound of the order of the lift acting on the anti-invariant cohomology sublattice of the Enriques involution. We use it to obtain some $\mod 2$ constraint on the original automorphism. As an application, we give a necessary condition for Salem numbers to be dynamical degrees on Enriques surfaces and obtain a new lower bound on the minimal value (joint work with Y. Matsumoto (Nagoya) and H. Ohashi (Tokyo)).

Joint work with Sebastian Barbieri. We will characterize entropy pairs on shadowing amenable group actions using the homoclinic relation. This will help up construct examples with completely positive entropy in order to answer a question by Pavlov.

We study correlation estimates of automatic sequences (that is, sequences computable by finite automata) with polynomial phases. As a consequence, we provide a new class of good weights for classical and polynomial ergodic theorems, not coming themselves from dynamical systems. We show that automatic sequences are good weights in $L^2$ for polynomial averages and totally ergodic systems. For totally balanced automatic sequences (i.e., sequences converging to zero in mean along arithmetic progressions) the pointwise weighted ergodic theorem in $L^1$ holds. Moreover, invertible automatic sequences are good weights for the pointwise polynomial ergodic theorem in $L^r$, $r>1$. This talk is based on joint work with Tanja Eisner.

We study the simplex of probability measures on a Bratteli diagram which are invariant with respect to the tail equivalence relation. We prove a criterion of unique ergodicity of a Bratteli diagram. In case when a finite rank $k$ Bratteli diagram $B$ has $1\leq l\leq k$ ergodic invariant measures, we describe the structures of the diagram and the subdiagrams which support these measures. This is a joint work with S. Bezuglyi and J. Kwiatkowski.

Rotation sets are generalizations of rotation number defined for homeomorphisms of a circle. We define rotation set for an arbitrary dynamical system on $X$ and a map from $X$ to $\mathbb R^n$ and consider the case of transitive subshift of finite type and a map constant on cylinders of length $2$. Then the rotation set is a convex hull of finite set and rotation vectors of periodic points are dense in it. Moreover vectors from the interior are rotation vectors of ergodic measures. Based on an article by Krystyna Ziemian (Fundametna Mathematicae 146).

The Cartwright-Littlewood-Bell fixed point theorem asserts that any planar homeomorphism fixes a point in any invariant continuum that does not separate the plane. I shall discuss the historical background of this result, and its more recent generalizations.

We study properties of the Weyl pseudometric in a dynamical system generated by an amenable group action. Specifically we obtain several results regarding the interaction of the resulting topology with the entropy function and other important objects. These generalize the work of Downarowicz and Iwanik. As a corollary we obtain an alternative proof of Krieger’s theorem.

Based on a paper by Z.Alexander, E.Bradley, J.D.Meiss and N.F.Sanderson (2015). Topology-based analysis of time-series data from dynamical systems is powerful: it potentially allows for computer-based proofs of the existence of various classes of regular and chaotic invariant sets for high-dimensional dynamics. Standard methods are based on a cubical discretization of the dynamics and use the time series to construct an outer approximation of the underlying dynamical system. The resulting multivalued map can be used to compute the Conley index of isolated invariant sets of cubes. In this paper we introduce a discretization that uses instead a simplicial complex constructed from a witness-landmark relationship. The goal is to obtain a natural discretization that is more tightly connected with the invariant density of the time series itself. The time-ordering of the data also directly leads to a map on this simplicial complex that we call the witness map. We obtain conditions under which this witness map gives an outer approximation of the dynamics and thus can be used to compute the Conley index of isolated invariant sets. The method is illustrated by a simple example using data from the classical Henon map.

Topology-based analysis of time-series data from dynamical systems is powerful: it potentially allows for computer-based proofs of the existence of various classes of regular and chaotic invariant sets for high-dimensional dynamics. Standard methods are based on a cubical discretization of the dynamics and use the time series to construct an outer approximation of the underlying dynamical system. The resulting multivalued map can be used to compute the Conley index of isolated invariant sets of cubes. In this paper we introduce a discretization that uses instead a simplicial complex constructed from a witness-landmark relationship. The goal is to obtain a natural discretization that is more tightly connected with the invariant density of the time series itself. The time-ordering of the data also directly leads to a map on this simplicial complex that we call the witness map. We obtain conditions under which this witness map gives an outer approximation of the dynamics and thus can be used to compute the Conley index of isolated invariant sets. The method is illustrated by a simple example using data from the classical Henon map.

In the book “What is Mathematics?” Richard Courant and Herbert Robbins presented a solution of a Whitney’s problem of an inverted pendulum on a railway carriage moving on a straight line. Since the appearance of the book in 1941 the solution was contested by several distinguished mathematicians. In the talk I present the history of the problem until its first formal solution published by Ivan Polekhin in 2014. Polekhin also proved a theorem on the existence of a periodic solution of the problem provided the motion of the carriage on the line is periodic. I indicate we how to obtain a similar result if the carriage moves periodically on the plane.

Recently Joel Moreira showed that any finite colouring of natural numbers admits a monochromatic configuration of the form $\{x, x + y, xy\}$, thereby (partially) solving a long-standing conjecture of Hindman. His proof proceeds by first translating the problem into the language of topological dynamics and then showing a relevant recurrence statement; an approach that goes back to Furstenberg and Weiss’ dynamical proof of van der Waerden’s theorem from 1978. During the talk we’ll present a unified approach to partition regular theorems including van der Waerden’s theorem, its polynomial extension obtained by Bergelson and Leibman and recent result of Moreira, higlight the role that ultrafilters play in partition regularity and offer some extension of Moreira’s result.

Rufus Bowen left a notebook containing 157 open problems and questions. Problem 32 on that list asks for a classification of shift spaces with the specification property. Unfortunately, there is no universally accepted agreement what does it mean “to classify” a family of mathematical objects and Bowen did not specify what he had in mind. I will describe one of the most commonly popular ways of making the problem formal based on the language of Borel equivalence relations. Inside that framework I will explain a conjecture saying that (roughly speaking) there is no reasonable classification for shift spaces with specification. In particular, if the conjecture holds true, then no classification using a finite set of definable invariants is possible. This will solve the problem provided that Bowen would agree with the set-theoretic notion of “classification”.

Podczas referatu wprowadzone zostanie pojęcie zbieżności leniwej dla ciągu zmiennych losowych - jest to pewien szczególny rodzaj zbieżności stochastycznej takiego ciągu do ustalonego zbioru. Przedstawione zostaną warunki wystarczające dla takiej zbieżności, sformułowane twierdzenie uzasadni kiepskie tempo zbieżności wielu bazujących na losowości metod algorytmicznych używanych w praktyce do znajdowania ekstremów globalnych funkcji.

The theory of isolated invariant sets has proven to play a central role in the study of the qualitative properties of differential equations and dynamical systems. In this talk we will focus on a special kind of isolated invariant sets, the class of isolated non-saddle sets, which in particular contains the class of attractors. We will expose some global properties of non-saddle sets such as the existence of homoclinic trajectories and dissonant points or the existence of a dual non-saddle set. In addition we will discuss some robustness properties.

The aim of this talk is to give a method to compute the Conley index of an isolated invariant continuum $K$ of a flow on a surface using only the topology of K and some knowledge about its unstable manifold. To do this we will show the existence of a special kind of isolating blocks, the so-called regular isolating blocks whose topology is related to the topology of $K$. Besides we will see some applications such as the classification of those isolated invariant continua without fixed points or a complete study its continuations.

Podczas referatu zamierzam przybliżyć pojęcie dychotomii wykładniczej (ED), które stanowi naturalne uogólnienie pojęcia hiperboliczności dla autonomicznych równań różniczkowych. Następnie przedstawię zastosowania ED do poszukiwania trajektorii homoklinicznych równań różniczkowych nieautonomicznych (zarówno ciągłych, jak i dyskretnych).

Given a subset A of integers we may identify the characteristic function $c_A$ of $A$ with a point in the full shift space of infinite 0-1 sequences. The closure of the orbit of $c_A$ with respect to the left-shift operator leads to a symbolic dynamical system, whose dynamical properties depend on combinatorial properties of $A$. This approach goes back at least to Furstenberg. Recently, Sarnak proposed to study square-free integers through dynamics of the shift space constructed in the above way. El Abdalauoi-Lemanczyk-De La Rue and Bartnicka-Kasjan-Kułaga-Przymus-Lemańczyk extended Sarnak’s approach and studied $B$-free integers generated by arbitrary subset of integers. Recall that an integer is $B$-free if it has no factor in $B$. Note that square free integers are generated by $B_sq=\{p^2\,:\,p \in \mathbb{P}\}$. These shift spaces and their higher dimensional analogs attracted recently much attention. During my talk I will describe a new approach to a related class of systems generated by rational subsets of integers. This class includes $B$-free shifts generated by sets $B$ possessing an asymptotic density. A set $B$ is rational if it can be arbitrary well approximated with respect to the upper asymptotic density by finite unions of arithmetic progressions. We study invariant measures of these systems and study their entropy. (This is a joint work with Jakub Konieczny and Michal Kupsa.)

Z końcem lat 90-tych XX wieku R. Forman zdefiniował kombinatoryczne pola wektorowe dla CW kompleksów i przedstawił wersję teorii Morse’a dla pól acyklicznych. Studiował również kombinatoryczne pola wektorowe bez założenia acykliczności, zdefiniował dla nich zbiór łańcuchowo powracający oraz udowodnił nierówności Morse’a. Prace Formana początkowo przyjęto z rezerwą, ale świat przekonał się do nich dzięki szybko rosnącej liczbie zastosowań: od topologii obliczeniowej poprzez analizę obrazów, zagadnienia wizualizacji po usuwanie z danych szumów. Jednak, co ciekawe, jak na razie brakuje zastosowań tam gdzie wydaje się to najbardziej naturalne: w dynamice obliczeniowej. W wykładzie przypomnimy wyniki Formana, przedstawimy uogólnienie w postaci kombinatorycznych pól wielowektorowych, a na jego bazie zaprezentujemy rozszerzenie teorii Morse’a-Formana w stronę dynamiki topologicznej.

W 1996 roku, Greg Kuperberg skonstruował skręconą wstawkę w celu wykonywania operacji Dehna na rozmaito±ciach trójwymiarowych zgodnej z danym układem dynamicznym. Tą metodę zastosował do udowodnienia, że każda rozmaitość trójwymiarowa bez brzegu posiada gładki, zachowujący miaer układ dynamiczny, z dyskretną rodziną okresowych trajektorii, które są jedynymi zbiorami minimalnymi. W tym odczycie zostaną przedstawione pokrótce operacje Dehna wycinania i wklejania pełnych torusów na rozmaitościach trójwymiarowych zgodne z dynamiką. Przedstawione będzie następujące uogólnienie powyższego twierdzenia: Każda rozmaitość trójwymiarowa bez brzegu posiada gładki, zachowujący miarę układ dynamiczny, z dyskretną rodziną okresowych trajektorii, które są jedynymi zbiorami minimalnymi i taki, że każda trajektoria jest ograniczona. Trajektoria jest ograniczona jeżeli jej domknięcie jest zwarte.

Oznaczmy przez $M_T(X)$ rodzinę borelowskich miar probabilistycznych niezmienniczych względem ciągłego odwzorowania $T$ działającego na zwartej przestrzeni metrycznej $X$. Wiadomo, że $M_T(X)$ wyposażone w *-słabą topologię jest przestrzenią zwartą. Dla dowolnego punktu $x$ przestrzeni $X$ niech $m(x,n)$ oznacza unormowaną sumę delt Diraca w punktach $x, T(x), ... T^{n-1}(x)$. Zdefiniujmy $om(x)$ jako rodzinę wszystkich punktów skupienia ciągu $\{m(x,n)\}_{n\in\mathbb N}$. Podczas referatu pokażemy, że jeśli układ dynamiczny $(X,T)$ spełnia własność AASP (pewną odmianę własności śledzenia pseudoorbit), to dla dowolnego kontinuum (zwartego spójnego i niepustego zbioru) $V$ zawartego w $M_T(X)$ istnieje taki punkt $x$ przestrzeni $X$, że $om(x)=V$. W szczególności oznacza to, że dla każdej miary $\mu$ w $M_T(X)$ istnieje punkt generyczny. W dowodzie wykorzystamy pseudometrykę Besichovitcha.

Układy kodowane to klasa układów symbolicznych stanowiąca naturalne uogólnienie shiftów soficznych. Wiele ”dobrych” własności tych ostatnich przenosi się na układy kodowane, ale nie jest to regułą. W czasie referatu omówię związki między różnymi rodzajami powracania (tranzytywność, mieszanie, specyfikacja) dla układów kodowanych. Referat będzie częściowo oparty na wynikach z preprintu, którego współautorami są J. Epperlein i P. Oprocha.

Maximal Entropy Random Walk (MERW), zwany także Ruelle-Bowen random walk, oznacza wybór najbardziej losowego z wszystkich dostępnych błądzeń losowych: maksymalizującego produkcję entropii. Podczas gdy standardowe modele dyfuzyjne okazują się być często rozbieżne z przewidywaniami mechaniki kwantowej (różne własności lokalizacyjne), modele oparte na MERW pozwalają zrozumieć i naprawić tą rozbieżność: ich stacjonarna gęstość prawdopodobieństwa jest identyczna jak dla kwantowego stanu podstawowego. Opowiem pokrótce też o innych zastosowaniach MERW: do maksymalizacji pojemności kanału informacyjnego przy zadanych więzach, oraz do analizy złożonych grafów, np. jako alternatywa dla PageRank lub do wyszukiwania wyróżniających się obszarów obrazka.

It is well known that the topology and dynamics of limit sets play its role in the theory and applications of dynamical systems. In first part of my talk I will briefly describe different kinds of limit sets, their topological, dynamical properties and mutual connections. Second part will be devoted to my recent work on a new criterium implying that the $\omega$-limit set, under some assumption on the space and asymptotic properties of an orbit, can be a topological continuum. As a result we can also prove, that the limit set is in fact some sort of attractor.

A shift space is intrinsically ergodic if it has a unique measure of maximal entropy (MME). Bowen introduced the specification property and proved that the shift spaces with the specification property are intrinsically ergodic. Pfister and Sullivan generalized Bowen’s notion and defined the $g$-almost product property, later coined the almost specification property by Thompson. Another generalization of specification is due to Marcus who introduced the weak specification property. I will describe a family of shift spaces showing that neither weak nor almost specification imply uniqueness of the MME, along with some other results about almost specified dynamical systems. I will relate these results to the work of Climenhaga and Thompson on intrinsic ergodicity of some shift spaces. Time permits I will discuss a related problem of intrinsic ergodicity of the subordinate shift spaces. The talk is based on a joint work with Piotr Oprocha (AGH Kraków) and Michał Rams (IM PAN Warszawa).

W wielu zagadnieniach w teorii układów dynamicznych istotną rolę pełnią arytmetyczne własności pewnych liczb związanych z tymi układami. W referacie przedstawię kilka twierdzeń ilustrujących to zjawisko (twierdzenie Siegela o linearyzacji perturbacji obrotu na płaszczyźnie zespolonej, wyniki teorii KAM) oraz pokażę jak można stosować metody probabilistycznej teorii liczb, aby uzyskać ilościowe wersje tych rezultatów.

Niech $x$ będzie niestałym rowiązaniem $T$-okresowym autonomicznego równania różniczkowego $x'=f(x)$, gdzie $f\colon X\to X$ jest lipschitzowska ze stałą $L$ oraz $X$ jest przestrzenią Banacha. Wcześniejsze wyniki pokazują, że $LT\geq 6$ dla dowolnej przestrzeni Banacha oraz $LT\geq 2\pi$ dla przestrzeni Hilberta. W czasie seminarium omówię wzmocnienie powyższych wyników. W szczególności, dla ściśle wypukłych przestrzeni Banacha zachodzi $LT>6$. Ponadto udowodnię, że dla dowolnej przestrzeni Banacha zachodzi $LT\geq 6$. Następnie pokażę, jak ze specjalnej postaci nierówności Wirtingera na przestrzeniach $W^{1,p}_{\text{per}}([0,T],X)$, gdzie $X =l^p(\mathbb R^n)$ lub $X=L^p(M,\mu)$ można wskazać przedział wartości $p$, dla którego $LT>6$.

Złożoność topologiczna układu dynamicznego względem pokrycia $\mathcal U$ to minimalna liczba elementów podpokrycia pokrycia $\bigvee_{i=0}^{n-1}T^{-i}\mathcal U$ (czyli jest to funkcja, której wykładniczy wzrost równoważny jest dodatniej entropii topologicznej). Podczas seminarium omówimy kilka własności złożoności topologicznej. Pokażemy między innymi, że układ ma ograniczoną (czyli od pewnego miejsca stałą) złożoność topologiczną wtedy i tylko wtedy, gdy jest równociągły oraz omówimy związek pomiędzy złożonością a mieszaniem i chaosem.

Przedstawię warunki implikujące, że sympleks miar niezmienniczych spełniającego je układu dynamicznego jest sympleksem Poulsena. Warunki te uogólniają własność specyfikacji. Podam także przykłady zastosowań.

W trakcie referatu przedstawione zostały wyniki zawarte w pracy “Noninvertible minimal maps” S. Kolyady, L. Snohy i S. Trofimczuka opublikowanej w Fund. Math. 168 (2001).

W referacie zostało zaprezentowane pojęcie rozpiętości kohomologicznej LS-indeksu Conleya. Zostały podane twierdzenia o istnieniu rozwiązań okresowych układów hamiltonowskich udowodnione z pomocą tego pojęcia. Została także przedstawiona metoda rachunkowa służąca do wyznaczania LS-indeksu punktów krytycznych funkcjonału stowarzyszonego z układami hamiltonowskimi.

A simplified proof of the Sharkovsky’s theorem has been presented.

Zreferowano następujący wynik z pracy J.-H. Mai: jeśli $X$ jest przestrzenią topologiczną, $f\in C(X,X)$, a $Y\subset X$ jest zbiorem zwartym takim, że $\partial_XY=\{v\}$ i $f(v)\in Y$, to w $Y$ istnieje zbiór minimalny $f$.

Podczas referatu zdefiniowałem okres asymptotyczny orbity jako możliwe uogólnienie klasycznego okresu dla orbit w układach dynamicznych z czasem ciągłym. Intuicyjnie okres asymptotyczny ma opisywać własności orbit, których dodatnimi zbiorami granicznymi są orbity okresowe lub punkty stałe. Definicja ta pokrywa się również z klasycznym okresem dla orbit okresowych. Przestawiłem podstawowe własności powyższego pojęcia. Między innymi pokazałem, że orbity o zerowym okresie asymptotycznym to dokładnie orbity, których zbiory omega-graniczne są jednopunktowe. Omówiłem również przypadek, gdy zbiór omega-graniczny jest orbitą okresową lub gęstą, a okres asymptotyczny jest skończony.

W trakcie referatu przedstawione zostaną globalne własności bifurkacji Hopfa. Rozważać będziemy trzy sytuacje: bifurkacje z punktu, orbity okresowej a także ogólniej z dowolnego atraktora, podczas której powstaje nowa rodzina atraktorów. Do badania własności tych ostatnich będziemy używać teorii kształtu Borsuka - uogólnienia teorii homotopii. Na samym początku referatu zostaną podane podstawe definicje i własności dotyczące kategorii i morfizmów kształtu, a także zaprezentowane inne podejście do tej teorii - klasyczne wg. Borsuka.

Kontinua metryczne zostały podzielone na cztery grupy w zależności od istnienia homeomorfizmów minimalnych i ciągłych niehomeomorfizmów minimalnych. Przedstawione zostały przykłady przestrzeni należących do każdej z klas. Udowodnione zostało, że na solenoidach istnieją minimalne niehomeomorfizmy ciągłe.

Problem zbieżności Markowowskich procesów optymalizacyjnych (z czasem dyskretnym) $X_t$ zadanych rekurencyjnym równaniem postaci $X_{t+1}=T_t(X_t,Y_t)$, gdzie $Y_t$, $t=0,1,\ldots$, to zmienne losowe niezależne, można sprowadzić do badania stabilności związanych z tym równaniem operatorów Foiasa, które zadają pseudo–układ dynamiczny na przestrzeni miar probabilistycznych na dziedzinie optymalizowanej funkcji. Celem referatu jest prezentacja oraz dowód ogólnego twierdzenia podającego warunki wystarczające by zbiór miar probabilistycznych skupionych na ekstremach globalnych optymalizowanej funkcji był globalnym atraktorem. Głównym narzędziem jest funkcja Lapunowa.

Ekstremalne tempo wzrostu orbit okresowych jest stałe dla równoważnych potoków bez punktów stałych. Istnieje jednak para równoważnych przepływów z punktami stałymi taka, że tempo wzrostu orbit okresowych jednego potoku jest nieskończone, a drugiego równe zero. W pracy W. Suna i C. Zhanga przedstawiona jest konstrukcja pary równowżnych potoków o podanych własnościach.

Przedstawiono propozycję uogólnienia pojęcia okresu na punkty nieokresowe, ale o regularnym granicznym zachowaniu. Przedyskutowano podstawowe własności nowego pojęcia. Poruszono problem związku prędkości wzrostu ilości orbit okresowych i prędkości wzrostu ilości orbit o skończonym uogólnionym okresie.

Praca O. Makarenkova i R. Ortegi Asymptotic stability of forced oscillations emanating from a limit cycle (JDE 2011, 39-52) poświęcona jest bifurkacjom cyklu granicznego w układach analitycznych oraz warunkom koniecznym i wystarczającym na asymptotyczną stabilność powstających rozwiązań okresowych - z wykorzystaniem metod topologicznych (stopień topologiczny). Podczas referatu przedstawiono główny wynik pracy, w oparciu o przykład podany przez autorów.

Let $\mathcal F$ be a flow given by a continuous action of a locally compact group $\Gamma$ on a compact space $X$. We are interested in the existence of a continuous cocycle $\mathcal C$ over $\mathcal F$ which takes its values in a given Polish group $G$ and satisfies a given property ((weak) topological ergodicity, weak topological mixing, a prescribed set of essential values etc.). We give sufficient conditions for such a cocycle $\mathcal C$ to exist and apply the obtained results to construct minimal fibre-preserving flows on fibre bundles.

Przedstawione zostały przykłady ilustrujące, że bez założenia zwartości przestrzeni nie ma wynikań w żadną stronę pomiędzy własnością śledzenia w średniej i własnością asymptotycznego śledzenia w średniej. Zostało udowodnione natomiast, że przy założeniu zwartości przestrzeni AASP implikuje ASP.

Podczas referatu omówiłem zagadnienia związane z twierdzeniem Perrona-Frobeniusa i jego nieskończenie wymiarowymi uogólnieniami (twierdzenie Kreina-Rutmana, Bonsalla i Nussbauma), ze szczególnym uwzględnieniem założenia styczności (zastępującego niezmienniczość stożka). Wiele uwagi poświęciłem wersjom tego twierdzenia w przypadku operatorów nieograniczonych, generujących silnie ciągłą półgrupę na przestrzeni Banacha wyposażonej w strukturę karty banachowskiej oraz dyskusję tzw. nierówności Kato w relacji do założenia styczności.

Zostały przypomniane pojęcia stopnia topologicznego oraz persystencji jedynego punktu stałego w sperturbowanym układzie autonomicznym. Dla lipschitzowskich pól wektorowych otrzymujemy pełną charakteryzację persystencji w wymiarze 2, jednak dla wyższych wymiarów problem pozostaje otwarty. Pokazano również i zilustrowano przykładami związki z istnieniem punktów izolowanych oraz odpowiednio gładkich dyfeotopii usuwających punkty stałe. Dla pola wektorowego ciągłego do pokazania, że punkt stały nie persystuje wystarczy, by stopień pola wynosił zero. Omówiono ideę dowodu, w którym zwraca uwagę wykorzystanie twierdzenia Kupki-Smale’a.

Zostały przypomniane pojęcia stopnia topologicznego oraz persystencji jedynego punktu stałego w sperturbowanym układzie autonomicznym. Dla lipschitzowskich pól wektorowych otrzymujemy pełną charakteryzację persystencji w wymiarze 2, jednak dla wyższych wymiarów problem pozostaje otwarty. Pokazano również i zilustrowano przykładami związki z istnieniem punktów izolowanych oraz odpowiednio gładkich dyfeotopii usuwających punkty stałe. Dla pola wektorowego ciągłego do pokazania, że punkt stały nie persystuje wystarczy, by stopień pola wynosił zero. Omówiono ideę dowodu, w którym zwraca uwagę wykorzystanie twierdzenia Kupki-Smale’a.

Jedną z podstawowych własności ciągu indeksów iteracji odwzorowania ciągłego $f\colon X\to X$ (dla $X$-zwartych ANR-ów) jest spełnianie relacji Dolda. W związku z tym powstaje naturalne pytanie, czy jeśli zadany z góry ciąg liczb całkowitych spełnia relacje Dolda, to potrafimy znaleźć odwzorowanie ciągłe, którego indeksy iteracji realizują dany ciąg? Celem referatu jest przedstawienie idei dowodu twierdzenia Graffa, Nowaka-Przygodzkiego opisującego wszystkie możliwe ciągi indeksów realizowalne przez odwzorowania klasy $\mathcal C^1$ dla odwzorowań $\mathbb R^3$ w siebie. Zaprezentowana idea konstrukcji (z wykorzystaniem specjalnych typów dynamiki) posłuży następnie do naszkicowania schematu budowy homeomorfizmu $f\colon\mathbb R^3\to\mathbb R^3$ zachowującego orientację, którego ciąg indeksów iteracji pokrywa się z zadanym a priori ciągiem liczb całkowitych, oraz dla którego $\{0\}$ jest izolowanym stabilnym punktem stałym.

Podczas referatu przedstawiony został problem istnienia lokalnie spójnego kontinuum skończonego wymiaru, które nie jest atraktorem żadnego iterowanego systemu funkcyjnego (IFS). W odpowiedzi na powyższe pytanie skonstruowaliśmy 1-wymiarowe Peano kontinuum, zwane ”zębami rekina”, które nie może być homeomorficzne z atraktorem IFS. Dowód opiera sie na własnościach S-wymiaru, ktory mozemy zdefiniować dla każdego Peano kontinuum. Udowodnione zostało, że atraktory IFS mają skończony S-wymiar, natomiast wspomniany przykład ”zębów rekina” jest nieskończenie wymiarowy.

Opisane zostały trzy przykłady kontinuów na płaszczyźnie, które nie są atraktorami żadnego iterowanego systemu funkcyjnego (dwa z nich były lokalnie spójne, a w tym jeden, który również nie jest atraktorem żadnego systemu słabych kontrakcji). Wprowadzone zostało pojęcie długości kontinuum i przy jego pomocy wprowadzony został warunek wystarczający na to, aby kontinuum mogło być włożone homeomorficznie w $\mathbb R^n$ w ten sposób, aby jego obraz nie był atraktorem żadnego IFSu.

W referacie przypomniane zostanie klasyczne twierdzenie o lokalnym dyfeomorfizmie (w wersji dla przestrzeni Banacha), które udowodnimy korzystając z tzw. metody Newtona. Następnie podane zostanie uogólnienie tego twierdzenia na przypadek przestrzeni funkcji klasy $\mathcal C^{\infty}$ wraz z ideą dowodu. Na koniec sformułujemy abstrakcyjne twierdzenie Nasha-Mosera korzystając z formalizmu “grzecznych” przestrzeni Frech´eta. Następnie zaprezentowane zostanie jedno z najbardziej spektakularnych zastosowań twierdzenia Nasha-Mosera — dowód twierdzenia Nasha o izometrycznym zanurzaniu rozmaitości riemannowskich w przestrzenie euklidesowe.

W referacie przypomniane zostanie klasyczne twierdzenie o lokalnym dyfeomorfizmie (w wersji dla przestrzeni Banacha), które udowodnimy korzystając z tzw. metody Newtona. Następnie podane zostanie uogólnienie tego twierdzenia na przypadek przestrzeni funkcji klasy $\mathcal C^{\infty}$ wraz z ideą dowodu. Na koniec sformułujemy abstrakcyjne twierdzenie Nasha-Mosera korzystając z formalizmu “grzecznych” przestrzeni Frech´eta. Następnie zaprezentowane zostanie jedno z najbardziej spektakularnych zastosowań twierdzenia Nasha-Mosera — dowód twierdzenia Nasha o izometrycznym zanurzaniu rozmaitości riemannowskich w przestrzenie euklidesowe.

Rozważamy nieliniowe zagadnienie z $p$-laplasjanem oraz warunkiem brzegowym typu Neumanna. Używając pierwszej wartości własnej $p$-laplasjanu, oraz zakładając monotoniczność prawej strony, podamy warunki wystarczające i konieczne do istnienia rozwiązania dodatniego rozważanego zagadnienia.

Badamy odwzorowanie Henona zachowujące pole $h(x, y) = R_{\alpha}(x, y − x^2)$, $\alpha=2\pi/3$, gdzie $R_{\alpha}$ jest obrotem o kąt $\alpha$. Zajmiemy się $\alpha=2\pi/3$ - przypadek rezonansu 3:1. Omówimy dowód wspierany komputerowo istnienia dynamiki symbolicznej i istnienia ”hiperbolicznego” zbioru niezmienniczego z intersującymi własnościami: - niektóre z punktów mają zerowe wykładniki Lapunowa, - inne mają niezerowe wykładniki Lapunowa, - a dla jeszcze innych granica w definicji wykładnika Lapunowa oscyluje między zerem i pewną wartością niezerową. To jest praca wspólna z Carlesem Simo (Barcelona) i Tomaszem Kapelą.

Niech $W$ i $V$ będą reprezentacjami zwartej grupy Liego $G$. Parę $(U, f)$ nazywamy niezmienniczym odwzorowaniem lokalnym, jeśli $U$ jest otwartym niezmienniczym podzbiorem $W$, $f\colon U\to V$ jest odwzorowaniem niezmienniczym oraz przeciwobraz zera $f^{-1}(0)$ jest zwartym podzbiorem $U$. W zbiorze niezmienniczych odwzorowań lokalnych wprowadza się relację otopii będącą prosta modyfikacją pojęcia homotopii. Omówimy konstrukcję pozwalającą na przyporządkowanie każdej parze $(U, f)$ pewnego elementu pierścienia Eulera $U(G)$. W ten sposób otrzymamy niezmiennik homotopijny (stały na klasach otopii), który jest naturalnym uogólnieniem klasycznego stopnia topologicznego. Celem badań prowadzonych w tym kierunku jest uzyskanie opisu niektórych składników prostych stabilnych niezmienniczych grup homotopii sfer. Wykład oparty jest na pracy P. Bartłomiejczyka, K. Gęby i M. Izydorka Otopy classes of equivariant maps, J. Fixed Point Theory Appl. 7 (2010), no. 1, 145-160.