Seminarium Układy dynamiczne

I will introduce a class of finitely generated groups defined by a quasi-isometry invariant property on their Cayley graphs. Every such group admits an effective but non-sofic subshift and this class covers all known examples with this property. I will show that many well-known classes of groups have this property. Based on joint work with K. Blot, M. Sablik and V. Salo. 

An equation is partition regular over its domain if, for any finite coloring of that domain, there exists a monochromatic nontrivial solution. In this talk, we will review the background of this topic, focusing on the ergodic theoretic tools used to tackle such problems and present a recent joint work with A. Koutsogiannis, A. Ferré Moragues and W. Sun, concerning the partition regularity problem of quadratic equations over some number fields. 

Motivated by applications in statistical physics, we study the induced dynamical systems (M(X),T) and (H(X),T), consisting respectively of Borel probability measures and non-empty closed subsets of a dynamical system (X,T). In this talk, we first survey the inheritance of various well-studied dynamical properties, including equicontinuity, distality, pointwise almost periodicity, transitivity, weak mixing, nullness, tameness, and zero entropy. We then turn to mean equicontinuity, diam-mean equicontinuity, and weak-mean equicontinuity, and discuss new inheritance results with respect to these notions. These results complete the picture and reveal another interesting asymmetry between the hyperspace and measure-theoretic constructions. The talk focuses on continuous surjective maps T: X → X, while the new results can also be extended to actions of locally compact σ-compact amenable groups.