Dynamical Systems Seminar

Random substitutions are generalisations of substitutions, where letters are mapped randomly and independently to one of a finite set of possible words. This typically gives rise to dynamical systems with a hierarchical structure, mixed spectral type, and a higher complexity than classical substitution systems. In fact, random substitution systems usually have positive entropy. Among the many ergodic measures, a special role is played by the measures that maximise the entropy - if there is a unique such measure the system is called intrinsically ergodic. We show that for certain random substitution systems, the measures of maximal entropy are precisely those that are invariant under the so called "shuffle group", introduced in previous work of Fokkink-Rust-Salo. This leads to an equivalent criterion for intrinsic ergodicity in terms of an associated Markov chain. Finally, we illustrate the richness of this class by providing an example with several measures of maximal entropy (joint work with A. Mitchell).

Given an irrational rotation, it is straightforward to see that for every Cantor set C, there is a dense (in fact, residual) set of points whose orbit doesn't intersect C. On the other hand, if C is a fat Cantor set (that is, of positive Lebesgue measure), almost every point visits C with a frequency equal to the measure of C. But what other frequencies of visits to C may occur? In the words of a MathOverflow post by Dominik [1], what is the Birkhoff spectrum of fat Cantor sets? We give a first answer to this question by showing that every irrational rotation allows for certain fat Cantor sets C whose Birkhoff spectrum is maximal, that is, equal to the interval [0,Leb(C)]. In this talk, I will focus on discussing some of the basic tools behind this result and extensions of it. [1] D. Kwietniak, Possible Birkhoff spectra for irrational rotations, MathOverflow (2020), https://mathoverflow.net/q/355860 (version: 2020-03-27).