One-Day Ergodic Theory Meeting University of Exeter, 15 June 2022

Titles and Abstracts

Surabhi Desai (University of Exeter)

Dynamical Borel-Cantelli lemmas on general target sets

Abstract: In probability theory, the Borel-Cantelli lemma can be used to determine the probability that infinitely many events occur in some sequence of events. In a dynamical setting, we may establish analogous properties on sequences of shrinking target sets. Typically, such target sets shrink to a point. In this talk, we shall focus on dynamical Borel-Cantelli properties in the context of sequences of sets which shrink to more general classes of sets.

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Jamie Walton (University of Nottingham)

Substitutions over infinite alphabets and unique ergodicity

Abstract: A substitution rule is a procedure for replacing tiles with larger patches of tiles (geometric or symbolic in one dimension) that can be iterated to define tilings of Euclidean space with aperiodic order. Famous examples include the Penrose tilings and, from symbolic dynamics, the Fibonacci, Thue-Morse and period doubling sequences. The theory is well understood when the set of prototiles or ‘alphabet’, in the symbolic case, is finite. In this talk I will present joint work with Neil Mañibo and Dan Rust which considers an extension of the theory to infinite alphabets. To retain some of the flavour of the finite setting, we choose to work with continuous substitutions on alphabets equipped with a compact Hausdorff topology. We find sufficient conditions on the substitution for the resulting tiling dynamical system to be uniquely ergodic. Many results rely on the theory of positive operators on Banach spaces, where the traditional substitution matrix is replaced with the substitution operator on continuous functions over the alphabet.

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Jonathan Fraser (University of St Andrews)

An expanded Sullivan dictionary via dimension interpolation

Abstract: The Sullivan dictionary provides a beautiful correspondence between Kleinian groups acting on hyperbolic space and rational maps of the extended complex plane. An especially direct correspondence exists concerning the dimension theory of the associated limit sets and Julia sets. I will demonstrate that by slightly expanding the family of dimensions considered, a richer and more nuanced correspondence arises. This is joint work with Liam Stuart.