# Ergodic Theory Network

This is the webpage for a network of collaborative meetings in ergodic theory, supported by a London Mathematical Society Scheme 3 grant. The universities in the network are Bristol, Exeter, Loughborough, Manchester, Queen Mary, St Andrews, Surrey and Warwick.

Next meeting: **Wednesday 28th November 2018** at Queen Mary University of London (Room W316 in the Queens' Building)

**Simon Baker (Warwick)**

Title: *The complexity of the set of codings for self-similar sets*

Abstract: In this talk I will discuss a recent paper with Derong Kong where we study the complexity of the set of codings for self-similar sets. In this paper we prove that if the similarities in our iterated function system have contraction ratios sufficiently close to $1$, then every interior point of the attractor has a coding containing all finite words. Similarly we prove that for any positive integer $k$, if the similarities in our iterated function system have contraction ratios sufficiently close to $1$ (in a way that depends upon $k$), then every interior point of the attractor has a coding such that all blocks of length $k$ occur with the same frequency. Our arguments make use of a well known construction of a normal number due to Champernowne, and techniques due to Erdos and Komornik.

3:15pm: Tea

**Ana Rodrigues (Exeter)**

Title: *On the dynamics of Translated Cone Exchange Transformations*

Abstract: In this talk I will introduce translated cone exchange transformations, a new family of piecewise isometries and renormalize its first return map to a subset of its partition. As a consequence I will show that the existence of an embedding of an interval exchange transformation into a map of this family implies the existence of infinitely many bounded invariant sets. Finally, I will prove the existence of infinitely many periodic islands, accumulating on the real line, as well as non-ergodicity of our family of maps close to the origin. (This is joint work with Pedro Peres.)

4:45pm: Break

**Jan Boronski (Ostrava)**

*Cantor set dynamics from inverse limits of graph covers*

Abstract: In my talk I shall discuss some recent applications of a method of constructing Cantor set dynamics by inverse limits of graph covers, that was introduced by Akin, Glasner and Weiss in 2008. This will include constructions of topologically mixing completely scrambled systems, as well as embeddings of minimal systems into the real line with vanishing derivative everywhere; based on a joint work with J. Kupka, and P. Oprocha.

**Jon Keating (Bristol)**

*Joint Moments of Characteristic Polynomials of Random Unitary Matrices*

Abstract: I shall describe recent results (obtained with E. Basor, R. Buckingham, A. Its, E. Its and T. Grava) relating the joint moments of the characteristic polynomial of a CUE random matrix and its derivative to a solution of the PainlevĂ© V equation. This connection can be used to derive explicit formulae and to show that in the large-matrix limit the joint moments are related to a solution of PainlevĂ© III equation.

The suggested venue for lunch is the Curve restaurant on the QMUL campus. A group of people will be there from 1:00pm but those attending Jon Keating's talk may wish to go earlier or make other arrangements. Directly following the final talk, there will be dinner at a local restaurant.