Ergodic Theory Workshop

All talks in Frank Adams - Alan Turing Building

15:30 Joel Moreira
University of Warwick
Partition Regularity of Pythagorean Pairs
An influential and still open question of Erdos and Graham in Ramsey theory asks whether any finite coloring of the natural numbers yields a monochromatic solution to the equation x^2+y^2=z^2. In recent joint work with Nikos Frantzikinakis and Oleksiy Klurman we answer a simpler question and show that in any finite coloring of the natural numbers there exists a solution to x^2+y^2=z^2 with two of the variables in the same color. The proof combines techniques from ergodic theory and number theory. In the talk I will explain how ergodic theory can be used in this kind of combinatorial problems, and I will outline the main dynamical steps in the proof.

16:10 Cagri Sert
University of Warwick
Stationary probability measures on projective spaces
We give a description of stationary probability measures on projective spaces for an iid random walk on PGL(d,R) without any algebraic assumptions. This is done in two parts. In a first part, we study the case (non-critical or block-dominated case) where the random walk has distinct deterministic exponents in the sense of Furstenberg-Kifer-Hennion. In a second part (critical case), we show that if the random walk has only one deterministic exponent, then any stationary probability measure on the projective space lives on a subspace on which the ambient group of the random walk acts semisimply. This connects the critical setting with the work of Guivarc'h-Raugi and Benoist-Quint. Combination of all these works allow to get a complete description. I will mention connections with other stationary measure classification results in homogeneous dynamics. Joint works with Richard Aoun.

16:50 Reem Yassawi
Queen Mary University of London
A dynamical view of Tijdeman's solution of the chairman assignment problem
In 1980, R. Tijdeman provided an on-line algorithm that generates sequences over a finite alphabet with minimal discrepancy, that is, such that the occurrence of each letter optimally tracks its frequency.

In this article, we define discrete dynamical systems generating these sequences. The dynamical systems are defined as exchanges of polytopal pieces, yielding cut and project schemes, and they code tilings of the line whose sets of vertices form model sets.

We prove that these sequences of low discrepancy are natural codings of toral translations with respect to polytopal atoms, and that they generate a minimal and uniquely ergodic subshift with purely discrete spectrum.

This is joint work with Valérie Berthé, Olivier Carton, Nicolas Chevallier and Wolfgang Steiner.

11:30 Mike Todd
University of St Andrews
Limit theorems for dynamics with heavy tails and clustering
We are interested in weak invariance principles for random walks coming from dynamical systems.  These can be represented by paths in Euclidean space: in the more classical case we see convergence to Brownian motion.  In our case there are two differences: i) large values of our variables cause jumps, so we are in the realm of Lévy processes ii) clustering of large values means that we have to enrich our Lévy processes and our path space.  I’ll introduce these ideas through simple examples.  This is joint work with A.C. Freitas and J.M. Freitas.

13:45 Henna Koivusalo
University of Bristol
Discrepancy estimates for the frequency of points in cut and project sets
Cut and project sets are obtained by taking an irrational slice through a lattice and projecting it to a lower dimensional subspace. This usually results in a set which has no translational period, even though it retains a lot of the regularity of the lattice. As such, cut and project sets are one of the archetypical examples of point sets featuring aperiodic order. In this talk I will give an overview of the definition and basic properties of cut and project sets, to demonstrate how they can be naturally studied in the context of dynamical systems, discrete geometry, harmonic analysis, or Diophantine approximation, for example, depending on one's own tastes and interests. As I will explain in the talk, all of these contexts are relevant to a new result of mine on discrepancy estimates for density of points in cut and project sets (the work is joint with Jean Lagacé, with an appendix by Tobias Hartnick and Michael Bjorklund).

14:25 Han Yu
University of Warwick
Number-theoretic properties of points on self-similar sets
In this talk, we will consider number-theoretic questions for points on self-similar sets. For example, are there normal/badly-approximable/well-approximable/irrational algebraic numbers in the middle third Cantor set (a self-similar set)? Those seemingly naive questions are often surprisingly difficult to answer. I will survey some classical results in this field as well as some challenging open problems and their recent breakthroughs.

11:30 Alexander Baumgartner
University of Warwick
Cusp Excursions on Bianchi Orbifolds

11:50 Shreyasi Datta
University of York
Bad is null

12:10 Boyuan Zhao
University of St Andrews
Countable Markov shifts with exponential mixing

14:00 Peej Ingarfield
University of Manchester
Thermodynamic Formalism of Self Similar Overlapping Measures

14:20 Jonguk Yang
University of Zurich
Self-similarity of 2D Dynamics at the Boundary of Chaos

14:40 Amlan Banaji
Loughborough University
When does the box dimension of a fractal exist?