Ergodic Theory and Dynamical Systems Seminar

The seminars are held on Tuesdays at 14:00 in B3.02.

On seminar days we meet for lunch at 12:30 and tea at 15:00 in the common room.

Term 1

Organizers: Steve Cantrell, Tom Rush, Cagri Sert, Han Yu

01/Oct/24: Speaker: Mike Hochman (Hebrew Univ.)

Title: New results on embedding of self-similar sets

Abstract: I will discuss the problem of affinely embedding self-similar sets in the line into other such sets. Conjecturally, embedding is precluded when the contraction ratios of the defining maps are incommensurable. This is a baby version of harder problems about intersections of fractals, but in the open cases even the embedding problem is challenging. I will describe recent joint work with Amir Algom and Meng Wu in which we confirm the conjecture whenever the contraction ratios are algebraic numbers, and also for a.e. choice of parameters. I will discuss the proof. If time permits, I will explain how this is related to the dimension of alpha-beta sets, and describe recent examples that show that the problem is more delicate than anticipated.

08/Oct/24: Speaker: Jens Marklof (Bristol)

Title: Directional Statistics of Lattice Points and Escape of Mass for Embedded Horospheres

Abstract: I will discuss escape of mass estimates for SL(d,ℝ)-horospheres embedded in the space of affine lattices, which depend on the Diophantine properties of the shortest affine lattice vector. These estimates can be used, in conjunction with Ratner's theorem, to prove the convergence of moments in natural lattice point problems, including the statistics of directions in lattices, inhomogeneous Farey factions, and the distribution of smallest denominators. Based on joint work with Wooyeon Kim (ETH).

15/Oct/24: Speaker: Firdavs Rakhmonov (St Andrews)

Title: Similar Point Configurations in Vector Spaces over Finite Fields and Quotients of Distance Sets

Abstract: I will discuss the existence of similar point configurations in subsets of finite-dimensional vector spaces over finite fields. Let $\mathbb{F}_q$ be the finite field with $q$ elements, and let $\mathbb{F}_q^d$ be the $d$-dimensional vector space over $\mathbb{F}_q$. If $E \subseteq \mathbb{F}_q^d$, then the distance set of $E$ is defined as $\Delta(E) := \{ \lVert x - y \rVert : x, y \in E \}$, where $\lVert \cdot \rVert : \mathbb{F}_q^d \to \mathbb{F}_q$ is defined by $\lVert \alpha \rVert = \alpha_1^2 + \dots + \alpha_d^2$ for $\alpha = (\alpha_1, \dots, \alpha_d) \in \mathbb{F}_q^d$. Iosevich, Koh, and Parshall proved that if $d \geq 2$ is even and $|E| \geq 9q^{\frac{d}{2}}$, then $\frac{\Delta(E)}{\Delta(E)} = \mathbb{F}_q$.

I will explore the geometric aspects of this result and consider its natural generalization. Additionally, I will explain the generalization of the Iosevich-Koh-Parshall result for arbitrary non-degenerate quadratic forms in $d$ variables over $\mathbb{F}_q$, where the main tool is discrete Fourier analysis.

22/Oct/24: Speaker: Thomas Jordan (Bristol)

Title: Countable Markov shifts, pressure at infinity and large deviations.

Abstract: For an ergodic transformation, large deviations measure the rate of the decay of the measure of the points where the Birkhoff average is away from the expected value. For many hyperbolic dynamical systems with Gibbs measures, this rate is exponential and the exponential rate can be determined using a suitable pressure function. We will show how these results can be adjusted to the setting of mixing subshifts where the alphabet is countably infinite. We show how the standard results can be adapted to this setting and related to the concept of pressure at infinity (recently introduced by Anibal Velozo). We show how the pressure at infinity interplays with the standard rate function and also show a version of large deviations where the rate functions come directly from the pressure at infinity. This is joint work with Godofredo Iommi and Anibal Velozo.

29/Oct/24: Speaker: Alexey Korepanov (Loughborough)

Title: Memory loss near the boundary of null recurrence for Harris recurrent Markov chains and infinite measure preserving intermittent dynamical systems

Abstract: I'll talk about our joint work in progress with Ilya Chevyrev. Memory loss is a quantification of how quickly an evolving system forgets its initial state. For example, for a Markov chain with transition operator P, given two probability measures mu and nu, we may want to know how quickly the distance between P^n mu and P^n nu decays in total variation. For Markov chains with slow (polynomial) recurrence, memory loss has been very well understood half a century ago (starting with Orey or Pitman) as long the chain is positive recurrent, yet we could not find any results in the null recurrent case (even though related questions are a subject of well developed Renewal Theory). A similar situation takes place in chaotic dynamical systems. I'll present (first?) results on memory loss that work for positive as well as null recurrent systems, taking a particular interest in proofs that survive the transition between positive and null recurrence.

05/Nov/24: Katy Loyd

Title: Pointwise ergodic averages along sequences of slow growth

Abstract: Following the Birkhoff Ergodic Theorem, it is natural to consider whether convergence still holds along subsequences of the integers. Bergelson and Richter showed that for uniquely ergodic systems, pointwise convergence holds along the number theoretic sequence $\Omega(n)$, where $\Omega(n)$ denotes the number of prime factors of $n$ counted with multiplicities. However, removing the assumption of unique ergodicity, a pointwise ergodic theorem does not hold along $\Omega(n)$. In this talk, we will classify the strength of this non-convergence by considering weaker notions of averaging. In particular, we will show that double logarithmic averaging recovers pointwise convergence. Additionally, we will introduce a criterion for identifying other slow growing sequences satisfying a similar non-convergence property (based on joint work with S. Mondal).

12/Nov/24: Timothee Benard

Title: Khintchine's theorem for fractal measures

Abstract: Khintchine's theorem is a key result in Diophantine approximation. Given a positive non-increasing function f defined over the integers, it states that the set of real numbers that are f-approximable has zero or full Lebesgue measure depending on whether the series of terms f(n))_n converges or diverges. I will present a recent work in collaboration with Weikun He and Han Zhang in which we extend Khintchine's theorem to any self-similar probability measure on the real line. The result provides an answer to an old question of Mahler, also asked by Kleinbock-Lindenstrauss-Weiss. The argument involves the quantitative equidistribution of upper triangular random walks on SL_2(R)/SL_2(Z).

19/Nov/24: Wael Bahsoun

Title: Linear response due to singularities

Abstract: It is well known that a family of tent maps with bounded derivatives has no linear response (the absolutely continuous invariant measure is not differentiable with respect to perturbation) for typical deterministic perturbations changing the value of the turning point. In this talk we consider a tent-like family with a cusp at the turning point and show that for such a family we recover the linear response.

26/Nov/24: Simon Machado (Note: this seminar will be the first introductory lecture for the 4h mini-course by Machado, see here for the details on the mini-course)

Title: The dynamics of point sets: local and global multiplicative structures

Abstract: The study of discrete point sets in homogeneous spaces has a long and rich history, with fascinating examples from statistical physics, aperiodic tilings, combinatorial number theory, and sphere packing. More recently, the theory of aperiodic point sets has found fruitful interactions with various branches of mathematics. This mini-course will focus on one emerging trend, exploring the relationship between global and local multiplicative structures from a dynamical perspective.

On the one hand, the concept embodying global structure will be a generalization of groups, known as approximate groups. This will lead us to approximate lattices (a.k.a. Meyer sets), a notion first explored by Yves Meyer.

On the other hand, the local structure will be embodied by notions of patterns studied dynamically. Analyzing pattern statistics dynamically is often beneficial, as evidenced by Furstenberg’s correspondence principle or the study of “tiling spaces”.

We will cover the following topics:

1. An introduction to approximate lattices, first defined by Yves Meyer and a discussion of their historical roots. We will focus on recent developments and how the global multiplicative structure arises from local patterns, such as 3-term arithmetic progressions.

2. We will discuss dynamical systems related to point sets, such as tiling spaces and Stone spaces. We will illustrate their usefulness by explaining a new proof of a recent breakthrough in approximate subgroups initially established through model theory.

3. Finally, I will briefly discuss exciting open problems that naturally fit this framework, including some related to Ulam stability and the Lorentz gas.

03/Dec/24: Nattalie Tamam

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