# Warwick ETDS Seminar 2022-23

##### Term 1
• October 11, 2022
Sabrina Kombrink (University of Birmingham)

Title: Renewal Theory in Symbolic Dynamics
Abstract: Renewal theory is concerned with the asymptotic behaviour of renewal functions. In this talk we will consider generalisations of S. Lalley's renewal functions in symbolic dynamics. We will explore the leading asymptotic term as well as asymptotic terms of lower order for such functions.

• October 18, 2022
Joe Thomas (Durham University)

Title: Poisson statistics, short geodesics and small eigenvalues on hyperbolic punctured spheres
Abstract: For hyperbolic surfaces, there is a deep connection between the geometry of closed geodesics and their spectral theoretic properties. In this talk, I will discuss recent work with Will Hide (Durham), where we study both sides of this relationship for hyperbolic punctured spheres. In particular, we consider Weil-Petersson random surfaces and demonstrate Poisson statistics for counting functions of closed geodesics with lengths on scales 1/sqrt(number of cusps), in the large cusp regime. Using similar ideas, we show that typical hyperbolic punctured spheres with many cusps have lots of arbitrarily small eigenvalues. Throughout, I will contrast these findings to the setting of closed hyperbolic surfaces in the large genus regime.

• October 25, 2022
Maryam Hosseini (Open University)
Title: About Minimal Dynamics on the Cantor Set
Abstract: Dimension group is an operator algebraic object related to minimal dynamical systems on the Cantor set. In this talk after a quick review of some definitions of dimension group,  the topological and algebraic rank of  Cantor minimal systems are considered and we will see how the rank of a Cantor system is dominated by the rank of its extensions.

• November 1, 2022
Carlos Matheus (École Polytechnique)
Title: Elliptic dynamics on certain SU(2) and SU(3) character varieties
Abstract: In this talk, we discuss the action of a hyperbolic element of SL(2,Z) on the SU(2) and SU(3) character varieties of once-punctured torii. This is based on a joint work with G. Forni, W. Goldman and S. Lawton.

• November 8, 2022
Donald Robertson (University of Manchester)
Title: Dynamical Cubes and Ergodic Theory
Abstract: In recent works Kra, Moreira, Richter and I showed that positive density sets always contain sums of any finite number of infinite sets, and a shift of the self-sum of an infinite set. The main step in our approach was to prove the existence of certain dynamical configurations described via limit points of orbits. In this talk I will describe what these configurations are, and explain how ergodic theory can be used to deduce their existence.

• November 15, 2022
Mar Giralt (Universitat Politecnica de Catalunya)
Title: Chaotic dynamics, exponentially small phenomena and Celestial Mechanics
Abstract: A fundamental problem in dynamical systems is to prove that a given model has chaotic dynamics. One of the methods employed to prove this type of motions is to verify the existence of transversal intersections between the stable and unstable manifolds of certain objects. Then, there exists a theorem (the Smale-Birkhoff homoclinic theorem) which ensures the existence of chaotic motions. In this talk we present a method to analyze the distance and transversality between certain stable and unstable manifolds when a small perturbation is added to an integrable system. In particular, we consider the case where the distance between manifolds is exponentially small. This implies that this difference cannot be detected by expanding the manifolds into a series with respect to the small perturbation parameter. Therefore, classical perturbation theory cannot be applied. Finally, we apply these techniques to a celestial mechanics problem. In particular, we study the Lagrange point $L_3$ in the restricted planar circular $3$-body problem.

• November 22, 2022
Timothée Bénard (Cambridge)
Title: The local limit theorem for biased random walks on nilpotent groups
Abstract: We prove the local limit theorem for biased random walks on a simply connected nilpotent Lie group G. The result allows to approximate at scale 1 the n-step distribution of a walk by the time n of a smooth diffusion process for a new group structure on G. We also show this approximation is robust under deviation. The proof uses a Gaussian replacement scheme, combining Fourier analysis and a swapping argument inspired by the work of Diaconis-Hough. As a consequence, we obtain a probabilistic version of Ratner's theorem: Ad-unipotent random walks on finite-volume homogeneous spaces equidistribute toward algebraic measures.

• November 22, 2022 at 3:30pm
Room: MS.01
Tim Austin (UCLA)
Title: Positive sofic entropy without relatively Bernoulli factors.
Abstract: The classical Kolmogorov–Sinai entropy is an invariant of probability-preserving transformations. Much of the resulting theory was successfully extended to actions of discrete amenable groups by Ornstein, Weiss and others.
Lewis Bowen’s more recent notion of sofic entropy extends the Kolmogorov–-Sinai definition to actions of sofic groups, a much larger class introduced by Gromov. A range of natural questions concern how entropy and its consequences differ between the sofic setting the amenable one.
After reviewing a special case of sofic entropy for certain free-product groups, this talk will present a new example of an action of such a group. The example has positive sofic entropy, but has no splitting as a direct product involving a Bernoulli factor. This contrasts with the world of amenable group actions, where many such splittings are guaranteed by the weak Pinsker theorem. The new example is an algebraic action, and its analysis depends on (slight modifications of) results
from the theory of random regular low-density parity-check codes.
This material is part of an ongoing joint project with Lewis Bowen, Brandon Seward and Christopher Shriver.
(This talk will be a continuation of the colloquium from Friday Nov 18th, and will assume some of the notions from that talk.)

• November 29, 2022
Ruxi Shi (Sorbonne Université)  cancelled
Title: TBA
Abstract: TBA

• December 6, 2022
Istvan Kolossvary (St. Andrews)
Title: A variational principle for box counting quantities
Abstract: The classical variational principle for topological pressure is an essential tool in thermodynamic formalism. The aim of the talk is to extend the framework into a non-conformal setting where we prove a variational principle for an appropriate pressure that is specifically tailored to calculate box counting quantities.  Similarities and differences between the variational principles will be highlighted. As an application, we can derive the $L^q$ spectrum of self-affine measures supported on planar carpets and higher dimensional sponges.