# Warwick ETDS Seminar 2022-23

* The seminars are held on Tuesdays at 14:00 in B3.02 (unless stated otherwise)*.

## Term 3

**May 2, 2023****Han Yu (Warwick)***Title: $p$-adic approximation on $q$-Cantor sets.*

Abstract: In this talk, I will introduce a series of problems regarding Diophantine approximation on Cantor sets. One example of such a problem was presented by Sanju Velani as follows: Take the middle-third Cantor set $C_3$. What can we say about approximating points on $C_3$ by dyadic rationals (i.e. rationals like $\mathbb{Z}/2^{\mathbb{N}}$)? We will see a few more related problems/conjectures in this direction and partial progress.

**May 9, 2023**

**Eddie Nijholt (Imperial College)**

*Title: Hidden Symmetry in Network Dynamical Systems*

Abstract: Network dynamical systems are prevalent throughout nature and engineering. A major obstacle in their analysis is the fact that most established techniques from dynamical systems theory are not suited to deal with them, as they involve coordinate transformations that destroy the network structure. We present a novel solution, by showing that many network properties --and sometimes the network structure itself-- can be realized as a form of symmetry. The key is to move away from the classical notion of group symmetry, and to consider more exotic algebraic structures such as quivers instead. This results in various reduction techniques that are now much better equipped to deal with the network setting, which in turn may lead to the classification of generic bifurcations in such systems.

**May 16, 2023**

**Angxiu Ni (Peking University)**

*Title: Adjoint method for chaos: sampling linear responses by an orbit*

Abstract: Physical measures encode the long-time statistics of chaotic dynamical systems, and the linear response is the parameter-derivative of physical measures. For physical measures of discrete-time hyperbolic chaotic systems, we derive an equivariant divergence formula for the unstable perturbation of measure transfer operators along unstable manifolds. With this new formula, the linear response can be sampled by recursively computing only $2u$ many vectors on one orbit, where $u$ is the unstable dimension. The numerical implementation of this formula is neither cursed by dimensionality nor the sensitive dependence on initial conditions. Our work generalizes conventional adjoint or backpropagation methods to chaotic systems.

**May 23, 2023**

**Dominic Veconi (Coventry University)**

*Title: Thermodynamics of smooth models of pseudo-Anosov diffeomorphisms*

Abstract: We discuss the thermodynamic and ergodic properties of a smooth realization of pseudo-Anosov surface homeomorphisms. In this realization, the pseudo-Anosov map is uniformly hyperbolic outside of a neighborhood of a set of singularities, and the trajectories are slowed down so the differential is the identity at the singularities. Using Young towers, we prove existence and uniqueness of equilibrium measures for the family of observables known as the geometric $t$-potentials. This set of equilibrium measures includes a unique smooth Sinai-Ruelle-Bowen physical measure and a measure of maximal entropy, the latter of which has exponential decay of correlations and the Central Limit Theorem.

**May 30, 2023**

**Andreas Mountakis (University of Warwick)**

*Title: Distinguishing sets of strong recurrence from van der Corput sets*

*Abstract:*Sets of recurrence (and variants thereof) form a class of sets that arises naturally in ergodic theory, while van der Corput sets (and variants thereof) arise in the theory of uniform distribution of sequences. Even though these classes look a-priori unrelated, it turns out that they are closely connected, and in fact it is difficult to produce examples showing that they are not the same class of sets. In this talk, we construct a set of strong recurrence (which is a natural strengthening of the notion of recurrence) which is not a van der Corput set. We will not assume any familiarity with those notions, and everything we need will be introduced.

**June 6, 2023**

**Nicholas Fleming (University of Warwick)**

*Title: Deterministic homogenisation of fast-slow systems and functional correlation bounds*

*Abstract:*In this talk, we consider homogenisation (convergence to a stochastic differential equation) for deterministic fast-slow systems. Kelly and Melbourne used rough path theory to show that homogenisation follows from certain statistical properties for the fast direction. However, previously it was not possible to check one of these properties for nonuniformly hyperbolic maps with slow decay of correlations (including certain classes of chaotic billiards).

**June 20, 2023**

**Gaétan Leclerc (Sorbonne University)**

*Title: Fourier dimension and dynamical fractals.*

*Abstract: Consider the triadic Cantor set equipped with the Cantor law. It happens that its cumulative distribution function, the devil’s staircase, is Hölder regular, and its best exponent of regularity is ln(2)/ ln(3), which is exactly the Hausdorff dimension of the Cantor set. Moreover, one can show that the Fourier transform of the Cantor law decay like |ξ|^{− ln 2/ ln 3} "on average". This is no coincidence, and hint for a deeper link between Fractal Geometry and Fourier Analysis. In this talk we will detail and explore this link through the notion of Fourier Dimension. We will introduce the Fourier dimension, compute it on some easy examples, quote some natural questions that arise, and then discuss a (partial) state of the art on the topic.*

**June 27, 2023**

**Roberto Castorrini (University of Pisa) CANCELLED**

*Title:*

Abstract:

## Term 2

**January 10, 2023****Cagri Sert (Warwick)***Stationary measures for SL(2,R)-actions on homogeneous bundles over flag varieties*

Abstract: Let X_{k,d} denote the space of rank-k lattices in R^d. Topological and statistical properties of the dynamics of discrete subgroups of G = SL(d,R) on X_{d,d} were described in the seminal works of Benoist--Quint. A key step/result in this study is the classification of stationary measures on X_{d,d}. Later, Sargent--Shapira initiated the study of dynamics on the spaces X_{k,d}. When k < d, the space X_{k,d} is of a different nature and a clear description of dynamics on these spaces is far from being established. Given a probability measure \mu Zariski-dense in a copy of SL(2,R) in G, we give a classifi cation of stationary measures on X_{k,d} and prove corresponding equidistribution results. In contrast to the results of Benoist--Quint, the type of stationary measures that \mu admits depends strongly on the position of SL(2,R) relative to parabolic subgroups of G. I will review the preceding works (Benoist--Quint, Eskin--Lindenstrauss, Sargent--Shapira) and discuss main cases and ideas. Joint work with Alexander Gorodnik and Jialun Li.

**January 17, 2023****Constantin Kogler (Oxford)***Random walks on symmetric spaces*Abstract: We discuss recent progress in describing the local behaviour of random walks on non-compact simple Lie groups such as SL(2,R). Indeed, we give the first examples of finitely supported measures satisfying a local limit theorem for the Lie group acting on its symmetric space. The employed techniques are related to Bourgain’s construction of a finitely supported measure on SL(2,R) with an absolutely continuous Furstenberg measure. In addition, we report on joint work with Wooyeon Kim establishing effective density of random walks on homogeneous spaces for measures supported on finitely many matrices with algebraic entries.

**January 24, 2023****Xiong Jin (Manchester)**

An extension of Hochman and Shmerkin’s projection theorem- Abstract: In this talk I will present an extension of Hochman and Shmerkin’s projection theorem on the product of integer-multiplication invariant measures on the unit circle. In the symbolic setting we extend these measures to left-shift invariant measures mapped through one-dimensional iterated function systems without any separation conditions. Consequently, we prove that Bernoulli convolutions with log-rationally independent parameters are dissonate, i.e., their convolution has the maximal possible dimension. If time allows, I will also mention the extension of H.-S. theorem from invariant measures to a class of random measures called Mandelbrot cascades. This leads to an extension of Furstenberg’s sumset conjecture (now a theorem by Hochman and Shmerkin) to some more general random fractal sets.

**January 31, 2023****Yann Chaubet (Cambridge)***Counting periodic trajectories under constraints*Abstract**:**On a closed negatively curved surface, Margulis gave the asymptotic growth of the number of closed geodesics of bounded length, when the bound goes to infinity. In this talk, we will present such asymptotic results for geodesics that satisfy certain (topological or geometric) constraints.**February 7, 2023****Terry Soo (University College London)***Sinai factors for nonsingular Bernoulli shifts*

Abstract:

**February 14, 2023****Caroline Series (University of Warwick)***Convergence of spherical averages for Fuchsian groups*Abstract:Suppose given a measure preserving action of a Fuchsian group on a probability space X, together with a real valued function f on X. We prove pointwise convergence of spherical averages, more precisely, averages of f(gx) over all words of length 2n in a fixed set of generators.We will briefly review previous results which involve either Cesàro averages or are restricted to free groups. The current proof is based on a new variant of the Bowen--Series symbolic coding for Fuchsian groups that simultaneously encodes all possible shortest paths representing a given group element. The resulting coding is self-inverse, giving a reversible Markov chain to which methods previously introduced by Bufetov in the free group case may be applied.This is joint work with Sasha Bufetov and Alexey Klimenko, to appear in Commentarii Math Helvetica.

**February 21, 2023****André Salles de Carvalho (University of São Paulo)**-*cancelled due to UCU strike*

**February 28, 2023****Álvaro Bustos Gajardo (The Open University)***Quasi-recognizability and continuous eigenvalues of torsion-free S-adic systems*Abstract: We discuss combinatorial and dynamical descriptions of S-adic systems (shift spaces obtained by iteration of an infinite sequence of morphisms between alphabets), focusing on the scenario where the morphisms have constant length and are defined in alphabets of bounded size. One of the key notions that recurrently appears in the theory of such systems is recognisability, which introduces a well-defined hierarchical structure on the points of the corresponding shift space. However, S-adic systems often lack this property, and so it is usually hard to obtain general conclusions on the properties of this type of systems without imposing strong restrictions on them. In order to overcome this issue, we introduce the notion of quasi-recognisability, a strictly weaker version of recognisability but which is indeed enough to reconstruct several classical arguments of the theory of constant-length substitutions in this more general context. Furthermore, we identify a large family of morphism sequences, which we call "torsion-free", for which quasi-recognisability is naturally guaranteed, and can be improved to actual recognisability with relative ease. Using these notions we give S-adic analogues of the notions of column number and height from the theory of substitutions, including dynamical and combinatorial interpretations of each, and give a general characterisation of the maximal equicontinuous factor of the identified family of S-adic shifts, showing as a consequence that in this context all continuous eigenvalues must be rational. We also briefly discuss the measurable scenario in this context.

**March 7, 2023****Rohini Ramadas (Warwick)***Degenerations and irreducibility problems in complex dynamics*Abstract:Per_n is a (nodal) Riemann surface parametrizing, up to conjugation, complex quadratic rational functions with an n-periodic critical point. The n-th Gleason polynomial G_n is a polynomial in one variable with Z-coefficients, whose roots are in bijection with conjugacy classes of complex quadratic polynomials with an n-periodic critical point. (The roots of G_n are the centers of the period-n components of the Mandelbrot set.)Two long-standing open questions in complex dynamics are: (1) Is Per_n connected? (2) Is G_n irreducible over Q? We show that if G_n is irreducible over Q, then Per_n is connected. In order to do this, we find a smooth point with Q-coordinates on a compactification of Per_n. (This will be essentially the same talk as I gave at Miles Reid’s birthday conference last term.)

**March 14, 2023**

**Paulo Varandas (Universidade do Porto)***Exponential mixing for Gibbs measures of certain Axiom A flows*In this talk I will consider the correlations decay rate for Gibbs measures associated to codimension one Axiom A attractors for flows. I will show that codimension-one Axiom A attractors whose strong stable foliation is C1+α either has exponential decay of correlations with respect to all Gibbs measures associated to Holder continuous potentials or its stable and unstable subbundles are jointly integrable. As a consequence, we derive the existence of C1-open sets of C3- vector fields generating Axiom A flows having attractors so that the first alternative above holds. Joint work with D. Daltro (UFRB)*Abstract:***April 4, 2023**

**Anders Öberg (Uppsala University)***Döeblin measures and continuous eigenfunctions of the transfer operator*

## 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 in the restricted planar circular -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 spectrum of self-affine measures supported on planar carpets and higher dimensional sponges.