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.

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: In joint work with Zemer Kosloff, we show that a totally dissipative system has all nonsingular systems as factors, but that this is no longer true when the factor maps are required to be finitary. In particular, if a nonsingular Bernoulli shift has a limiting marginal distribution p, then it cannot have, as a finitary factor, an independent and identically distributed (iid) system of entropy larger than H(p); on the other hand, we show that iid systems with entropy strictly lower than H(p) can be obtained as finitary factors of these Bernoulli shifts, extending Keane and Smorodinsky's finitary version of Sinai's factor theorem to the nonsingular setting.

  ---

• 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
  Abstract:  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)
• 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.