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Titles and Abstracts

Titles and abstracts will be posted here as they become available.

Timothée Bénard (Cambridge)

Title: Inverse problem for the harmonic measure

Abstract: Sample paths of a random walk on a hyperbolic group are known to converge to the Gromov boundary. The distribution of the exit point is called the harmonic measure. I will discuss the following inverse problem: how much information on the law of increment of the walk can be deduced from the harmonic measure at infinity? I will explain how one can address this question by studying some winding statistics of a typical geodesic ray on the group chosen randomly by the harmonic measure.

Rhiannon Dougall (Durham)

Title: Group extensions and unitary representations

Abstract: I will motivate the study of group extensions (for instance with dynamics and geometry) from which you will inevitably be led to thinking in terms of unitary representations. (And possibly vice-versa.) If I am feeling ambitious I will discuss mixing for a tower of finite sheeted covers.

Maik Gröger (Jagiellonian University)

Title: Continuity of Følner averages

Abstract: The notion of generic/mean points goes back to the seminal work of Krylov and Bogolyubov. The first to investigate the question of what happens when all points of a dynamical system are generic for some invariant measure seem to be Dowker and Lederer in 1964. As it turns out, combining this property with other topological regularity criteria yields measure-theoretic rigidity results of the dynamical system. For example, minimality of the system implies its unique ergodicity in this setting. Another natural topological criterion in place of minimality is to assume that the map, which assigns each point its invariant measure to which it is generic, is continuous. By several recent works by different authors, the following picture emerges for abelian group actions in this setting: each point is generic for some ergodic measure and even stronger, each orbit closure is uniquely ergodic. In my talk, I will show that this is no longer the case for general actions by topological amenable groups, providing concrete counter examples involving the group of all orientation preserving homeomorphisms on the unit interval as well as the Lamplighter group. Moreover, in the course of the talk I will elaborate on the recently introduced notion of weak mean equicontinuity if time permits. This is joint work with G. Fuhrmann and T. Hauser.

Anders Karlsson (Geneva and Uppsala)

Title: Metric spaces, an ergodic theorem and theoretical aspects of deep learning

Abstract: The composition of random transformations appears in many scientific contexts, such as the theory of dynamical systems, random walks on groups or the time evolution of various systems subject to some randomness. It also appears in deep learning in several ways. There is a general ergodic theorem for such noncommutative random products that was developed and established in joint works with Ledrappier and with Gouëzel. Special cases (in view of some separate geometric arguments) include Oseledets’ multiplicative ergodic theorem for invertible matrices, Thurston’s spectral theorem for surface homeomorphisms, and random mean ergodic theorems. The metric methods involved are part of what could be called metric functional analysis, in analogy with ordinary (linear) functional analysis. In joint work with B. Avelin, we display semi-invariant metric for some of the more standard choices of deep neural networks. Then the noncommutative ergodic theorem can be applied to the common random initialization procedure and guarantee a certain stability and regularity that recent research suggests might be advantageous for the learning.

Sabrina Kombrink (Birmingham)

Title: Asymptotic expansions of functions related to the geometry of fractals

Abstract: We will discuss asymptotic expansions of functions related to the geometry of attractors of certain Graph-Directed Systems, such as parallel volumes and spectral counting functions. The proofs of the asymptotic expansions rely on spectral properties of Ruelle-Perron-Frobenius operators.

Ian Morris (QMUL)

Title: A variational principle relating self-affine measures and self-affine sets

Abstract: A breakthrough result of ‪Bárány‬, Rapaport and Hochman published in 2019 showed that if a two dimensional affine iterated function system is strongly irreducible and satisfies the strong open set condition, then the Hausdorff dimension of its attractor is equal to a value defined by Falconer in 1988. Their result applied a Ledrappier Young formula established by Bárány and Käenmäki and a variational principle due to Morris and Shmerkin in combination with deep new results on projections of self-affine measures. This Ledrappier-Young formula has since been extended to higher dimensions by Feng, and projections of self-affine measures in higher dimensions are currently being studied by Rapaport. In this talk I will describe an extension of the variational principle to higher dimensions. In combination with a recent preprint of Rapaport this implies that the above theorem of Bárány‬, Rapaport and Hochman is also valid in dimension three. This is joint work with Çağrı Sert.

Çağrı Sert (Warwick)

Title: Stationary measures on projective spaces

Abstract: We show that if a probability measure μ on GL_d(R) has only one deterministic exponent in the sense of Furstenberg-Kifer, then and μ-stationary measure on the projective space P(R^d) lives on a projective subspace of R^d on which the support of μ acts completely reducibly. This connects the works of Furstenberg-Kifer and Guivarc'h-Raugi & Benoist-Quint. Together with our first work on topic and the aforementioned ones, it yields a classification of stationary measures on projective spaces without any algebraic assumptions (but under a moment assumption). Joint works with Richard Aoun.

Manuel Stadlbauer (Universidade Federal do Rio de Janeiro)

Title: The Poincaré series of a regular cover with a hyperbolic group of deck transformations

Abstract: Let G be a convex-cocompact Kleinian group and N < G be a normal subgroup such that G/N is non-amenable. It is known that in this setting the abscissa of convergence δ of the Poincaré series P(s) of N is strictly smaller than the one from G and, moreover, that P(δ) is finite. If, in addition, G/N is word hyperbolic, it is possible to determine the asymptotics of the derivative of P(s) as s tends to δ from above.

Dalia Terhesiu (Leiden)

Title: Strong mixing for Z^d extensions of hyperbolic dynamical systems

Abstract: In the first part I will recall known results on strong mixing for Z^d extensions of discrete hyperbolic dynamical systems (including the Lorentz gas map with finite and infinite horizon). In the second part I will highlight the difficulties in the continuous time (flows) and present the newest results in joint work with F. Pène on strong mixing for the Lorentz gas flow with infinite horizon.

Mike Todd (St Andrews)

Title: Almost Anosov flows

Abstract: I will present a new class of perturbed Anosov flows which are uniformly hyperbolic except at one point. The non-uniform hyperbolicity at this point leads to the SRB measure satisfying stable laws or non-standard CLTs for natural observables. These properties are linked to the local behaviour of the pressure function. The techniques involve reducing the system to a suspension flow and building a functional analytic framework for the base system. This is joint work with H. Bruin and D. Terhesiu.