Speaker: Boyuan Zhao (St Andrews)

Title: Almost sure cover times for dynamical systems

Abstract:  Given a topologically transitive system on the unit interval, one can investigate the time for an orbit to reach certain level of resolution in the repeller. Under mixing conditions, the asymptotics of typical cover times are determined almost surely by Minkowski dimensions when they are finite, or by stretched Minkowski dimensions otherwise. For application, we show that for countably full-branched affine maps, results using the usual Minkowski dimensions fail to produce a finite log limit of cover times whilst the stretched version gives an finite limit. In addition, cover times of irrational rotations are explicitly calculated as counterexamples, due to the absence of mixing.

Speaker: Inhyeok Choi (Cornell Univeristy)

Title: Dynamics of semigroups of circle homeomorphisms

Abstract: The circle homeomorphsim group is a huge group that contains a variety of subgroups and subsemigroups. Because of this ampleness, Tits alternative for the circle homeomorphism group does not hold in general. Still, there is a dynamical Tits alternative, asked by Ghys and first proved by Margulis, that reads as follows: every subgroup of Homeo(S1) either preserves an invariant probability measure, or it contains a Schottky pair that enables ping-pong argument. A similar dichotomy holds for subsemigroups of Homeo(S1) thanks to Malicet’s theory of random walks on Homeo(S1). In this talk, I will explain an alternative approach to dynamical Tits alternative for semigroups, motivated by Margulis’ original approach.

Speaker: Michael Hochman (Hebrew Univerisity)

Title: Equidistribution for toral endomorphisms

Abstract: I will discuss a version of Host's equidistribution theorem on multidimensional tori, giving an essentially optimal result that removed some unnecessary restrictions that appeared in Host's work on the subject. In the commuting case this gives a new proof and an extension of the measure classification theorem of Einsiedler and Lindenstrauss.

Speaker: Reem Yassawi (Queen Mary University London)

Title: Two applications of normalisation of Pisot expansions

Abstract:  A linear recurrence with constant coefficients, whose characteristic polynomial is the minimal polynomial of a Pisot number, defines a positional numeration system L for the natural numbers. While these numeration systems do not define an additive group, Frougny and Solomyak showed that L + L ⊂ L + F for some finite set F. Furthermore using the concept of normalisation, they showed that these discrepancies can be computed by a finite automaton with bounded delay. We give two applications of normalisation. The first is to define and partly characterise Mahler equations for irreducible Pisot recurrences. The interest in this is that in the more classical reducible case, solutions to Mahler equations define transcendental numbers. The second application is to give an alternative proof of a result of Vershik, who shows that the completion of what he calls the Fibonadic numeration is the circle. This is joint work with Olivier Carton and Jake Sudbery.

Speaker: Riccardo Giannini (University of Glasgow)

Title: Large Kernel Property for Geometric Homomorphisms and Applications

Abstract: Artin groups have presentations given in terms of labelled graphs and some of them can be embedded in mapping class  groups, the groups of orientation-preserving homeomorphism of finite-type surfaces up to homotopy. The embeddings are given by some natural homomorphisms called geometric, used to describe group-theoretic properties of some Artin groups that, despite being easy to define, proved to be intricate to study. In this talk, we show that the kernels of these geometric homomorphisms can usually be huge, as they contain a non-abelian free group of rank 2 detected by a Ping-Pong argument. Nevertheless, the consequences of the result are deeper, shedding some light on the topology of some moduli spaces. These moduli spaces are called strata of translation surfaces and parametrize closed Riemann surfaces with a particular flat metric on the complement of some finite number of points. 

Speaker: Tuomas Sahlsten (University of Helsinki)

Title: Harmonic analysis in parabolic dynamics

Abstract: High frequency Fourier asymptotics of equilibrium states for chaotic systems is a tool to study the intersections of attractors, rates of mixing, and quantizations (i.e. quantum chaos) of these systems. The Fourier asymptotics seems to be intimately connected to the algebraic features of the dynamics, e.g. on middle third Cantor set (invariant set for times 3 mod 1 that is affine and “algebraic”) equilibrium states cannot have decaying Fourier coefficients at high frequencies but on badly approximable numbers with prescribed digits (invariant set for Gauss map that is non-linear) they can. We discuss how intermittency (variation between regular and chaotic phases of dynamics)  e.g. for Manneville-Pommeau maps manifests in the Fourier transform of equilibrium states for such systems.

Speaker: Kasun Fernando (Brunel University London)

Title: Asymptotic properties of estimators for parameters in chaotic dynamical systems

Abstract: In this talk, I will give a brief introduction to parameter estimation in dynamical systems and discuss a recent result (joint with Nan Zou, Macquarie University) on the second order accuracy of the bootstrap for dynamical systems, which provides a method to quantify the uncertainty of estimators based on finite orbit data. I will also discuss an alternative to the bootstrap called “self-normalisation” adapted from time series analysis into dynamical systems setting (on going joint work with Bryan Bajar, Georgy Sofranov, and Nan Zou, Macquarie University).

Speaker: Stephanie Zbinden (Heriot Watt University)

Title: Dynamics on the Morse boundary

Abstract:  The action of hyperbolic groups on their Gromov boundary has been proven to be a very useful in the study of said groups. In this talk, we introduce a generalization of the Gromov boundary, called the Morse boundary, which is defined for all finitely generated groups, not just hyperbolic groups. We then show that the action of a group on its Morse boundary can be used akin to the Gromov boundary to gather information about the group. 

Speaker: Anton Zorich (University of Paris 7: Paris Diderot)

Title: Random square-tiled surfaces and random multicurves in large genus

Abstract: Moduli spaces of Riemann surfaces and related moduli spaces of quadratic differentials are parameterized by a genus g of the surface. Considering all associated hyperbolic (respectively flat) metrics at once, one observes more and more sophisticated diversity of geometric properties when genus grows. However, most of metrics,

on the contrary, progressively share certain rules. Here the notion of “most of” has explicit quantitative meaning, for example, in terms of the Weil-Petersson measure. I will present some of these recently discovered large genus universality phenomena.

I will use count of metric ribbon graphs (after Kontsevich and Norbury) to express Masur-Veech volumes of moduli space of quadratic differentials through Witten-Kontsevich correlators. Then I will present Mirzakhani's count of simple closed geodesics on hyperbolic surfaces. We will proceed with description of random

geodesic multicurves and of random square-tiled surfaces in large genus. I will conclude with a beautiful universal asymptotic formula for the Witten-Kontsevich correlators predicted by Delecroix, Goujard, Zograf and myself and recently proved by Amol Aggarwal.