# Ergodic Theory and Dynamical Systems Seminar

###### The seminars will take place in Room B3.02 (unless stated otherwise)

##### Organiser: Selim Ghazouani

Term 1

- 8 October

**Pablo Shmerkin**(Torcuato di Tella)

Title:**A nonlinear version of Bourgain's projection theorem, with applications.**

Abstract:*Bourgain's projection theorem is an extension of his celebrated discretized sum-product estimate that has found striking applications in ergodic theory and other areas. I will discuss a generalization of the projection theorem from the family of linear projections to parametrized families satisfying a technical but mild condition. I will present some applications to distance sets and direction sets, as well as a relaxation of the hypotheses of the theorem that is new even in the linear case. The proofs are based on applying the original version of the theorem to measures in a suitable multiscale decomposition, that I will describe if time allows.*

- 15 October
(Imperial)

Andrew Clarke

Title:**Arnold Diffusion in Multi-Dimensional Convex Billiards**

Abstract:*Consider billiard dynamics in a strictly convex domain, and consider a trajectory that begins with the velocity vector making a small positive angle with the boundary. Lazutkin proved in the 70’s that in two dimensions, it is impossible for this angle to tend to zero along trajectories. Using the geometric techniques of Arnold diffusion, we show that in three or more dimensions, assuming the geodesic flow on the boundary of the domain has a hyperbolic periodic orbit and a transverse homoclinic, the existence of such trajectories is a generic phenomenon in the real-analytic category.*

- 22 October

**Andy Hammerlindl**(Monash)

Title:Abstract:**Partially hyperbolic surface endomorphisms***Partially hyperbolic surface**endomorphisms are a family of not necessarily invertible surface maps which are associated with interesting dynamics. The dynamical behaviour of these maps is less understood than their invertible counterparts, and existing results show that they can exhibit properties not possible in the invertible setting. In this talk, I will discuss recent results regarding the classification of partially hyperbolic surface endomorphisms. We shall see that either the dynamics of such a map is in some sense similar to a linear map, or that the map falls into a special class of interesting examples. This is joint work with Layne Hall*

- 29 October

**Joël Moreira**(Warwick)

Title:**Multicorrelation sequences, nilsequences and primes**

Abstract:*Multicorrelation sequences are fundamental objects of study in ergodic Ramsey theory. The structure of multicorrelation sequences for a single transformation was described by Bergelson, Host and Kra, up to a negligible error. A similar description was later obtained by Frantzikinakis for several commuting transformations, with an arbitrarily small (but no longer negligible) error. Recently Le proved an analogue of the Bergelson-Host-Kra theorem for averages over the set of prime numbers. In this talk I will explain how to obtain a similar analogue of Frantzikinakis theorem for averages over the primes.*

*This talk is based on joint work with Anh Le and Florian Richter.*

- 5 November

**Davoud Cheraghi**(Imperial)

Title:**Complex Feigenbaum phenomena with degenerating geometries**

Abstract:*The renormalisation is one of the main focus of the theory of one-dimensional complex dynamics. It is connected to the central conjectures on the density of hyperbolicity and the local connectivity of the Mandelbrot set. For quadratic polynomials, there are two different types of renormalisations — the primitive and satellite types. The primitive renormalisation has been successfully studied over the past few decades; the corresponding maps exhibit tame dynamical behaviour. The satellite type has a very different nature and remained mostly mysterious until recently. In this talk, we discuss the wide range of possibilities for the dynamics in presence of infinitely many satellite renormalisation structures.*

- 12 November

**Adrien Boulanger**(Marseille)

Title:**Counting problems in infinite measure.**

Abstract:*Given a group acting properly and discontinuously on a metric space, one would like to measure how big is the orbit of a point in counting how many points of such an orbit lie in some ball of large radius.**A simple asymptotic of the above quantity when the radius of the ball goes to infinity would be called a counting result.**In the setting of Kleinian groups, e.g discrete groups of isometries of some hyperbolic space, the question was extensively studied for decades. There is mainly two different approaches to this problem: the analytical one, relying on the Selberg's pre-trace formula, and the dynamical one which relies on the mixing of the geodesic flow. Both of them need a finite measure assumption somewhere to work.**During this talk we shall see how to 'merge' the two methods through the introduction of the Brownian motion in order to drop the finite measure assumption for a weaker one.* - 19 November

**Paul Verschueren**(Imperial)

Title:

**Anergodic Birkhoff Sums**

Abstract:*Anergodic Birkhoff Sums lie in the growing intersection between Number Theory and Dynamical Systems.**A paradigmatic example is given by the final paper by Hardy & Littlewood on Diophantine Approximation. In this paper Hardy & Littlewood deployed many of the advanced techniques they had previously developed in order to analyse the growth of the Birkhoff sum of cosecants. Davenport wrote In his introduction to The Collected Papers of GH Hardy : "The proof of this remarkable result is curiously indirect; it involves contour integration and the use of Cesaro means of arbitrarily high order". He included it in his list of the top 5 unsolved problems from Hardy's work, adding: "The problem is to give a simpler and more direct proof of these results".**A breakthrough was made in 2009 by Sinai and Ulcigrai who proved a result on a related series of cotangents using the "cut and stack" technique of interval dynamics. Although "elementary", the proof was far from simple! We will present another new approach based on circle dynamics, and discuss the challenges remaining.*

- 26 November

**Mehdi Yazdi**(Oxford)

Title:**The Perron-Frobenius degrees of Perron algebraic integers**

Abstract:*A real algebraic number p \geq 1 is called Perron if it is strictly larger than its other Galois conjugates. An important theorem of Lind classified the set of possible entropies of mixing shifts of finite type as the set of logarithms of Perron numbers. Given p, the smallest size of such shift of finite type is defined as the Perron-Frobenius (PF) degree of p. For a Perron number p that is not totally-real, we give a lower bound for its PF degree, in terms of the layout of the Galois conjugates of p in the complex plane. As an application, we obtain a result known to Lind, McMullen and Thurston namely there are cubic Perron numbers whose PF degrees are arbitrary large. I will discuss connections to certain mapping class group problems.*

- 3 December

**Yan Mary He**(Luxembourg)

Title:

**A Riemannian metric and Hausdorff dimension on the Mandelbrot set**Abstract:

*In this talk, we introduce a Riemannian metric on hyperbolic components of the Mandelbrot set which is conformal equivalent to the pressure metric. As an application, we show that the Hausdorff dimension function has no local maximum on any hyperbolic component. Along the way, we introduce multiplier functions for invariant probability measures on Julia sets, which is a key ingredient in the construction of our metric. This is joint work with Hongming Nie.*

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### Term 2

- 7 January

**Ian Melbourne**(Warwick)

Title:

**Anomalous diffusion in deterministic systems**Abstract:

*In sufficiently slowly mixing dynamical systems, the classical central limit theorem breaks down and may lead to convergence to a superdiffusive Levy process.**After reviewing older work on Pomeau-Manneville intermittent maps, we will describe recent results on**(i) convergence to Levy processes in billiards,**(ii) convergence to superdiffusive stochastic processes in fast-slow systems.**The results cast light upon, and raise questions about, the various Skorokhod topologies used in the analysis of superdiffusive phenomena.*

- 14 January

**Disheng Xu**(Imperial)

Title:**Pathology and asymmetry: centralizer rigidity for partially hyperbolic system**Abstract:*In this talk we will discuss some results and some open problems on the subject the classifications of the centralizer of partially hyperbolic systems. For example, conservative perturbation of discretized geodesic flow over negatively curved surface, partially hyperbolic skew product or DA system on tori, etc. Joint work with D. Damjanovic and A. Wilkinson and joint work with S. Gan, Y. Shi and J. Zhang.*

- 28 January

**Arturo Viero**(Barcelona)

Title:**TBA**

- 4 February

**Olivier Glorieux**(IHES)

Title:**TBA**

- 18 February

**Catherine Bruce**(Manchester)

Title:**TBA**

- 25 February

**Natalia Jurga**(Surrey)

Title:**TBA**

- 10 March

**Jon Chaika**(Utah)

Title:**TBA**

See also:

Mathematics Research Centre

Mathematical Interdisciplinary Research at Warwick (MIR@W)

Past Events

Past Symposia

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**E-mail:**

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