# 2019-20

### 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
Andrew Clarke
(Imperial)
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: Partially hyperbolic surface endomorphisms
Abstract: 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
Joel 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
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: Hamiltonian-Hopf bifurcation under a periodic forcing.
Abstract: We consider a Hamiltonian with an equilibrium having a Hopf
bifurcation such that it becomes a complex saddle with compact invariant
manifolds. We add a very simple non-autonomous periodic perturbation
that contains infinitely many terms.

We study the splitting of the invariant 2-dimensional stable/unstable manifolds
after the perturbation using a first order Poincaré-Melnikov approach. Due to
the interaction of the intrinsic angle and the periodic perturbation the
splitting behaves quasi-periodically on two angles. We are interested in the
relevant harmonics of the splitting functions and in the detection of harmonics
associated to best approximants which never dominate the splitting (hidden
harmonics). Different frequencies of the time periodic perturbation will be
considered: quadratic irrationals, frequencies having continuous fraction
expansion with bounded and unbounded quotients, and "typical" frequencies in
measure theoretical sense.

This is a joint work with Ernest Fontich and Carles Simó.
• Victor Kleptsyn (Rennes). This seminar will take place at 4pm in room MS.02

Title:The Furstenberg theorem: adding a parameter and removing the stationarity.

Abstract: The classical Furstenberg theorem describes the (almost sure) behaviour of a random product of independent matrices from SL(n,R); their norms turn out to grow exponentially. In our joint work with A. Gorodetski, we study what happens if the random matrices from SL(2,R) depend on an additional parameter. It turns out that in this new situation, the conclusion changes. Namely, under some natural conditions, there almost surely exists a (random) "exceptional" set on parameters where the lower limit for the Lyapunov exponent vanishes.

Another direction of the generalization of the classical Furstenberg theorem is removing the stationarity assumption. That is, the matrices that are multiplied are still independent, but no longer identically distributed. Though in this setting most of the standard tools are no longer applicable (no more stationary measure, no more Birkhoff ergodic theorem, etc.), it turns out that the Furstenberg theorem can (under the appropriate assumptions) still be generalized to this setting, with a deterministic sequence replacing the Lyapunov exponent.
The two generalizations can be mixed together, providing the Anderson localization conclusions for the non-stationary 1D random Schrödinger operators.
• 4 February
Olivier Glorieux (IHES)

Title : Random paths on negatively curved manifolds.

Abstract : Simple random walks have been studied in many different contexts. However they often rely on a fixed measure on a group that one iterates. In this talk we will describes a random process on Hadamard manifolds, defined through the use of Dirichlet fundamental domains, that does not come from a simple random walk.
We will explain how the spectral theory can help us to understand the problem and will show converging properties for this Markov process.

• 11 February
Polina Vytnova (Warwick)

Title : Dimension of Bernoulli convolutions: computer assisted estimates.

Abstract: Bernoulli convolutions is a family of self-similar probability measures on the unit interval. The question of computing the Hausdorff dimension for specific parameter values has recently received lots of attention. We will present several computer-assisted methods addressing tricky values, such as Salem numbers. Based on the joint work with M. Pollicott and V. Kleptsyn

• 18 February
Catherine Bruce (Manchester)

Title: Projections of random measures on products of $\times m,\times n$-invariant sets and a random Furstenberg sumset conjecture

Abstract: In 2012 Hochman and Shmerkin proved that, given Borel probability measures on [0,1] invariant under multiplication by 2 and 3 respectively, the Hausdorff dimension of the orthogonal projection of the product of these measures is equal to the maximum possible value in every direction except the horizontal and vertical directions. Their result holds beyond multiplication by 2,3 to natural numbers m,n which are multiplicatively independent. We discuss a generalisation of this theorem to include random cascade measures on subsets of [0,1] invariant under multiplication by multiplicatively independent m,n. We will define random cascade measures in a heuristic way, as a natural randomisation of invariant measures on symbolic space. The theorem of Hochman and Shmerkin fully resolved a conjecture of Furstenberg originating in the late 1960s concerning sumsets of these invariant sets. We apply our main result to present a random version of this conjecture which holds for products of percolations on $\times m, \times n$-invariant sets.

• 25 February
Natalia Jurga (Surrey)
Title: Dimension spectrum of non-conformal iterated function systems
Abstract: Given an infinite iterated function system (IFS) $\Phi=\{\phi_i: \mathbb{R}^d \to \mathbb{R}^d\}_{i \in \N}$ of contractions, one can define the dimension spectrum $D(\Phi):=\{\dim F_I : I \subset \mathbb{N}\}$, where $F_I$ denotes the attractor of the subsystem $\{\phi_i\}_{i \in I}$.

Dimension spectra were introduced by Kessebohmer and Zhu (Journal of Number Theory, 2006) alongside their proof of the Texan conjecture, which asserted that $D(\Phi)=[0,1]$ when the associated IFS is given by the inverse branches of the Gauss map. Recently there has been interest in the topological properties of $D(\Phi)$ in the case that the maps in $\Phi$ are conformal, (eg. Chousionis, Leykekhman & Urbanski, Transactions of the AMS, 2019 & Selecta Mathematica, to appear; Das & Simmons, preprint, 2019).

In this talk we'll discuss the challenges that arise when investigating the dimension spectrum of IFS consisting of non-conformal maps and interesting new phenomena that emerge in this setting.

• 3 March
Reem Yassawi (Open University)
Title: The Ellis semigroup of some nontame shifts
Abstract: Let (X, σ) be a topological dynamical system, where X is a compact metric space
and σ : X → X is a homeomorphism. Its Ellis semigroup is the compactification
of the group action generated by σ in the topology of pointwise convergence on the
space $X^X$. The Ellis semigroup is, typically, a huge beast, and its computation has
been restricted mainly to systems (X, σ) which are metrically equicontinuous; such
systems are called tame.
In this talk we describe a complete description of the Ellis semigroup for a
family of nontame shifts (X, σ). These are shifts defined by bijective substitutions,
of which the Thue-Morse shift is an example. (X, σ) admits an equicontinuous
factor π : (X, σ) → (Y, δ), and so the Ellis semigroup E(X) is an extension of
Y by its subsemigroup $E^{fib}(X)$ of elements which preserve the fibres of π; this
includes all idempotents. We give a complete description of $E^{fib}(X)$, expressing it
as an uncountable product of the finite group G, defined to be the normal closure
of the group generated by the idempotents, with a semigroup Σ. We illustrate
with examples all the possibilities that can occur. This is joint work with Johannes
Kellendonk.
• 10 March
Jon Chaika (Utah)

Title: There exists a weakly mixing billiard in a polygon.

Abstract: This main result of this talk is that there exists a billiard flow in a polygon that is weakly mixing with respect to Liouville measure (on the unit tangent bundle to the billiard). This strengthens Kerckhoff, Masur and Smillie's result that there exists ergodic billiard flows in polygons. The existence of a weakly mixing billiard follows, via a Baire category argument, from showing that for any translation surface the product of the flows in almost every pair of directions is ergodic with respect to Lebesgue measure. This in turn is proven by showing that for every translation surface the flows in almost every pair of directions do not share non-trivial common eigenvalues. This talk will explain the problem, related results, and approach. Time permitting it will present a bit of the argument to rule out shared eigenvalues. The talk will not assume familiarity with translation surfaces. This is joint work with Giovanni Forni.

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### Term 3

• 28 April
Cagri Sert (Zürich) Zoom link
Title: The joint spectrum and spectral radius of random products of matrices

Abstract: I will survey some recent progress on random matrix products theory. In a first part, I will talk about the recently introduced notion of joint spectrum, an asymptotic limit shape associated to a bounded set of invertible matrices, and its relevance in random matrix products theory. In a second part, I will focus on the spectral radius of random products and mention some recent limit theorems. Based on joint works with Richard Aoun and Emmanuel Breuillard.

• 5 May
Florian Richter (Northwestern)
Title: Dynamical generalizations of the Prime Number Theorem and disjointness of additive and multiplicative actions

Abstract: One of the fundamental challenges in number theory is to understand the intricate way in which the additive and multiplicative structures in the integers intertwine. We will explore a dynamical approach to this topic. After introducing a new dynamical framework for treating questions in multiplicative number theory, we will present an ergodic theorem which contains various classical number-theoretic results, such as the Prime Number Theorem, as special cases. This naturally leads to a formulation of an extended form of Sarnak's conjecture, which deals with the disjointness of actions of $(\mathbb{N},+)$ and $(\mathbb{N},\times)$. This talk is based on joint work with Vitaly Bergelson.

• 12 May
Jungwon Lee
(Paris)

Title: Dynamics of continued fractions and conjecture of Mazur-Rubin Zoom link
Abstract: Mazur and Rubin established several conjectural statistics for modular symbols. We show that the conjecture holds on average. We plan to introduce the approach based on spectral analysis of transfer operator associated to a certain skew-product Gauss map and consequent result on mod p non-vanishing of modular L-values with Dirichlet twists (joint with Hae-Sang Sun).
• 19 May
Han Yu
(Cambridge)

Title: Arithmetical independence between doubling sequences and distal sequences.
Abstract: On the one-dimensional torus, consider the doubling map and the irrational rotation map. Those two maps behave very differently. In this talk, we shall see that these two maps are in some sense independent to each other. More precisely, given any pair of numbers (a,b) with b being irrational, the closure of $\{2^n a+nb\}_{n\geq 0}$ has full Hausdorff dimension.

• 2 June
Charles Fougeron
(Paris)
Title: Dynamics of multidimensional continued fraction algorithms and simplicial systems Zoom link

Abstract: By using a randomized approach to multidimensional continued fraction algorithms we can study their dynamics using random walks methods.

These techniques enables us to introduce a graph criterion to show ergodicity of such algorithms. Moreover it provides us with a convenient tool to prove exponential tail results and start a thermodynamical study of their dynamics. I will start by explaining how one can give a graph description of a multidimensional continued fraction algorithm called a simplicial system. Then I will explain the random walk point of view on these objects and how it implies dynamical results.

I will finish by explaining some applications on fractal dimensions (e.g. for the Rauzy gasket) using thermodynamic formalism.

• 9 June
Daniel Glasscock
(University of Massachusetts, Lowell)
Title: Uniformity in the dimension of sumsets of p- and q-invariant sets, with applications in the integers Zoom link

Abstract: : Harry Furstenberg made a number of conjectures in the 60’s and 70s seeking to make precise the heuristic that there is no common structure between digit expansions of real numbers in different bases. Recent solutions to his conjectures concerning the dimension of sumsets and intersections of times p- and q-invariant sets now shed new light on old problems. In this talk, I will explain how to use tools from fractal geometry and uniform distribution to get uniform estimates on the Hausdorff dimension of sumsets of times p- and q-invariant sets. I will explain how these uniform estimates lead to applications in the integers: the dimension of a sumset of a p-invariant set and a q-invariant set in the integers is as large as it can be. This talk is based on joint work with Joel Moreira and Florian Richter

• 16 June
(Paris)

Title : Multilple Borel Cantelli lemma and applications Zoom link

Abstract : The Borel Cantelli lemma is a classical tool to decide whether almost surely an infinite number of rare events are realized. It does not give however any information on the repartition in time of the realizations. We extend the Borel Cantelli lemma to characterize the realization infinitely many times, of multiple events in a same time scale.

Dynamical versions of this extension allow to characterize multiple recurrence and multiple hitting of shrinking targets for exponentially dynamical mixing systems. Among the applications are multilog laws for geodesic excursions (generalizing Sullivan's logarithm law) as well as for Diophantine approximations (generalizing the Khintchine-Groshev theorems).

This is a joint work with Dmitry Dolgopyat et Sixu Liu.

• 23 June
Sylvain Crovisier (Orsay)
Title: Renormalization of Hénon maps Zoom link

Abstract: For surface diffeomorphisms, a strictly positive entropy is associated with the existence of 'horseshoes'. I will focus on surface diffeomorphisms with zero entropy: can the dynamics of these 'simple' systems be described? how does it bifurcate to positive entropy systems? These questions will be discussed for a class of volume-contracting surface diffeomorphisms whose dynamics is intermediate between one-dimensional dynamics and general surface dynamics. It includes the dynamics of any Hénon diffeomorphism with Jacobian smaller than 1/4.
De Carvalho, Lyubich and Martens have built a renormalization for the (real) Hénon maps with Jacobian close to 0 and described the boundary of the parameters with zero entropy. With E. Pujals and C. Tresser we extend some of these results up to Jacobian 1/4: any Hénon map with zero entropy can be renormalized. As a consequence, we obtained a two-dimensional version of Sharkovsky’s theorem about the set of periods of interval maps.

• 2 July (exceptionally on Thursday, 2pm)
Anders Karlsson
(Genève)
Title: An ergodic theorem for noncommuting random products Zoom link

Abstract: The composition of random maps that do not commute was perhaps first discussed by Poincaré, but also studied from the 1950s and onward with the need to solve linear difference equations with random coefficients, giving rise to products of random matrices. It also appears in differentiable dynamics, with the derivative cocycle. In both these settings a basic fundamental theorem is the multiplicative ergodic theorem of Oseledets from 1968 asserting the existence of Lyapunov exponents. But there are also several non-linear settings of interests, such as random walks on groups, surface homeomorphisms, holomorphic maps, and deep learning. In joint works, first with Margulis, then with Ledrappier and finally with Gouëzel, we establish a general extension of the multiplicative ergodic theorem, that in a way responds to a question raised by Furstenberg in 1963. It is given in terms of metric functionals (which include horofunctions) and suggests a (non-linear) metric functional analysis and spectral theory.