# Ergodic Theory and Dynamical Systems Seminar

###### Starting from the 12th of February 2019, the seminars will take place in Room B3.03 (unless stated otherwise)

##### Organizers: Polina Vytnova and Selim Ghazouani

### Term 1

- 2 October

**Richard Sharp**(Warwick)

Title:*Periodic orbit growth on covers of Anosov flows*

Abstract: Suppose we have the lift of an Anosov flow to a regular cover. By considering periodic orbits intersecting some bounded region in the cover, one may define a Gurevic entropy which is less that or equal to the topological entropy of the Anosov flow. In the special case of a geodesic flow over a compact manifold with negative sectional curvatures, we have equality if and only if the cover is amenable. This result fails for other Anosov flows but we will discuss a natural generalisation. This is joint work with Rhiannon Dougall. - 9 October

**Michele Triestino**(Dijon)

Title:*Smoothening singular group actions on manifolds*

Abstract: Motivated by the recent results around Zimmer’s program, we study group actions on manifolds, with singular regularity (we require that every element is differentiable at all but countably many points). The groups under considerations have a fixed point property, named*FW*, which generalizes Kazhdan’s property (*T*) (in particular we can consider actions of lattices in higher rank simple Lie groups).

The main result is that if a group*G*has property*FW*, any singular action of*G*on a closed manifold

1) either has a finite orbit,

2) or is conjugate to a differentiable action, up to changing the differentiable structure of the manifold.

This is a joint work with Yash Lodha and Nicolas Matte Bon. - 16 October

**Etienne Le Masson**(Universite de Cergy-Pontoise)

Title:*Quantum ergodicity and random waves in the Benjamini-Schramm limit*

Abstract: One of the fundamental problems in quantum chaos is to understand how high-frequency waves behave in chaotic environments. A famous but vague conjecture of Michael Berry predicts that they should look on small scales like Gaussian random fields. We will show how the notion of Benjamini-Schramm convergence of manifolds (originally defined for graphs) can be used to formulate Berry’s conjecture precisely. The Benjamini-Schramm convergence includes the high-frequency limit as a special case but provides a much more general framework that will lead us to consider a case where the frequencies stay bounded and the size of the manifold increases instead. In this alternative setting we will explain how the ergodicity of the Gaussian wave and the mixing of the geodesic flow can be used to prove weaker forms or consequences of the random wave conjecture.

Joint work with Miklos Abert and Nicolas Bergeron. - 17 October (2:00pm, B3.02)

**Hillel Furstenberg**(Jerusalem)

Title:*Affine group actions*

Abstract: When*X*is a compact, convex space, and a group*G*acts on*X*preserving the affine structure, we speak of an "affine action", or, representation. In analogy with linear representation theory, one would like to describe all minimal — or irreducible — affine actions. We develop the theory for Lie groups and focus on*PSL(2,R)*, noting that every bounded harmonic function in the unit disc leads to an irreducible affine representation. It turns out surprisingly that up to equivalence, this group has a unique irreducible affine representation. - 23 October
**Stephen Cantrell**(Warwick)

Title:*Comparing word length with displacement for actions on CAT(-1) spaces*

Abstract: Suppose a hyperbolic group acts sufficiently nicely on a complete CAT(-1) geodesic metric space . Fix an origin for . Each group element in displaces this origin by a distance comparable to the word length of . In this talk we discuss various ways in which we can make an averaged comparison between the word length and displacement associated to the action of on . We will also discuss other geometrically interesting real valued functions on hyperbolic groups, for which these comparison results apply. - 30 October
**Dmitry Turaev**(Imperial)

Title:*On wandering domains near a homoclinic tangency*

Abstract: Given a map, we define a wandering domain as an open region such that the diameter of its images by the iterations of the map shrinks to zero but the corresponding limit set is not a periodic point. It is known that many finitely smooth two-dimensional diffeomorphisms have wandering domains while it is not known if a polynomial diffeomorphism of a plane can have one. We discuss wandering domains whose limit sets is a homoclinic tangency. We show the existence of real analytic planar diffeomorphisms with wandering domains and discuss how to find wandering domains for polynomial diffeomorphisms of the three-dimensional space.

**Gabriella Keszthelyi**(Rényi Institute, Budapest) — 3:00pm, B3.02

Title:*Dynamical properties of biparametric skew tent maps*

Abstract: pdf - 6 November
**Federico Rodriguez-Hertz**(Penn State/Lille)

Title:*Classification of Anosov actions and first cohomology*

Abstract: In the late 60’s and early 70’s it was conjectured that Anosov diffeomorphisms have an algebraic origin, indeed that Anosov diffeomorphisms are topologically conjugated to algebraic automorphisms of infranilmanifolds. In this direction, J. Franks and A. Manning showed that if the underlying manifold is an infranilmanifold the Anosov diffeomorphism is topologically conjugated to an algebraic one. When the acting group is higher rank, strong rigidity results are expected and it is conjectured that actions of higher rank lattices with an Anosov element should be on infranilmanifolds and smoothly equivalent to algebraic. In this talk I will discuss some recent advances in this direction, proving results analogous to Franks-Manning in the setting of higher rank abelian actions and higher rank lattices on semisimple Lie groups actions. This is a joint project with A. Brown and Z. Wang. - 13 November
**Jose Alves**(Porto/Loughborough)

Title:*Entropy formula and continuity of entropy for piecewise expanding maps*

Abstract: We consider some classes of piecewise expanding maps in finite dimensional spaces with invariant probability measures which are absolutely continuous with respect to Lebesgue measure. We derive an entropy formula for such measures. Using this entropy formula, we present sufficient conditions for the continuity of that entropy with respect to the parameter in some parametrized families of maps. We apply our results to some families of piecewise expanding maps. Joint work with Antonio Pumariño. - 20 November
**Ariel Rapaport**(Cambridge)

Title:*Dimension of planar self-affine sets and measures.*

Abstract: A compact is called self-affine if there exist affine contractions such that . In the 1980s, Kenneth Falconer introduced a value , called the affinity dimension, which is the ''expected'' value for the dimension of . I will discuss a recent project with Mike Hochman, building on our joint work with Balázs Bárány, in which we establish the equality under mild assumptions. - 21 November (2:00pm, MS.05)

**Yotam Smilansky**(Einstein Institute, Jerusalem)

Title:*Multiscale substitution schemes and Kakutani sequences of partitions.*

Abstract: Substitution schemes provide a classical method for constructing tilings of Euclidean space. Allowing multiple scales in the scheme, we introduce a rich family of sequences of tile partitions generated by the substitution rule, which include the sequence of partitions of the unit interval considered by Kakutani as a special case. In this talk we will use new path counting results for directed weighted graphs to show that such sequences of partitions are uniformly distributed, thus extending Kakutani's original result. Furthermore, we will describe certain limiting frequencies associated with sequences of partitions, which relate to the distribution of tiles of a given type and the volume they occupy. - 27 November

**Björn Winckler**(Imperial)

Title:*Instability of renormalization*

Abstract: In this talk I will discuss renormalization of low-dimensional dynamical systems and associated phenomena such as universality and rigidity. The first part of the talk will be an introduction to the classical results in this field. In the second part I will discuss new phenomena that appear in the article 'Instability of Renormalization' (arXiv:1609.04473). In particular, I will introduce the renormalization operator acting on Lorenz maps (these are one-dimensional maps associated with three-dimensional flows undergoing a homoclinic bifurcation). This operator turns out to have non-trivial dynamics inside topological classes of stationary type which in turns leads to the phenomena of coexistence and as dimensional discrepancy. All of these notions will be explained in the talk. - 4 December

**Mike Whittaker**(Glasgow)

Title:*Correspondences for Smale spaces*

Abstract: Correspondences provide a notion of a generalised morphism between Smale spaces. Using correspondences we define an equivalence relation on all Smale spaces which reduces to shift equivalence for shifts of finite type. In this talk I will outline our construction and introduce the key ingredients. This is joint work with Robin Deeley and Brady Killough.

### Term 2

- 8 January

**Matt Galton**(Warwick)

Title:*Limit theorems and the iterated weak invariance principle for slowly mixing dynamical systems.*

Abstract: The iterated weak invariance principle (WIP) is an ingredient in showing solutions of certain ODEs converge to a solution of some SDE. In this talk, we will consider dynamical systems given by a non-invertible map undergoing some mild mixing assumption with respect to bounded observables. The iterated WIP involves approximating iterated integrals of Brownian motion by iterated integrals of some process associated to the dynamical system. I will introduce the central limit theorem, WIP and the iterated WIP, and discuss sufficient conditions for these to hold. - 15 January

**Igors Gorbovickis**(Bremen)

Title:*Critical points of the multiplier map.*

Abstract: The multiplier of a non-parabolic periodic orbit of a map can be extended by means of analytic continuation to a multiple-valued algebraic function on the space of all quadratic polynomials . Information about the location of the critical points of this function might shed light on the question of possible geometric shapes of hyperbolic components of the Mandelbrot set. We show that as the period of the periodic orbits increases to infinity, the critical points of the multiplier map equidistribute on the boundary of the Mandelbrot set.

This is joint work with Tanya Firsova. - 17 January (3:00pm, MS.03; joint with Geometry and Topology seminar)

**Artem Dudko**(Warsaw)

Title:*On computational complexity of Cremer Julia sets.*

Abstract: Informally speaking, a compact subset of a plane is called computable if there is an algorithm which can draw arbitrarily good approximations of this set. Computational complexity measures how long does it take to draw these approximations. It is known that for some classes of rational maps (e.g. hyperbolic, parabolic and Collet-Eckmann) the Julia sets have polynomial complexity. For others (e.g. Siegel) the Julia sets can have arbitrarily high computational complexity and even may be uncomputable. However, not much is known in the case of presence of Cremer periodic points. We show that there exist abundant Cremer quadratic polynomials with Julia sets of arbitrarily high complexity.

The talk is based on joint work with Michael Yampolsky. - 22 January

**Anatoly Neishtadt**(Loughborough)

Title:*On destruction of adiabatic invariance in a magnetic billiard*

Abstract: Billiard in a magnetic field is a popular model in nonlinear dynamics. In this model, motion of a charged particle in a plane region with perfectly reflected smooth boundary is considered. A magnetic field is directed perpendicular to the plane. We assume that the magnetic field is strong enough and nonuniform. In the particle motion, two modes of motion alternate: the skipping along the boundary of the billiard and the drift in the interior part of the billiard. Motion at each mode has an adiabatic invariant. Change of mode of motion results in a small jump of the adiabatic invariant. We demonstrate that the accumulation of these jumps leads to the destruction of the adiabatic invariance.

This is joint work with A. Artemyev. - 24 January (3:00pm, MS.03; joint with Geometry and Topology seminar)

**Dmitry Jakobson**(McGill)

Title:*On small gaps in the length spectrum.*

Abstract: We discuss upper and lower bounds for the size of gaps in the length spectrum of negatively curved manifolds. For manifolds with algebraic generators for the fundamental group, we establish the existence of exponential lower bounds for the gaps. On the other hand, we show that the existence of arbitrary small gaps is topologically generic: this is established both for surfaces of constant negative curvature, and for the space of negatively curved metrics. While arbitrary small gaps are topologically generic, it is plausible that the gaps are not too small for almost every metric; we discuss one result

This is joint work with Dima Dolgopyat. - 29 January

**Ale Jan Homburg**(Amsterdam)

Title:*On-off intermittency*

Abstract: I'll consider skew product systems with low dimensional maps as fiber maps, driven by shifts or by expanding circle maps . The fiber maps will have a common fixed point and I will consider the case where Lyapunov exponents in the fiber direction vanish. I will explain that this can lead to intermittent dynamics. - 30 January (2:00pm, MS.04)

**Victor Kleptsyn**(Rennes)

Title:*Interacting Polya urns*

Abstract: The classical Polya urn process is a reinforcement process, in which there are balls of different color in the urn, we take out a ball at random, and the color that was just out of it gets an advantage for all future turns: we return this ball to the urn and add another one of the same color.

However, in this process on every step all the colors are competing. What will happen if on different steps there will be different subsets of competing colors? For instance, if there are companies that compete on different markets, or if a signal is choosing its way to travel?

Some questions here have nice and simple answers; my talk will be devoted to the results of our joint project with Mark Holmes and Christian Hirsch on the topic. - 5 February (2:00pm, B3.02)

**John Smillie**(Warwick)

Title:*Counting closed geodesics in polygonal billiard tables*

Abstract: I will describe how ideas involving renormalisation can be used to count the number of families of closed geodesics of a given length in planar billiard tables where the angles are rational multiples of . If time permits I will explain the conjectural connection with Ratner theory. - 12 February

**Yuri Lima**(UFC)

Title:*Markov partitions for billiards*

Symbolic dynamics is a tool that simplifies the study of dynamical systems in various aspects. One approach to obtain symbolic dynamics is constructing Markov partitions: if properly constructed, then the system has properties similar to Markov chains in probability. In this talk, we will discuss the recent developments of this approach for non-uniformly hyperbolic planar billiards. A consequence is a new estimate on the number of periodic trajectories for the billiard map of some Sinai billiards. - 14 February (1:00pm, MS.05)

**Italo Cipriano**(Santiago)

Title:*On the Wasserstein distance between stationary probability measures*

Abstract: I will present the answers to some of the questions proposed by Jon Fraser in [First and second moments for self-similar couplings and Wasserstein distances, Mathematische Nachrichten, 288, (2015), 2028—2041]. This is in part joint work with Mark Pollicott. - 19 February

**Leticia Pardo Simón**(Liverpool)

Title:*Escaping dynamics of a class of transcendental functions*

Abstract: As a partial answer to Eremenko's conjecture, it is known for functions with bounded singular set and of finite order that every point in their escaping set can be connected to infinity by an escaping curve. Even if those curves, called "hairs" or "rays" not always land, this has been positively proved for some functions with bounded postsingular set by showing that their Julia set is structured as a Cantor Bouquet. In this talk I will consider certain functions with bounded singular set but unbounded postsingular set whose singular orbits escape at some minimum speed. In this setting, some hairs will split when they hit critical points. We show that the existence of a map on their parameter space whose Julia set is a Cantor Bouquet guarantees that such hairs, if maybe now with split ends, still land.

**Tony Samuel**(3:00pm, B1.01) (Birmingham)

#### Cancelled

~~Title:~~*On the regularity of Sturmian words and other aperiodic sequences*

Abstract: The theory of aperiodic order is a relatively young field of mathematics, which has attracted considerable attention in recent years. It has grown rapidly over the past three decades; on the one hand, due to the discovery of quasicrystals; and on the other hand, due to intrinsic mathematical interest in describing the very border between crystallinity and aperiodicity. While there is currently no axiomatic framework for aperiodic order, various types of order conditions have been and are still being investigated.

At the turn of the century Durand and Lagarias & Pleasants established key order conditions to be studied. In this talk, we will discuss these order conditions, as well as generalisations and extensions thereof, for two classes of aperiodic sequences: Sturmian words and a new family of aperiodic sequences stemming from Grigorchuk's infinite 2-group. We will also show that (exact) Jarník sets naturally give rise to a classification of Sturmian words in terms of such order conditions. - 26 February

**Valery Gaiko**(National Academy of Sciences of Belarus, Minsk)

Title:*Limit cycles of planar polynomial dynamical systems*

Abstract: We discuss new bifurcational geometric methods based on the Wintner-Perko termination principle for the global qualitative analysis of planar polynomial dynamical systems. This is related to the solution of Hilbert’s sixteenth problem on the maximum number and distribution of limit cycles. - 5 March

**Vassili Gelfreich**(Warwick)

Title:*Energy transfer in slow-fast systems: ergodicity vs non-ergodicity*

Abstract: In this talk we discuss the role of ergodicity in the equilibration in slow-fast systems. In classical statistical mechanics the justification of the low of equipartition relies on ergodicity of dynamics. Second, in a slow-fast system ergodicity of the fast sub-system impedes the equilibration of the whole system due to the presence of adiabatic invariants. We show that the violation of ergodicity in the fast dynamics effectively drives the whole system to equilibrium. We use some models based on non-automous billiard to illustrate the ideas.

This talk is based on joint works with V. Rom-Kedar, K. Shah and D. Turaev. - 12 March

**Jens Marklof**(Bristol)

Title:*Pair correlation and equidistribution on manifolds*

Abstract: A series of recent papers [1, 2, 4] has established that if a given deterministic sequence on a circle has a Poisson pair correlation measure, then the sequence is uniformly distributed. Analogous results have been proved for point sequences on higher-dimensional tori [3, 5]. The purpose of this lecture is to describe a simple statistical argument that explains this observation and furthermore permits a generalisation to sequences (and more general triangular arrays) in bounded Euclidean domains as well as compact Riemannian manifolds.

*References*- C. Aistleitner, T. Lachmann, and F. Pausinger, Pair correlations and equidistribution, Journal of Number Theory 182 (2018), 206–220.
- S. Grepstad and G. Larcher, On pair correlation and discrepancy, Arch. Math. 109 (2017), 143–149
- A. Hinrichs, L. Kaltenböck, G. Larcher, W. Stockinger and M. Ullrich, On a multi-dimensional Poissonian pair correlation concept and uniform distribution, arXiv:1809.05672
- S. Steinerberger, Poissonian pair correlation and discrepancy, Indagationes Math. 29 (2018), 1167– 1178
- S. Steinerberger, Poissonian pair correlation in higher dimensions, arXiv:1812.10458

### Term 3

- 30 April
**Pierre Berger**(Paris) and**Oliver Butterley**(Nova Gorica) - 7 May
**James Robinson**(Warwick) - 14 May
**Kenneth Falconer**(St Andrews) - 21 May
**Toby Hall**(Liverpool) - 28 May
**Norbert Peyerimhoff**(Durham) - 4 June
**Jeroen Lamb**(Imperial) and**Tony Samuel**(Birmingham) - 11 June
**Carl Dettmann**(Bristol) - 18 June
**Simon Baker**(Warwick) - 25 June
**No seminar**— One Day ETDS meeting in Birmingham

See also:

Mathematics Research Centre

Mathematical Interdisciplinary Research at Warwick (MIR@W)

Past Events

Past Symposia

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