# Ergodic Theory and Dynamical Systems 2013-14

##### The seminars are held on Tuesdays at 14:00 in Room B3.02 - Mathematics Institute

Term 3

• April 15, 2014 [Room MS.03]
Dan Thompson (Ohio State University)

Title: Equilibrium states for certain robustly transitive diffeomorphisms
Abstract: We establish results on uniqueness of equilibrium states for the well-known Mañé examples of robustly transitive diffeomorphisms. This is an application of machinery developed by Vaughn Climenhaga and myself, which applies when systems satisfy suitably weakened versions of expansivity and the specification property. The Mañé examples are partially hyperbolic maps of the 3-torus, and I'll explain why these maps satisfy our hypotheses. This is joint work with Vaughn Climenhaga (Houston) and Todd Fisher (Brigham Young).

• April 17, 2014 [Room MS.05]
Abel Farkas
(St Andrews University)
Title: Projections and other images of self-similar sets
Abstract: We investigate how the Hausdorff dimension and measure of a self-similar set $K \subseteq \mathbb{R}^{d}$ behave under linear images. It turns out that this depends on the nature of the group generated by the orthogonal parts of the defining maps for $K$. We prove our results without assuming any separation conditions and introduce a new method that leads to a similarity dimension-like formula for Hausdorff dimension.
• April 29, 2014
Peter Balint (Technical University of Budapest)
Title: Rare interaction limit of two particle system
Abstract: A system is considered consisting of two particles one of which follows chaotic motion while the other is integrable, and interaction takes place via occasional collisions resulting in energy exchanges. In the rare interaction limit the evolution of the energies is expected to converge to a jump Markov process. This modification of the popular Gaspard-Gilbert model seems to be easier to access by rigorous methods than the original version. In my talk I would like to describe the model, the limiting Markov process and some of the technical aspects encountered. This is work in progress, joint with Péter Nándori, Imre Péter Tóth and Domokos Szász.
• May 6, 2014
Matthieu Arfeux (University of Toulouse Paul Sabatier)
Title: Deligne-Mumford compactification and Berkovich spaces
Abstract: The space of rational map of degree d>1 modulo conjugacy by Moebius transformations is not compact. When you consider a diverging sequence, we can see some phenomena called "rescaling limits". During this talk, I will present and compare two ways to study these phenomena. The first way uses the Deligne-Mumford compactification of the moduli space of marked spheres (my thesis work). The second way uses non-archimedian tools, the Berkovich spaces. I will not expect people to know anything about these two topics.
• May 20, 2014
Luchezar Stoyanov (University of Western Australia)
Title: Spectra of Ruelle transfer operators and large deviations
Abstract: The talk concerns Ruelle transfer operators for contact Axiom A flows on basic sets satisfying certain regularity conditions. First, strong spectral estimates for such operators will be described and some consequences of these, e.g. exponential decay of correlations for Hoelder continuous potentials with respect to any Gibbs measure, existence of a non-trivial analytic continuation of the Ruelle zeta function, a Prime Orbit Theorem with an exponentially small error. We will then discuss a sharp large deviation principle concerning intervals shrinking with sub-exponential speed for certain models involving the Poincare map related to a Markov family for the basic set.
• May 27, 2014
Jean-Pierre Marco (University of Paris VI) CANCELLED
Title: A geometric proof of the Arnold conjecture for 3 degree-of-freedom convex nearly integrable systems
Abstract: We will give a short overview of a geometric proof of the existence of global diffusion for cusp-generic 3 degree-of-freedom convex nearly integrable Hamiltonian systems. The main ingredients are:
1) the existence of compact normally hyperbolic cylinders visiting most of the phase space;
2) the Birkhoff theory of twist maps on the annulus applied to suitable sections inside the cylinders;
3) the generic properties of the homoclinic correspondence;
4) a general simple shadowing lemma for compact normally hyperbolic manifolds.

Term 2

• January 7, 2014
Weixiao Shen (National University of Singapore)
Title: The topological complexity of attractors for unimodal interval maps
Abstract: We study the topological complexity of a unimodal map restricted to a Cantor attractor. We show that for any open cover $U$ of this attractor, the complexity function $p(U,n)$ is of order $n\log n$. We shall also show that there is a non-renormalizable map with a Cantor attractor for which $p(U,n)$ is bounded from above for any open cover $U$. This is a joint work with Simin Li.
• January 14, 2014
David Marti Pete (Open University)
Title: Escaping points of transcendental self-maps of $\mathbb C^*$
Abstract: Transcendental dynamics studies the iteration of holomorphic functions that have an essential singularity. One of the main problems in this area is understanding the geometry of the escaping set: points which tend to the essential singularities under iteration. I am interested in self-maps of $\mathbb C^*= \mathbb C \setminus \{0\}$ for which both infinity and zero are essential singularities. I will introduce this setting and then I will focus on two problems: the existence of dynamic rays and the possible rates of escape of these points.
• January 21, 2014
Wael Bahsoun (Loughborough University)
Title: Decay of correlations for random intermittent maps
Abstract: We study a class of random transformations built over finitely many intermittent maps sharing a common indifferent fixed point. Using a Young-tower technique, we show that the map with the fastest relaxation rate dominates the asymptotics. In particular, we prove that the rate of correlation decay for the annealed dynamics of the random map is the same as the sharp rate of correlation decay for the map with the fastest relaxation rate.
• January 28, 2014
Luna Lomonaco (Roskilde University)
Title: Parabolic-like mappings
Abstract: A polynomial-like mapping is a proper holomorphic map $f : U' \to U$, where $U'$, $U \approx \mathbb D$, and $U' \subset \subset U$. This definition captures the behaviour of a polynomial in a neighbourhood of its filled Julia set. A polynomial-like map of degree $d$ is determined up to holomorphic conjugacy by its internal and external classes, that is, the (conjugacy classes of) the restrictions to the filled Julia set and its complement. In particular the external class is a degree $d$ real-analytic orientation preserving and strictly expanding self-covering of the unit circle: the expansivity of such a circle map implies that all the periodic points are repelling, and in particular not parabolic. We extended the polynomial-like theory to a class of parabolic mappings which we called parabolic-like mappings. A parabolic-like mapping is thus similar to a polynomial-like mapping, but with a parabolic external class; that is to say, the external map has a parabolic fixed point, whence the domain is not contained in the codomain. In this talk we give a sketch of the proof of the Straightening Theorem, which states that every parabolic-like mapping of degree 2 is hybrid equivalent to a member of the family of quadratic rational maps of the form $P_A(z) = z + 1/z + A$, $A \in \mathbb C$. Then we will consider families of parabolic-like mappings, state the main result in this setting and give an application.
• February 4, 2014
Robert Simon (LSE)
Title: Ergodic Theory and Games of Incomplete Information
Abstract: There is a connection between ergodic theory and the equilibria of games where the players hold private information and the possibilities of what they can know are uncountably infinite. These are like games of poker, but with infinitely many cards. The knowledge of the players can correspond to measure preserving transformations, and the group of these transformations can act on any equilibrium in a way so that they cannot be measurable. We explore what is known so far in the last ten years and where this line of research may go in the future.
• February 11, 2014 (in Room B3.03)
Eric Olson (University of Nevada Reno)
Title: The Weak Separation Condition and Assouad Dimension
Abstract: This talk presents joint work with J. Fraser, X. Henderson and J. Robinson which shows the Assouad dimension is greater or equal to one for the invariant set of an iterated function system of contracting similarities for which the weak separation condition of Zerner is not satisfied. In particular, we suppose for every positive epsilon there is a sequence of maps followed by a sequence of inverse maps of the iterated function system whose composition is epsilon close to the identity but not equal to the identity. From this we are able to construct a weak tangent of the attractor that contains a line segment. We also show that if the weak separation property is satisfied, then the Assouad dimension equals the Hausdorff dimension. This provides a precise dichotomy for self-similar sets in the real line.
• February 18, 2014
Ilies Zidane (University of Toulouse III)
Title: On the bifurcation locus of cubic polynomials and the size of Siegel disks
Abstract: Yoccoz gave a sufficient arithmetical condition of linearization of fixed points of holomorphic germs with multiplier $\exp(\textbf{i} 2 \pi \alpha)$ where $\alpha$ is an irrational number: $f(z) = \exp(\textbf{i} 2 \pi \alpha) z + {\mathcal O}(z^2)$. He also proved that this condition is optimal for quadratic polynomials. We will discuss this optimality for cubic polynomials and quadratic rational maps. We will see how is it related to the size of Siegel disks and parabolic implosion/renormalization. This leads to the study of slices of bifurcation locus where some surprising, unexpected and complicated phenomenons occur due to the interaction between the two critical points. We also investigate some virtual slices arising as geometric limits (parabolic enrichment) of dynamical systems.
We seek analogues of Zakeri curves (the locus where the two critical points lie at the boundary of the Siegel disk) in these slices, when the rotation number is not of bounded type, and even, for Cremer slices. Given a Siegel slice, the logarithm of the conformal radius of the Siegel disk is a subharmonic function, whose Laplacian is therefore a measure which gives a new viewpoint as well as a lot of information.
• February 25, 2014
Ian Melbourne (Warwick University)
Title: Smooth approximation of stochastic processes
Abstract: In this talk, we present an ergodic theory approach to one of the classical questions in stochastic analysis -- the interpretation of stochastic integrals. Roughly speaking, it is well-known that Brownian motion $W$ can be smoothly approximated by processes $W_n$ arising from continuous time dynamical systems (flows). Now consider processes X_n and X satisfying $dX_n=a(X_n)dt+b(X_n)dW_n$ and $dX=a(X)dt+b(X)dW$ respectively. The aim is to show that the sequence of smooth processes $X_n$ converges to the Ito process $X$, but simultaneously (and more significantly) to give the correct interpretation to the stochastic integral $b(X)dW$. In joint work with David Kelly, we give a definitive answer to this question using a mixture of rough path theory and smooth ergodic theory. In this talk we focus on the ergodic theory aspects, and do not assume a background in stochastic analysis.
• March 4, 2014
Alexey Korepanov (Warwick University)
Title: Spatial structure of SRB measures for billiards
Abstract: Sinai-Ruelle-Bowen measures are the only physically observable invariant measures for billiard dynamical systems under small perturbations. These measures may be singular with respect to Liouville measure. I will talk about the behavior of physically relevant observables, which appear to have nice properties despite irregularity of SRB measures.
• March 11, 2014
Ana Rodrigues (Exeter University)
Title: Periodic orbits for real planar polynomial vector fields of degree $n$ having $n$ invariant straight lines
Abstract: In this talk, we will study the existence and non-existence of periodic orbits and limit cycles for planar polynomial differential systems of degree $n$ having $n$ real invariant straight lines taking into account their multiplicities. (This is joint work with Jaume Llibre.)

Term 1

• October 8, 2013
Jonathan Fraser (University of Warwick)
Title: Scenery flows for non-conformal measures
Abstract: Using ergodic theory to study problems in geometry is not new, however, there have recently been some major advances in the fields of fractal geometry and geometric measure theory made by studying the dynamics of the process of 'zooming in' on fractal sets and measures. In particular, Hochman and Hochman-Shmerkin have recently developed ideas of Furstenberg to produce a rich and ripe theory. The dynamics of the blow-ups can be modeled using a 'CP-chain', which records both the point where we zoom-in, and the scenery which we then see. Thus far CP-chains have proved a powerful tool in studying geometric properties of self-similar measures, with applications to projection theorems and distance set problems.
The aim of this talk is to motivate the study of CP-chains and attempt to extend the theory beyond the conformal setting. This will be done in the context of Bernoulli measures on self-affine Bedford-McMullen carpets.
This talk is based on joint work with Andrew Ferguson and Tuomas Sahlsten.
• October 15, 2013
Terry Soo (University of Warwick)
Title: Ergodic universality
Abstract: A topological dynamical system is universal if for every invertible ergodic measure-preserving dynamical system, we can endow the topological system with an invariant measure that makes it isomorphic to the measure-preserving system, provided that the necessary entropy constraints are satisfied. In joint work with Anthony Quas, we prove that toral automorphisms and time-one maps of geodesic flows on compact surfaces of negative curvature are universal.
• October 22, 2013
Tuomas Sahlsten (University of Bristol)
Title: Ergodic flows and geometric measure theory
Abstract: We apply the machinery of M. Hochman on scenery flows to problems in local geometry of measures. We establish that the classical conical density results by Besicovitch and Marstrand are manifestations of rectifiability and the dimension theory of the measure theoretical porosity introduced by Eckmann and Järvenpääs can be reduced back to the set theoretical analogue. Proofs are based on exploiting the regular microscopic geometry the scenery flow generates and invoke classical tools in ergodic theory to pass information back to the global scale. Joint work with Antti Käenmäki (Jyväskylä) and Pablo Shmerkin (Buenos Aires).
• October 29, 2013
Anders Oberg (Uppsala University)
Title: Ergodic theory of Kusuoka measure
Abstract: There has been recent advances in field of analysis on fractals (and more generally: graph limits) using a certain energy Laplacian which is defined weakly, given a Dirichlet form, with respect to the Kusuoka measure (a combination of energy measures with respect to a basis of harmonic functions). We define a g-function, for which the Kusuoka measure is a g-measure. It is known that the Kusuoka measure is an ergodic invariant measure. We prove (in this preliminary study) that the Kusuoka measure has strong ergodic properties. For instance, the iterates of the transfer operator, defined via the g-function, converges. This is joint work with Anders Johansson and Mark Pollicott.
• November 5, 2013
Dave Sixsmith (Open University)
Title: The size of the Julia set of functions outside the Eremenko-Lyubich class
Abstract: We briefly review some well-known results regarding the size of the Julia sets of some transcendental entire functions. Most of these results concern functions in the Eremenko-Lyubich class. We then discuss two recent results which concern functions outside this class. The first generalises a result of McMullen, and gives a class of transcendental entire functions for which the Julia set has positive area and is a spider's web. The second concerns a class of functions with zeros in a certain sector for which the Julia set has Hausdorff dimension equal to 2.
• November 12, 2013
Sandro Vaienti (CPT, Marseille)
Title: On the loss of memory for non uniformly expanding (and hyperbolic) systems
Abstract: We show that for some non-uniformly expanding maps of the interval the loss of memory is polynomial. We also show that a similar result persist for non-uniformly hyperbolic diffeomorphisms with singularities.
• November 19, 2013
Adam Epstein (University of Warwick)
Title: Rescaling limits of quadratic rational maps
Abstract: Let $f_k$ be a sequence of quadratic rational maps diverging in moduli space. Under certain circumstances, there may exist a sequence of Moebius conjugate maps $F_k$ and some $q \geq 2$ such that the iterates $F_k^q$ converge algebraically to a rational map of degree 2 or more. It has been conjectured that, up to a suitable notion of equivalence, there can be at most two such "rescaling limits", the first a quadratic rational map with a fixed point of multiplier 1, the second a quadratic polynomial. We discuss partial results obtained with Carsten Petersen.
• November 21, 2013 (in Room D1.07) (Extra seminar)
Sjoerd Verduyn Lunel (Utrecht University)
Title: Transfer operators, Hausdorff dimension and the spectral theory of positive operators
Abstract: The dimension of an invariant set of a dynamical system is one of the most important characteristics. In this talk we present a new approach to compute the Hausdorff dimension of conformally self-similar invariant sets. The approach is based on a direct spectral analysis of the transfer operator associated with the dynamical system. This operator theoretic approach relies on the theory of positive operators and the theory of trace class operators and their determinants. In the case that the maps defining the dynamical system are analytic, our method yields a sequence of successive approximations that converge to the Hausdorff dimension of the invariant set at a super-exponential rate. This allows us to estimate the dimension very precisely. We illustrate our approach with examples from dynamical systems and from number theory via Diophantine approximations. This talk is based on joint work with Roger Nussbaum and Amit Priyadarshi, see Trans. Amer. Math. Soc. 364 (2012), 1029-1066.
• November 26, 2013
Yakov Pesin (Penn State University)
Title: Essential Coexistence of Regular and Chaotic Dynamics
Abstract: I will describe an interesting phenomenon of essential coexistence of regular (zero Lyapunov exponents) and chaotic (nonzero Lyapunov exponents) in volume preserving systems. In particular, I will describe an example of a $C^\infty$ volume preserving topologically transitive diffeomorphism of a compact smooth Riemannian manifold which is ergodic (indeed is Bernoulli) on an open and dense subset $U$ of not full measure and has zero Lyapunov exponent on the complement of $U$. I will also discuss a continuous-time version of this example.
These constructions provide examples of "complete" KAM picture in the volume preserving category (in both discrete and continuous-time cases): the system has a Cantor set of positive volume of invariant tori, where it acts as a translation or is simply the identity, which is surrounded by chaotic "sea", where the system is ergodic (indeed is Bernoulli) and has nonzero Lyapunov exponents almost everywhere.
• December 10, 2013 (in Room B3.03) (Joint ETDS and Geometry & Topology seminar)
Francoise Dal'bo (Rennes 1 University)
Title: Growth of nonuniform lattices in pinched negatively curved manifolds
Abstract: Let X be a complete and simply connected Riemannian manifold with sectional curvature bounded between two negative constants. If X admits a uniform lattice, the volume entropy h(X) of X coincides with the critical exponent of the Poincaré series associated to the lattice, and the growth function f(x,R) = Vol(B(x,R)) is asymptotically equivalent to exp(h(X)R). What happens for nonuniform lattices?