We are planning a number of talks on Wednesday 12th and Thursday 13th April 2017.
University of Exeter, Harrison Building, Harrison 203.
Wednesday 12th April 2017
14.00-15.00 Alexander Adam (UPMC)
Title: Resonances for Anosov diffeomorphisms
Abstract: Deterministic chaotic behavior of invertible maps T is appropriately described by the existence of expanding and contracting directions of the differential of T. A special class of such maps are Anosov diffeomorphisms. Every hyperbolic matrix M with integer entries induces such a diffeomorphism on the 2-torus. For all pairs of real-analytic functions on the 2-torus, one defines a correlation function for T which captures the asymptotic independence of such a pair under the evolution T^n as $n\to\infty$.
What is the rate of convergence of the correlation as $n\to\infty$, e.g. what is its decay rate? The resonances for T are the poles of the Z-transform of the mereomorphic continued correlation function. The decay rate is well-understood if T=M. There are no non-trivial resonances of M. In this talk I consider small real-analytic perturbations T of M where at least one non-trivial resonance of T appears. This affects the decay rate of the correlation.
15.05-16.05 Thomas Jordan (Bristol)
Title: Measures of maximum dimension for self-affine sets
Abstract: (Joint work with Jonathan Fraser and Natalia Jurga) Self-affine sets are sets which made up of affine copies of themselves. The Hausdorff dimension of such sets has been a long standing research topic. For self-similar and self-conformal systems a standard approach has been to use the thermodynamic formalism to find a Gibbs measure on the associated shift space which projects to a measure which will have the same dimension as the set. By using the subadditive thermodynamic formalism the same approach can be used for self-affine sets but with much greater difficulties. It is known due to Kaenmaki that a large class of self-affine sets will have such a measure of maximal dimension (such measures are often called Kaenmaki measures). In this talk we’ll give this background before looking at the properties this measure will have in particular cases (diagonal, positive, irreducible). The new work covered will be what measure we would expect to be the measure of maximal dimension in a situation where the iterated function system is made up of diagonal and anti-diagonal matrices and how the structure of such systems can be used to show it is indeed often a measure of maximal dimension.
16.30-17.30 Ian Melbourne (Warwick)
Title: Singular hyperbolic flows
Abstract: The classical Lorenz attractor is an important example of a singular hyperbolic attractor and its statistical properties are very well understood. However, many of these results rest heavily on the fact that certain stable foliations are smooth.
In this talk, we discuss the general situation where the stable foliation need not be smooth. In addition to clarifying existing results on existence of spectral decompositions and SRB measures, we extend many of the statistical properties for the classical Lorenz attractor to general three-dimensional singular hyperbolic flows. Our results hold also in higher dimensions (for codimension two singular hyperbolic flows). This is joint work with Vitor Araujo.
Thursday 13th April 2017
11.30-12.30 Tomas Persson (Lund)
Title: Shrinking targets in parametrised families
Abstract: I will talk about a joint work with Magnus Aspenberg. We consider a parametrised family of piecewise expanding interval maps $T_a$ and a point $x(a)$, and study the following shrinking target problem: For which parameters $a$ is $T_a^n (x(a))$ inside a shrinking neighbourhood of a point $y$ for infinitely many $n$? We give upper and lower bounds for the set of such parameters. Our results are generalisations of several previous results for specific families. The proofs rely on techniques originating from Benedicks and Carleson, and in particular on a result by Schnellmann on typical points.
14.00-15.00 Viviane Baladi (CNRS, IMJ-PRG, UPMC)
Title: Linear response for discontinuous observables
Abstract: Linear response formulas describe how the physical measure of a dynamical system reacts to perturbations of the dynamics. For hyperbolic dynamics, linear response is usually stated for differentiable observables only. Discontinuous observables involving thresholds (Heaviside functions) appear naturally in extreme value theory. We present our recent results with Kuna and Lucarini giving sufficient conditions, on observables allowing thresholds, ensuring linear response. Our proof uses the fine properties of anisotropic Banach spaces. This will also be an opportunity to give a survey talk on anisotropic spaces suitable for transfer operators of hyperbolic dynamical systems.
15.10-16.10 Damien Thomine (Paris Sud)
Title: Hitting probabilities, potential kernel and ergodic theory
Abstract: Given a recurrent random walk, there is a simple relationship between the probability that an excursion from the origin hits a given site, and the (symmetrized) potential kernel of the random walk. The classical proof draws from harmonic analysis. We give here a new proof of this relationship, which uses tools from ergodic theory rather than from harmonic analysis. As a consequence, we are able to generalize these results to contexts for which no simple harmonic objects are available. This is joint work with Françoise Pène (Université de Brest).