PLENÁRIAS
Title: Hedgehogs and pseudo-foliations for periodic point free homeomorphisms in dimension 2
Abstract: The dynamics of circle homeomorphisms without periodic orbits can be rather easily described: such a map has an irrational rotation number, admits a unique minimal set and is a topological extension of the corresponding irrational rotation.
In dimension 2, the torus is the only connected orientable compact manifold admitting periodic point free homeomorphisms, but the dynamical description of these systems is much more complicated. In this talk we shall discuss the rotation theory of these maps paying especially attention to minimal homeomorphisms. We will also introduce two families of topological objects, hedgehogs and pseudo-foliations, that play a fundamental role in these works.
Title: About conformal symplectic billiards.
Abstract: In a joint work with Martin Leguil and Olga Bernardi, we study the dynamics of conformally symplectic convex billiards. In these billiards, the usual elastic reflection law is replaced with a new law where the angle bends towards the normal after each collision. For such billiard dynamics there exists a global attractor; we focus on an invariant subset of this attractor, the so-called Birkhoff attractor, which was first introduced by Birkhoff in a general framework and whose study has been pursued in several works, in particular by Charpentier and Le Calvez. Our goal is to understand how complex the Birkhoff attractor may be for conformally symplectic convex billiards.
Title: On centralizer rigidity for affine maps
Abstract: I will describe the semi-local questions concerning centralisers of perturbations of partially hyperbolic affine maps, and some results we obtained in this direction. This is joint work with A. Wilkinson and D. Xu.
Title: Automorphic Measures and Invariant Distributions for Circle Dynamics
Abstract: This talk is based on joint work with Pablo Guarino and Bruno Nussenzveig. Let f : M\to M be a C^1 diffeomorphism of a smooth compact manifold M. Given s \in \mathbb{R}, an automorphic measure of exponent s for f is a Borel probability measure \nu on M whose pullback under f is equivalent to \nu, with Radon-Nikodym derivative given by the s-power of the Jacobian of f. In the 1980's, R. Douady and J-C. Yoccoz proved that every C^{1+BV} diffeomorphism of the unit circle without periodic points admits a unique automorphic measure of exponent s for each s>0.
In this talk I will show that the same holds for multicritical circle maps, and will also give two applications of this result. The first is to prove that the space of invariant distributions of order 1 of any given multicritical circle map without periodic points is one-dimensional, spanned by the unique invariant measure (this first application is inspired by the work of Avila and Kocsard on invariant distributions for circle diffeomorphisms). The second is an improvement over the Denjoy-Koksma inequality for multicritical circle maps and absolutely continuous observables (this second application is inspired by work of Navas and Triestino).
Title: Lyapunov exponents of linear cocycles.
Abstract: In the context of linear cocycles defined over an invertible map, Bochi proved that cocycles with non-zero exponents do not form an open set in the C^0 topology. On the other hand, Viana and Yang showed examples of cocycles defined over a non-invertible map, where Bochi's conclusion is not true. In this talk we will discuss different results on the continuity of Lyapunov exponents for linear cocycles in the contexts of invertible/non-invertible base.
Title: Shadowing and hyperbolicity for delay equations
Abstract: The shadowing property is known to be strongly connected with several interesting dynamical properties like stability and hyperbolicity. For instance, for finite dimensional linear systems it is known to be equivalent to hyperbolicity, while for infinite dimensional systems this may fail to be true. In the present talk we will show that for a class of delay differential equations, which are examples of infinite dimensional systems, shadowing property is again equivalent to hyperbolicity. This is a joint work with Davor Dragičević and Mihály Pituk.
Title: Physical measures close to the time one map of an Anosov flow.
Abstract: We prove arbitrarily close to the time one map of a C^r Anosov flow, r>1, there exists C^1 open sets of diffeomorphisms with a unique Physical measure. Moreover, this measure has total basin and negative center exponent.
This is a joint work with Sylvain Crovisier.
Title: Renormalization operators and Arnold's tongues
Abstract: Many studies in circle dynamics are devoted to Arnold tongues: level sets of the rotation number in parametric families of circle maps.
E. Risler proved in 1999 that in analytic families of circle diffeomorphisms, Arnold tongues that correspond to Herman rotation numbers are analytic curves. In contrast to this result, Llave and Luque observed in 2011 using numerical investigations that these Arnold tongues are only finitely smooth at critical circle maps.
With M.Yampolsky, we provided explanations of these effects in terms of renormalization operators. I am going to outline main ideas of our proofs.
Title: Absolutely continuous invariant measures for partially hyperbolic endomorphisms satisfying a transversality condition
Abstract: We study physical measures for partially hyperbolic attractors for local difeomorphisms whose central direction is neutral and satisfy a geometric transversality condition between the unstable directions. We prove existence and finiteness of physical measures, we also prove that these measures are absolutely continuous and the union of their basins has total volume in the basin of attraction of the attractor.
Furthermore, we verify that if the attractor has neutral central direction and that the action of the derivative on the unstable direction is close to conformal, then it is possible to perturb the dynamics in a way to satisfy the geometric condition of transversality and, therefore, has finite absolutely continuous physical measures