Abstracts

Minicourse

Davi Obata

"Stable ergodicity"

In this 3 days mini course I will talk about stable ergodicity in different settings. The goals of this mini course will be to give an introduction on this topic, explain some recent results and leave some open problems.

Day 1: On the first day I will give a gentle introduction to stable ergodicity. I will tell some of the history behind it, explain Hopf argument and give the state of the art in the partially hyperbolic setting, especially for the Pugh-Shub's conjectures.

Day 2: The goal of this presentation will be to explain the proof of the stable ergodicity for the Berger-Carrasco example (which I will introduce). This is a very interesting example of a partially hyperbolic system with center dimension 2, it has no domination of the center, and it presents both expansion and contraction along the center.

Day 3: The goal of this presentation will be to explain the proof of a recent work of Gabriel Nuñez, Jana Rodriguez-Hertz and myself about examples of stably ergodic systems in dimension 3. We proved that in the isotopy class of any partially hyperbolic system in dimension 3, there is a non partially hyperbolic system stably ergodic.

Talks

Christian Bonatti

"Anosov flows on 3-manifolds, the surgeries and the foliations: a dynamical game"

What is the effect of Dehn-surgeries along periodic orbits on the stable and unstable foliation of an Anosov flow? With my phd student Ionanis Iakovoglou, we exhibit a very elementary dynamical game describing the effect of the surgeries on the holonomies. I will present the game, then I will try to explain the relation with the surgeries, and finally I will explain how we bypass the resolution of the game for getting directly qualitative information on the foliations.

Stefano Galatolo

"The existence of Noise Induced Order, a computer aided proof"

Dynamical systems perturbed by noise appear naturally as models of physical systems. In several interesting cases the mathematical understanding of these systems can be approached rigorously by computational methods (computer aided proofs). We show the existence of noise induced order in the model of chaotic chemical reactions where it was first discovered numerically by Matsumoto and Tsuda in 1983. We show that in this random dynamical system the increase of noise causes the Lyapunov exponent to decrease from positive to negative, stabilizing the system.

Dominik Kwietniak

"How to measure the distance between orbits and what comes from that?"

I will describe methods for measuring distance between orbits of a dynamical system (X,T). As a result of such a measurement one usually obtains a pseudo-metric on the phase space X. These pseudo-metrics are called dynamically defined pseudo-metrics. The pseudo-metrics of Besicovitch, Weyl, Feldman-Katok, and mean-orbital pseudo-metric are examples of dynamically defined pseudo-metrics. Furthermore, I will explore what happens when a dynamically defined pseudo-metric is continuous with respect to the initial metric on the phase space. This condition usually singles out an interesting class of systems. I will illustrate the theory with examples and applications. The talk will be based on the results obtained in collaboration with several mathematicians (F. Cai, F. García-Ramos, J. Li, M. Łącka, P. Oprocha, H. Pourmand).

Maria José Pacífico

"Beyond geometric Lorenz attractors"

We present a slightly modified version of the well known "geometric Lorenz attractor". It consists in a C1 open set O of vector fields in R3 having an attracting region U containing a unique singular attractor Λ and the maximal invariant set in U contains at most 2 chain recurrence classes, which are Λ and (at most) one hyperbolic horseshoe. The horseshoe and the singular attractor have a collision along the union of 2 co-dimension 1 submanifolds which divide O in 3 regions. By crossing this collision locus, the attractor and the horseshoe may merge in a two-sided Lorenz attractor, or they may exchange their nature: the Lorenz attractor expels the singular point σ and becomes a horseshoe and the horseshoe absorbs σ becoming a Lorenz attractor. This corresponds to a joint work with Diego Barros and Christian Bonatti.

Daniel Smania

"Transfer operators, atomic decomposition and Besov spaces"

We use the method of atomic decomposition to study the action of transfer operators associated to piecewise expanding maps. It turns out that these transfer operators are quasi-compact even when the associated potential, the dynamics and the underlying phase space have very low regularity. In particular it is often possible to obtain exponential decay of correlations, the Central Limit Theorem and almost sure invariance principle for fairly general observables, including unbounded ones. Indeed the class of observables for which we obtain such results often coincides with certain Besov spaces. Joint work with Alexander Arbieto (UFRJ-Brazil)

Marcelo Viana

"Two issues in partially hyperbolic dynamics"

José L. Vieitez

"Lyapunov exponents, expansiveness and entropy for set valued maps"

We give a notion of expansiveness for set valued maps F : X ( X different from that given by Richard Williams in the early seventies of the last century. We prove that the topological entropy of an expansive set valued map defined on a Peano space of positive dimension is greater than zero. Furthermore we define Lyapunov exponents for set valued maps and prove that positiveness of its maximal Lyapunov exponent on a non-trivial continuum implies positive topological entropy. Finally we introduce the definition of (Lyapunov) stable points for set valued maps and prove a dichotomy for the set of stable points of set valued maps defined on Peano spaces: either it is empty or is the whole space.