Artur Avila   (IMPA, Brazil and University of Zurich, Switzerland)

“TBA”

Lucas Backes   (Federal University of Rio Grande do Sul, Brazil)

“A variational principle for the metric mean dimension of level sets”

In this talk we will explore the notion of metric mean dimension introduced by E. Lindenstrauss and B. Weiss. This is a well suited quantity to study systems with infinite entropy. We will present a result relating the upper metric mean dimension of level sets $$ \left\{x\in X: \lim_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1}\varphi(f^{j}(x))=\alpha\right\} $$ associated to continuous potentials $\varphi:X\to \mathbb R$ and continuous dynamics $f:X\to X$ defined on compact metric spaces and exhibiting the specification property with the growth rates of measure-theoretic entropy of partitions decreasing in diameter associated to some special measures. Moreover, we will present several examples to which our result may be applied to. This is based on a joint work with F. Rodrigues.

Alexandre Baraviera   (Federal University of Rio Grande do Sul, Brazil)

“Stationary measures for infinite graphs”

The purpose of this talk is to describe some recent work, joint with Pedro Duarte and Joana Torres, where the technique of isospectral reduction, introduced by L. Bunimovich and B. Webb, is applied in the context of certain tridiagonal operators related to stochastic processes in order to show the existence of stationary measures and also get some approximation of the measure by means of the iteration of a dynamical procedure.

Pierre Berger   (Institut Mathématique de Jussieu, France)

“Analytic pseudo-rotations”

We construct analytic symplectomorphisms of the cylinder or the sphere with zero or exactly two periodic points and which are not conjugated to a rotation. In the case of the cylinder, we show that these symplectomorphisms can be chosen ergodic or to the opposite with local emergence of maximal order. In particular, this disproves a conjecture of Birkhoff (1941) and solves a problem of Herman (1998).

Leonid Bunimovich   (Georgia Tech, USA)

“Wild Rose, Narcissus and other Elliptic flowers Billiards”

I will talk about a new class of billiards with three invariant sets in a phase space. Two sets (called tracks) consist of orbits which move clockwise or counter-clockwise around the third, called a core. All four possibilities of coexistence of chaotic or non-chaotic dynamics in tracks and in the core are realized. Moreover, these billiards demonstrate various bifurcations which create singularities of the wave fronts as well as new wave trains.

Pablo Carrasco   (Federal University of Minas Gerais, Brazil)

“A robust 𝒞¹ class of endomorphisms: questions and problems”

A central result is smooth ergodic theory is that for diffeomorphisms of compact surfaces, either they are Anosov or they can be approximated by maps having zero exponents. This is consequence of the Bochi-Mañé’s theorem from around 2000, and is a cornerstone of several developments. But what about endomorphisms? Does something similar hold for say, non-singular endomorphisms of surfaces?

Together with M. Andersson and R. Saghin we found out that the answer is negative, and in fact we established the existence of a large 𝒞¹ open class 𝒰 of area preserving endomorphisms of the 2-torus (the only orientable surface that admits endomorphisms without critical points) satisfying

- every f ∈ 𝒰 is non-uniformly hyperbolic: one positive and one negative Lyapunov exponent a.e. - essentially every homotopy class intersects 𝒰 (in particular, there are expanding maps that can be deformed in order to "flip" one exponent) - the (integrated) exponents vary continuously in 𝒰 - if the linear part of f ∈ 𝒰 us transitive and f is 𝒞², then f is Bernoulli.

In view of the above, 𝒰 behaves much like the uniformly hyperbolic class for diffeomorphisms, and I’d like to propose that this is a natural analogue for endomorphisms of that class. I’m going to present a quick description and then focus and what remains to be done.

João Lopes Dias   (University of Lisbon, Portugal)

“Polygonal billiards with contracting reflection laws”

Billiards on polygons are known to be conservative and non-chaotic. By considering a reflection law that contracts the reflected angles towards the normal, a hyperbolic attractor emerges. We show the existence of finitely many ergodic SRB measures whose basins cover a full Lebesgue measure set in phase space. Joint work with P. Duarte, G. Del Magno, J. P. Gaivão and D. Pinheiro.

Xiao-Chuan Liu   (Federal University of Alagoas, Brazil)

“Livsic type theorems”

We discuss some aspects in Livsic type theorems and emphasize the relationship of this theory with the recent developments of the theory of linear cocycles. We also try to discuss possible further generalizations or limitations in more general settings. Joint work with Artur Avila and Alejandro Kocsard.

Karina Marin   (Federal University of Minas Gerais, Brazil)

“Continuity of Lyapunov exponents for cocycles with values in \({\it SL}(2,\mathbb{R})\)”

In this lecture we will present results on the continuity of Lyapunov exponents in the Hölder topology for cocycles with values in \({\rm SL}(2,\mathbb{R})\) defined over a Bernoulli shift. In particular, we will prove that the Bocker-Viana discontinuity example is not typical among cocycles with sufficiently small upper Lyapunov exponents. This is a joint work with Catalina Freijo.

Vilton Pinheiro   (Federal University of Bahia, Brazil)

“On the thermodynamical formalism for expanding measures”

Let $f:M\to M$ be a non-flat $C^{1+\alpha}$ map defined on a Riemannian manifold $M$. A $f$-invariant probability $\mu$ is called expanding if all its Lyapunov exponents are positive, i.e., $\lim_{n\to+\infty}\frac{1}{n}\log|Df^n(x)\vec{v}|>0$ for $\mu$-almost every $x\in M$ and every $\vec{v}\in T_xM\setminus\{0\}$. Let $\mathcal{E}(f)$ be the set of all $f$-invariant expanding probability. A $f$-invariant probability $\mu$ is called an expanding equilibrium state for $\varphi$ if $\mu\in\mathcal{E}(f)$ and $$h_{\mu}(f)+\int\varphi d\mu=sup\bigg\{h_{\nu}(f)+\int\varphi d\nu\,;\,\nu\in\mathcal{E}(f)\bigg\}.$$ The map $f$ is called strongly transitive if $\bigcup_{n\ge0}f^n(A)=M$ for every open set $A\subset M$.

Theorem. Suppose that $f$ is strongly transitive.

1. $f$ has at most one expanding equilibrium state for a given Hölder potential $\varphi$.

2. If $f$ has an expanding equilibrium state for $\varphi\equiv 0$ then $f$ has one and only one expanding equilibrium state $\mu$ for any given Hölder potential $\psi$ with small variation.

Corollary. A Viana map has one and only one equilibrium state for every Hölder potential with small variation.

Mauricio Poletti   (Federal University of Ceará, Brazil)

“Hölder continuity of the Lyapunov exponents for cocycles over hyperbolic maps”

Given a hyperbolic homeomorphism on a compact metric space, consider the space of linear cocycles over this base dynamics which are Holder continuous and whose projective actions are partially hyperbolic dynamical systems. We prove that locally near any typical cocycle, the Lyapunov exponents are Holder continuous functions relative to the uniform topology. This is a joint work with P. Duarte and S. Klein.

Longmei Shu   (Dartmouth College, USA)

“Isospectral Reductions and the Stationary Measure of Stochastic Matrices”

Isospectral reductions are defined in two ways, from a graph perspective and a matrix perspective. I will introduce isospectral reductions first and then show some of its properties, the preservation of eigenvalues and eigenvectors. At last I will talk about some partial results from discussions with Pedro Duarte, Joana Torres and Alexandre Baraviera. We have seen isospectral reductions decrease the computation time for the stationary measure for some stochastic matrices and don’t have the full explanation yet.

Sergey Tikhomirov   (Saint-Petersburg State University, Russia and IMPA, Brazil)

“Shadowing and gamblers ruin problem”

It is well-known that shadowing holds in a neighborhood of a hyperbolic set. It is known that shadowing can hold for non hyperbolic systems, but due to results of Sakao, Abdenur, Diaz, Pilyugin, Tikhomirov shadowing is "almost" equivalent to structural stability. At the same time numerical experiments by Hammel-Grebogi-Yorke for logistics and Henon maps shows that shadowing holds for relatively long pseudotrajectories. It poses a question which type of shadowing holds for systems, which are not necessarily hyperbolic. We consider probabilistic approach for the shadowing. We study the shadowing property for non uniformly hyperbolic cocycles and shows that shadowing is closely related to gamblers ruin problem. The approach allows to show that polynomials long pseudotractories can be good shadowed. We also study generalisation to the case of pseudotrajectories with nonuniform jumps. The talk is based on joint works with G. Monakov, V. Priezhev, P. Priezhev.

Marcelo Viana   (IMPA, Brazil)

“TBA”

Caroline Wormell   (Sorbonne University, France)

“The chaotic hypothesis and linear response”

Rigorously, not much is known about complex chaotic systems that lack strong geometrical conditions such as uniform hyperbolicity. Instead, they are typically hypothesised to behave like Axiom A, i.e. uniformly hyperbolic, systems, at least at large scales: the so-called chaotic hypothesis of Gallavotti and Cohen. In this talk I will discuss a physically relevant class of examples which can nevertheless at large scales approximate arbitrary dynamics; on the other hand, I will present mechanisms by which large enough systems can nevertheless recover the nice statistical properties of uniformly hyperbolic systems, specifically in the realm of response theory.