Program

Speakers

• Ali Tahzibi (ICMC)

Title: Disintegration of non-hyperbolic ergodic measures along center foliation - slides

• Aline Melo (UFC)

Title: Continuity of the Lyapunov exponent for Markov cocycles 

Abstract: An important problem in ergodic theory concerns the regularity of the Lyapunov exponents  as a function of the data. In this talk, we establish the joint Hölder continuity of the maximal Lyapunov  exponent as a function of the Markov cocycle and the transition kernel. Our approach provides a more  computable Hölder exponent. This is a joint work with Ao Cai, Marcelo Durães and Silvius Klein. 

• Carlos Vázquez (PUCV)

Title: Minimal invariant foliations for certain classes of partially hyperbolic diffeomorphisms 

Abstract: Consider the class of partially hyperbolic diffeomorphisms defined on $\mathbb{T}^4$ that are isotopic to Anosov, dynamically coherent, with two-dimensional compact central leaves. In this talk, we will discuss the minimality of the strong stable (or unstable) foliation. This is a work in progress in collaboration with Sebastian Perez-Opazo and Radu Saghin (PUCV). 

• David Burguet (Université Sorbonne)

Title: An entropic approach for building SRB measures 

Abstract: There are various ways to build SRB measures for uniformly and non uniformly hyperbolic systems : by pushing the Lebegue measure on an unstable discs (the so called-geometrical method), by coding with the use of Markov partitions, by inducing with Young towers, by taking zero-noise limits,... I will present an entropic variant of the so called geometrical method and explain how it can be applied in different new and old settings (C^r smooth or partially hyperbolic systems). 

• Edhin Castillo (UFMG)

Title: On the uniqueness of the measure of maximal entropy for n-dimensional compact visibility manifolds without conjugate points 

Abstract: The geodesic flow of a compact Riemannian manifold of negative curvature is a classical example of Anosov flow of geometric origin. Many dynamical and ergodic properties are well-known for this class of Riemannian manifolds. In particular, the existence and uniqueness of the measure of maximal entropy was proved by Margulis and Bowen in the 1970s. There was interest in extending this ergodic property to Riemannian metrics more general than negative curvature metrics. In 1998, Knieper extended this property to compact rank-1 manifolds of non-positive curvature using the so-called Patterson Sullivan measure. In 2018, Gelfert and Ruggiero proved the same conclusion for compact higher genus surfaces without focal points using an expansive factor flow of the geodesic flow. We extend Gelfert-Ruggiero's approach to the n-dimensional compact manifolds without conjugate points assuming the so-called gap-entropy and a special global geometry property of the manifold called visibility condition. This is a joint work with Rafael Ruggiero.  - slides

• Federico Rodriguez-Hertz (PSU)

Title: Bessel, Hankel, Lifshitz, Salem and Veech

Abstract: In this talk I plan to give a proof of a theorem by Veech solving the Lifshitz cohomological equation in the case of linear automorphisms of the torus associated with Salem numbers. The method of proof uses Bessel functions and Hankel transform instead of using Chevalley's theorem about profinite group topologies of subsets of algebraic fields.

• François Ledrappier (Notre Dame)

Title: Dimension of limit sets for Anosov representations 

Abstract: We consider the action of discrete finitely generated subgroup of matrices on the space of flags. Under hyperbolicity and  non-degeneracy conditions, we can estimate the dimension of minimal invariant sets. The proofs use properties of random walks on the group, in particular the Lyapunov dimension of the random walk. This is joint work with Pablo Lessa (Montevideo)  - slides

• Graccyela Salcedo (UMK Poland)

Title: Large deviations of Lyapunov exponents for IFSs on the circle. 

Abstract: We consider an Iterated Function System (IFS) of diffeomorphisms on S1. Assuming that there is no an invariant measure by all maps in the IFS, we study the behavior of random orbits relative to the IFS. Under the proximality hypothesis, by using techniques of metric change, we obtain certain limit laws, such as a Central Limit Theorem. We also describe the deviation of finite-time Lyapunov exponents associated with the IFS. This is joint work with Katrin Gelfert. 

• Karina Marin (UFMG)

Title: Continuity of Lyapunov exponents for linear cocycles 

Abstract:  In this lecture we will present results on the continuity of Lyapunov exponents in the Hölder topology for linear cocycles with values in SL(2,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. 

• Lorenzo Diaz  (PUC Rio)

Title: Heterodimensional Cycles of hyperbolic measures: settings, consequences, and robustness (?) 

Abstract: We will introduce and discuss the concept of a heterodimensional cycle between hyperbolic ergodic measures (of different indices). In the setting of partially hyperbolic dynamics with a one-dimensional center, we study the impact of such cycles on the topological properties of the space of invariant measures. We also present some settings for the occurrence of such cycles. This is joint work with Ch. Bonatti and K. Gelfert 

• Luis Pedro Piñeyrua (UFC)

Title: Dynamical coherence in isotopy classes of fibered lifted partially hyperbolic diffeomorphisms 

Abstract: We introduce the notion of fibered lifted partially hyperbolic diffeomorphisms and we prove that any partially hyperbolic diffeomorphism isotopic to a fibered lifted one where the isotopy take  place inside partially hyperbolic systems is dynamically coherent. Moreover we prove some global stability result: every two partially hyperbolic diffeomorphisms in the same connected component of a fibered lifted partially hyperbolic diffeomorphisms, are leaf conjugate. This is a joint work with Martín Sambarino.  - slides

• Martin Andersson (UFF)

Title: Non-uniformly hyperbolic endomorphisms of the torus 

Abstract: I will present a recent result obtained in collaboration with Pablo Carrasco and Radu Saghin about Lyapunov exponents of conservative non-invertible maps of the torus. Consider a non-invertible linear endomorphism of the two torus (e.g. an expanding map). Is it possible to deform it through homotopy to obtain a map which preserves the Haar measure and has a negative Lyapunov exponent almost everywhere? If so, can the Lyapunov exponents be as far away from zero as we want? We give a positive answer to both questions, provided that the topological degree is not too small. - slides

• Martin Leguil (Université  de Picardie Jules Verne)

Title: Birkhoff attractors of dissipative billiards 

Abstract: We consider a particle moving within some convex planar billiard according to a modified reflection law, where collisions become inelastic; more precisely, at each (non-orthogonal) collision with the boundary, the (unoriented) outgoing angle of reflection is strictly smaller than the incoming angle of incidence, both being measured with respect to the normal. The resulting dissipative billiard map has a global attractor. In a joint work with A. Florio and O. Bernardi, we study the topological and dynamical complexity of an invariant subset of this attractor, the so-called Birkhoff attractor, whose study goes back to Birkhoff, Charpentier, and more recently, the work of Le Calvez. We show that for a generic convex billiard, the Birkhoff attractor will be « simple » (typically, a normally contracted manifold) when the dissipation is strong; on the contrary, we show that if the dissipation is mild, then the Birkhoff attractor is topologically « complicated » (an indecomposable continuum) and has rich dynamics (rotation set with non-empty interior, presence of horseshoes…). 

• Nicolas Gourmelon (Université  de Bordeaux)

Title: Diffeomorphisms isotopic to Identity "are" dynamically diffeomorphisms close to Identity 

Abstract: In this talk, I will introduce a notion of total renormalization - an example of which is the Rauzy induction - then I will build open sets of smooth diffeomorphisms that are totally renormalizable. I will explain how, concatenating such diffeormorphisms by surgery, we build a group $P$ of diffeormophisms that are total renormalizations of close to identity maps. Its tangent space at Identity is an (infinite dimensional) Lie algebra whose rigidity properties ultimately imply that $P$ is the group of diffeomorphisms isotopic to identity. This means that the smooth dynamics isotopic to identity are, in substance, the smooth dynamics close to identity. This answers questions by Takens-Ruelle, Turaev, Thouvenot. This is a common work with  Pierre Berger and Mathieu Helfter. 

• Pablo Barrientos (UFF)

Title: Mostly contracting random maps 

Abstract: We are interested in understanding the long-term behavior of the iteration of a random map f given by the compositions $f^n_\omega = f_{\omega_{n-1}} \circ \dots \circ  f_{\omega_0}$, where each $f_{\omega_i}$ is a Lipschitz transformation of compact metric space independently and randomly selected according to the same probability measure. Such a random map is said to be mostly contracting if all the Lyapunov exponents associated with the stationary measures are negative. This requires the introduction of the notion of (maximal) Lyapunov exponent in this general context of Lipschitz transformations on compact metric spaces. We will prove:

(1) This class is open with respect to a coarser topology than the $C^1$-topology.

(2) The global Palis' conjecture holds, i.e., for any mostly contracting random map, there exist finitely many physical measures with disjoint support and whose union of basins covers almost everywhere.

(3) The annealed Koopman operator is quasi-compact, which implies many statistical properties such as the central limit theorems, large deviations, and Hölder continuity (of exponents), etc. Examples of this class of random maps include random products of circle $C^1$ diffeomorphisms, interval $C^1$ diffeomorphisms onto their images, and $C^1$ diffeomorphisms of a Cantor set on a line. All of these are considered under the assumption of no common invariant measure. Moreover, as a consequence of a comprehensive general theory, one can extend (1), (2), and (3) above to the setting of Markovian random maps (i.e., random compositions driven by Markov measures). One of the main tools to prove the above results requires the generalization of Kingman's subadditive ergodic theorem and uniform Kingman's subadditive ergodic theorem for general Markov operators. These two results are of independent interest since they may have many other applications in other contexts. This is a work in collaboration with Dominique Malicet from Université Gustave Eiffel. - slides

• Sebastián Perez (PUCV)

Title: Equilibrium states for a class of skew products 

Abstract: In this talk we discuss some results on the existence and uniqueness of equilibrium states for a certain class of skew products and continuous potentials. These results apply in particular to the classical non-hyperbolic examples of Abraham-Smale and Shub. If there is time, we will also discuss the statistical stability of these states. This is joint work with Maria Carvalho (CMUP, Portugal).  - slides

• Sylvain Crovisier (Université Paris-Saclay)

Title: Non-uniform Hyperbolicity and Strong Positive Recurrence 

Abstract: I will introduce the strong positive recurrence for diffeomorphisms, a strengthened form of non-uniform hyperbolicity which implies quantitative ergodic properties for the measures maximizing the entropy. I will explain how it is related to the continuity of the Lyapunov exponents. This is a joint work with Jérôme Buzzi and Omri Sarig.  - slides

• Yuri Lima (UFC)

Title: Symbolic dynamics for large non-uniformly hyperbolic sets of three dimensional flows 

Abstract: In a joint work with Jérôme Buzzi and Sylvain Crovisier, we construct symbolic dynamics for three dimensional flows with positive speed. More precisely, for each χ > 0, we code a set of full measure for every invariant probability measure which is χ–hyperbolic. These include all ergodic measures with entropy bigger than χ as well as all hyperbolic periodic orbits of saddle-type with Lyapunov exponent outside of [−χ, χ]. This contrasts with previous work of Lima & Sarig which built a coding associated to a given invariant probability measure. As an application, we code χ–hyperbolic measures in homoclinic classes of measures by irreducible countable Markov shifts.