Program

Schedule:

Talks:

• Aline Melo (UFC-UFRJ)

Title: Analiticity of Lyapunov exponent of i.i.d. multiplicative process

Abstract: In this talk, we extend the results and methods of Y. Peres from a finite to an infinite (but compact) space of symbols. In other words, we establish the analiticity of the maximal Lyapunov exponent for i.i.d. random product of matrices as a function of the transition probabilities. Our approach combines the spectral properties of the associated Markov operator with the theory of holomorphic functions in Banach spaces. This is a joint work with Artur Amorim and Marcelo Durães. 

• Andrés Navas (USACH)

Title: Connecting commuting 1-D diffeomorphisms

Abstract: We will examine some of the main ideas involved in the proof of the path-connectedness of the space of Z^d actions on 1-D manifolds by C^1+ac diffeomorphisms. We will particularly concentrate on the notion of asymptotic distortion, which is a kind of Lyapunov exponent at the level of second derivatives and plays a key role in the proof. This is joint work with Hélène Eynard-Bontemps (Inst. Fourier, Grenoble). 

• Anibal Velozo (PUC Santiago)

Title:Pressure at infinity and multifractal analysis on countable Markov shifts 

Abstract: The pressure at infinity quantifies the free energy outside compact regions of phase space in a dynamical system endowed with a potential function, generalizing the concept of entropy at infinity. In this talk, I will discuss the pressure at infinity for potentials in countable Markov shifts, its relationship with the upper semi-continuity of the pressure map, and its implications for the existence of equilibrium states for potentials with low regularity. Furthermore, there is a close relationship between entropy at infinity and the Hausdorff dimension of points that are recurrent and diverge on average. In joint work with Godofredo Iommi (PUC) and Thomas Jordan (Bristol), we study multifractal analysis for suspension flows over countable Markov shifts, focusing on orbits that are recurrent and diverge on average. It turns out that the relevant rate function is analogous to the usual rate function when replacing the pressure with the pressure at infinity.

• Cristina Lizana (UFBA)

Title: Intrinsic ergodicity for a certain class of Derived from Anosov 

Abstract: We will talk briefly about some classic examples of Derived from Anosov (DA), that is, homotopic maps to an Anosov diffeomorphism, whose dynamics are partially hyperbolic. We will address some known results related to entropy invariance and the existence (and uniqueness) of measures of maximal entropy  for this class of diffeomorphisms. Finally, we will present recent results in collaboration with L. Parra (PUCV) and C. Vásquez (PUCV) for a certain class of DA generated after a Hopf bifurcation, previously introduced by [M. Carvalho'93].

• Felipe Riquelme (PUCV)

Title: Dimension of geodesics that diverge on average 

Abstract: A classical theorem by C. J. Bishop and P. W. Jones (1997) states that the Hausdorff dimension of the radial limit set of a Kleinian group coincides with its critical exponent. From the perspective of the dynamics of the geodesic flow on the quotient manifold, and thanks to a theorem by J. P. Otal and M. Peigné (2004), this theorem tells us that the dimension of recurrent directions in compact sets is the topological entropy of the flow. In this talk, we will present an analogous result in terms of orbits that escape to infinity on average. An orbit is said to be divergent on average if the average visit time on any compact set asymptotically decreases to 0. We will show that the dimension of geodesics that diverge on average is the entropy at infinity of the flow. As a consequence, we will find an upper bound for the measure theoretical entropy of infinite ergodic measures. This is a joint work with Aníbal Velozo (PUC Chile).

• Javier Correa (UFMG)

Title: Dispersion of orbits with sub-exponential rates

Abstract: In a joint work with Enrique Pujals we developed the notion of generalized entropy that allow us to quantify the dispersion of orbits of a system in the space of orders of growth. This object, allow us to classify families of dynamical systems with classical 0 entropy. The idea of this talk is to give an introduction, discuss results obtained so far (with Enrique and Hellen de Paula) and also new projects I have in mind.

• Ioannis Iakovoglou (Sorbonne Université)

Title: On the 3-manifolds supporting transitive Anosov flows

Abstract: Despite the progress that has been made in the understanding of Anosov flows in dimension 3, the question of which 3-manifolds support Anosov flows remains surprisingly still open. During this talk, I will explain how one can recursively construct the fundamental groups of all 3-manifolds supporting transitive Anosov flows. This result relies on some previous work on the classification of Anosov flows by geometric types and a work in common with François Béguin.

• Karina Marin (UFMG)

Title: Lyapunov spectrum of volume-preserving partially hyperbolic maps

Abstract: The Lyapunov spectrum of a map is said to be simple if every Lyapunov exponent has multiplicity one. That is, if the Oseledets decomposition is given by one dimensional subspaces. This problem has been extensively studied in the context of linear cocycles. In this talk, we discuss the simplicity of the Lyapunov spectrum for partially hyperbolic volume-preserving diffeomorphisms with two dimensional center bundle. This is a joint work with D. Obata and M. Poletti. 

• Katrin Gelfert (UFRJ)

Title: Full flexibility of entropies among ergodic measures

Abstract: For partially hyperbolic diffeomorphisms with minimal strong foliations and unstable/stable blender-horseshoes, we establish restricted variational principles for entropy, fixing a specified center Lyapunov exponent and varying the metric entropies among ergodic measures. We prove that for each exponent value in the interior of the spectrum (including value 0), every possible entropy value can be achieved by some ergodic measure. This is joint work with LD Díaz, M Rams, and J Zhang.

• Lorenzo Díaz (PUC Río de Janeiro)

Title: To be or not to be (hyperbolic): A conservative dilemma

Abstract: In dimension three, we present three conservative partially hyperbolic settings (derived from Anosov systems, time one-maps of geodesic flows on negatively curved surfaces, and skew-product with circle fibers) where the following dichotomy holds: the diffeomorphism is either Anosov or it supports and (ergodic) nonhyperbolic measure. Join work with J, Yang (UFF, Niterói, Brazil) J. Zhang (Beihaan, Beijing, China).

• Maria Isabel Cortez (PUC Santiago)

Title: Symbolic dynamics, invariant measures and residually finite groups

Abstract: In 1969, Jacobs and Keane introduced Toeplitz subshifts in the context of Z-actions. Since then, different works have shown that there are dynamical systems within this family of subshifts with varied behavior. For example, Downarowicz has shown in 1991 that any possible set of invariant probability measures is realizable by some Toeplitz subshift in {0,1}^Z.  The flexibility of Toeplitz subshifts motivated further generalizations beyond Z-actions, providing examples of group actions on the Cantor set with exciting properties. Another consequence of the generalization of the Toeplitz subshift concept is the characterization of infinite residually finite groups as those admitting actions corresponding to non-periodic Toeplitz subshifts. In this talk, we gather some results about Toeplitz subshifts and their relationship with minimal equicontinuous systems. For example, applying results concerning Furstenberg-Weiss type almost 1-1 extensions, we will show that Toeplitz subshifts are a test family for the amenability of residually finite groups.

• Martin Leguil (Ecole polytechnique & Université de Picardie Jules Verne)

Title:  Rigidity of transitive Anosov flows in dimension 3

Abstract: In a joint work with Andrey Gogolev and Federico Rodriguez Hertz, we study when two transitive Anosov flows X,Y in dimension 3 which are topologically conjugated are actually smoothly conjugated. By the work of de la Llave, Marco, Moriyón and Pollicott from the late 80s, a necessary and sufficient condition for that to happen is that stable and unstable eigenvalues at corresponding periodic points match. In our work we show that for generic transitive Anosov flows X,Y in dimension 3, the latter condition is already implied by the existence of a topological conjugacy; in particular, the conjugacy is smooth, unless the conjugacy swaps positive and negative SRB measures of the two flows. This complements a recent work of Gogolev and Rodriguez Hertz in the volume preserving case. I will try to explain how this rigidity problem is connected with other notions, in particular, the so-called templates introduced by Tsujii and Zhang to study the regularity of stable and unstable distributions, and a positive proportion Livschits Theorem recently shown by Dilsavor and Marshall Reber.

• Mauricio Poletti (UFC)

Title: Measures of maximal entropy for non-uniformly hyperbolic maps

Abstract: For $C^(l+)$ maps, possibly non-invertible and with singularities, we prove that each homoclinic class of an adapted hyperbolic measure carries at most one adapted hyperbolic measure of maximal entropy. In this talk I will give two the applications: Uniqueness of MME for finite horizon dispersing billiards and the robustly non-uniformly hyperbolic volume-preserving endomorphisms introduced by Andersson-Carrasco-Saghin. This is a joint work with Y. Lima and D. Obata.

• Nelda Jaque Tamblay (U de Chile)

Title: Examples of Semiflows with Expansive Behavior in the Inverse Limit

Abstract: In this talk, we will present examples of semiflows that exhibit expansive behavior in the inverse limit. Expansiveness has been extensively studied in the context of continuous flows, and recently, definitions have been proposed for semiflows on compact spaces. However, these definitions typically focus on the semiflow itself, often leading to systems characterized as finite unions of periodic orbits. 

In contrast, this talk introduces a novel perspective by examining expansiveness in the inverse limit. Specifically, we will demonstrate that suspensions of positively expansive endomorphisms give rise to expansive inverse limits.

This is joint work in progress with Juan Peña and Wellington Cordeiro.

• Nicolas Gourmelon (Université Bordeaux)

TBA

• Pierre Berger (Sorbonne Université)

Title: Analytic pseudo-rotations on spheres, disks and annuli

Abstract: We introduce a way to perform the approximation by conjugacy method of Anosov-Katok among surface analytic symplectomorphisms. This produces transitive analytic symplectomorphisms of the sphere, the disk and the cylinder with a finite number of periodic points. This disproves a conjecture of Birkhoff (1941), and solves problems of Birkhoff (1927),  Herman (1998), Fayad-Katok (2004) and Fayad-Krikorian (2018).

• Sylvain Crovisier (CNRS & Université Paris-Saclay)

Title: Robust transitivity in dimension 3

Abstract: I will consider partially hyperbolic diffeomorphisms in dimension 3 and prove a dichotomy: up to a C1-perturbation, the dynamics either exhibits a non-trivial attractor, or is robustly transitive. Along the way, I will discuss the SH property of strong unstable foliations, which ensures some minimal expansion along the center direction and show that this property is frequent.

This is a joint work with R. Potrie. 

• Yuki Yayama (U de Bío-Bío)

Title: Some sequences of continuous functions associated with relative pressure functions

Abstract: Given an almost additive sequence of continuous functions with bounded variation $\F=\{\log f_n\}_{n=1}^{\infty}$ on a subshift $X$ over finitely many symbols, we study properties of a function $f$ on $X$ such that $\lim_{n\to\infty}\frac{1}{n}\int \log f_n d\mu=\int f d\mu$ for every invariant measure $\mu$ on $X$. Under some conditions we construct a function $f$ on $X$ and study a relation between the property of $\F$ and some particular types of $f$.  We also study the case when $\F$ is weakly almost additive. As applications we consider sequences of continuous functions associated with relative pressure functions and study the images of Gibbs measures for continuous functions under one-block factor maps. 

• Yuri Lima (UFC)

Title: Limit Theorems for geodesic flows in nonpositive curvature 

Abstract: While geodesic flows in negative curvature exhibit exponential decay of correlations for the Liouville measure and satisfy classical limit theorems, the study of similar properties for geodesic flows in nonpositive curvature is far from a global understanding. In this talk, I will discuss results and an ongoing project regarding these topics for certain classes of geodesic flows in surfaces of nonpositive curvature. Joint work with Carlos Matheus and Ian Melbourne.

Short talks:

• Edhin Castillo (UFMG)

Title: Equilibrium measures of certain potentials for geodesic flows

Abstract: The thermodynamic formalism was introduced into Dynamical Systems by Sinai, Bowen and Ruelle. An important problem is to study the equlibriums states for certain family of potential (tipically Holder continuous) on a particular dynamical system. That is, the existence, finiteness or uniqueness of equlibrium measures as well as their ergodic properties. In this talk, for the geodesic flow of a compact manifold with continuous Green bundles and a hyperbolic closed geodesic and for a certain family of potentials, we show the uniquenes of their equilibrium measures and some ergodic properties of them.

• Graccyela Salcedo (Ecole Polytechnique)

Title: Maximal exponent for IFS of circle homeomorphisms

Abstract: We study iterated systems of homeomorphism functions of the circle. We assume proximality and the absence of finite orbits. We explore the maximal exponent to conclude interesting properties such as: existence and uniqueness of stable and unstable directions, and the dimensional accuracy of the stationary measure. This is a joint work with Jamerson Bezerra. 

• Juan Carlos Mongez (UFC)

Title: Measure of Maximal Entropy for Certain Partially Hyperbolic Diffeomorphisms

Abstract: We investigate the number of measures of maximal entropy for certain partially hyperbolic diffeomorphisms. Our main result demonstrates that these maps have a finite number of ergodic measures of maximal entropy when the central Lyapunov exponents of all ergodic measures with large entropy are uniformly bounded away from zero. Additionally, we explore the relationship between the number of maximal entropy measures for these diffeomorphisms within an open set. This is a joint work with Maria Jose Pacifico and Mauricio Poletti. 

• Leonardo Dinamarca (PUC Santiago)

Title: Quadratic growth for the derivatives of iterates of interval diffeomorphisms

Abstract: 20 years ago, in the seminal work of Polterovich and Sodin shows a (short but) surprising result: if the iterates of derivatives of a diffeomorphisms of the interval of class C^2 growth are subexponential, then it grows at most quadratic. In this talk we will discuss a stronger result that asserts that the lim_n max Df^n / n^2  exists. We briefly discuss the modern tools who allow us to obtain the results. 

• Odylo Costa (Sorbonne Université) 

Title: A new example of a flow with high emergence

Abstract: In the 1970s, three distinct counterexamples to the so-called Periodic Orbit Conjecture were presented. Sullivan, Thurston, and Epstein-Vogt each constructed examples of flows on compact manifolds where every orbit is closed, yet the period can be arbitrarily large. In this short talk, we present a modification of one of these constructions to obtain a system with high emergence.

• Raquel Salgado (UFMG)

Title: Example of discontinuity for the Lyapunov exponents of SL(2,R)-valued cocycles

Abstract: In 2010, Bocker and Viana presented an example of a discontinuity point for the Lyapunov exponents as a function of the cocycle relative to the α-Holder topology. The linear cocycle taking values in SL(2, R) is defined over a Bernoulli shift and the perturbations are constructed under some conditions related to α and the norm of the cocycle. Butler, in 2018, improved the hypothesis of Bocker-Viana but he restricted the probability measure considered. He constructs locally constant cocycles arbitrarily close to the original one with small Lyapunov exponents. In this talk, with a stronger condition than Butler but without restricting the measure, we show that the original cocycle can be approximated by locally constant cocycles with zero Lyapunov exponents. In particular, this extends Bocker-Viana result. This is a joint work with Edhin Mamani.

• Yann Delaporte (Sorbonne Université)

Title: Analytic dynamics displaying ergodic behaviour or emergence on the sphere, cylinder and disk

Abstract: Pierre Berger recently introduced a principle, based on the Approximation by Conjugacy (AbC) method, allowing to build analytic dynamics displaying specific properties on the sphere, the disk and the cylinder. In this talk, I will use it to obtain analytic symplectomorphisms which are ergodic or display a maximum order of local emergence on the above surfaces. To that extend, in the spirit of the AbC method, I build specific symplectomorphisms to reach these properties. Then, since it seems that the original principle does not encompass minimal ergodicity, I will generalize this principle so that it does on these surfaces. As a result, I obtain analytic symplectomorphisms with exactly 3 ergodic measures on the sphere, the disk and the cylinder.

Poster:

• Ana Cristina Barreto Sabino de Araújo (IMPA)

Title: Hölder Continuity of Lyapunov Exponents in Random Products of Invertible Matrices

Abstract: Consider a probability measure $\nu$ with compact support in $GL(d)$, the linear space of invertible $d\times d$ matrices, and a linear cocycle  $\mathcal{F}:GL^\mathbb{N}(d)\times \mathbb{R}^d\rightarrow GL^\mathbb{N} (d)\times \mathbb{R}^d$ defined as $\mathcal{F}((g_j)_j,v)=((g_{j+1})_j,g_0 v)$. The cocycle associated with a random product of $2\times 2$ invertible matrices (\textit{i.e.}, with $d=2$) under the probability distribution $\nu$ has two (possibly equal) Lyapunov exponents $\lambda_1(\nu)\ge\lambda_2(\nu)$. When $\lambda_1>\lambda_2$ we can prove that those two exponents are pointwise Hölder continuous with respect to the probability measure $\nu$. A natural question arises:  does this result generalize to higher dimensions? In this poster, we'll explore the concept of random $GL(d)$-cocycle, define their Lyapunov exponents and investigate the above-mentioned generalization. This is a work in progress with Adriana Sánchez, El Hadji Yaya Tall and Marcelo Viana. 

• Artur Assis Amorim (IMPA) 

Title: Analicity of Lyapunov Exponents on the case of Random Matrix Products and its consequences

Abstract: We aim to present a generalization of a theorem of Yuval Peres, regarding the analicity of the Lyapunov Exponents with respect to the transitional probabilities when equipped with the total variation norm. We then use this result and a generalization of the Implicit Function Theorem to investigate the zeroes of the Lyapunov Exponents on the space of complex probabilities. 

• Filiphe Veiga (UFMG) 

Title: GIKN construction of non-hyperbolic measures

Abstract: In this poster, we study the GIKN construction, where it is proven that the space of skew products over a Bernoulli shift with a circle as the fiber contains a nonempty open set of maps, each admitting an invariant ergodic measure with a zero Lyapunov exponent. 

• Jõao Henrique Lírio da Silva (UFRJ) 

Title: An approach to rigorously calculating diffusion coefficients for expanding maps

Abstract: The study of the statistical behavior of a dynamic system is important in various areas of research. We use Ulam's method to provide rigorous approximation diffusion coefficients for uniformly expanding maps.  And thus obtain Central limite theorem for some examples.

• Miguel Pineda (UFRJ)

Title: Horseshoe and ASH systems

Abstract: A classical result in dynamical systems, due to Katok, establishes that for any $C^{1+\alpha}$ diffeomorphism on a surface with positive topological entropy, the entropy can be approximated by a horseshoe. In this poster, we extend this result to asymptotically sectional-hyperbolic (ASH) $C^1$ vector fields on Riemannian manifolds. ASH is a weak form of hyperbolicity, where the tangent bundle admits a dominated splitting such that one subbundle is contracting, while the other asymptotically expands area for points outside the stable manifold of the singularities. 

• Nestor Jara (U de Chile)

Title: Spectrum invariance dilemma for nonuniformly kinematically similar systems

Abstract: It is known that if two systems are kinematically similar, then they have the same exponential dichotomy spectrum. This is not the case when we consider nonuniformly kinematically similar systems, even if we consider the nonuniform dichotomy spectrum, unlike what was previously thought in literature. In order to explore the theoretical foundations of this lack of invariance, we discern the pivotal influence of the parameters involved in the definition of nonuniform dichotomy. Although spectra invariance might not be achieved, we present a weaker notion which can be verified and depends on nontrivial neighborhoods of the nonuniform dichotomy spectrum. 

• Sebastián Ramírez (PUC Santiago) 

Title: Stable Ergodicity for Maps on $\mathbb{T}^2$

Abstract: We prove that the homotopy class of non-hyperbolic elements (with either 1 or -1 as an eigenvalue) on the 2-torus, provided its degree is sufficiently large, contains non-uniformly hyperbolic endomorphisms that are also \( C^2 \) stably ergodic. These results provide partial answers to certain questions posed by M. Andersson, P. Carrasco, and R. Saghin in their work on non-uniformly hyperbolic endomorphisms. 

• Vinícius Pinheiro (UFMG) 

Title: Removing Zero Lyapunov Exponents

Abstract: I present part of a technique, presented by Baraviera and Bonatti, to perturb an C1-stably ergodic partially hyperbolic diffeomorphism and remove a null Lyapunov exponent. The hypothesis of stably ergodicity arises because the analysis of Lyapunov exponents of perturbations becomes simpler when maintaining ergodicity. The technique is based on transferring some expansion to the central directions. 

• Ygor Arthur Cesar de Jesus (Universidade Estadual de Campinas) 

Title: Construction of Partially Hyperbolic Geodesic Flows

Abstract: In this work we present techniques to produce examples of partially hyperbolic geodesic flows. Some of the produced examples are rank 1 and the set of vectors with rank bigger than one has measure zero, therefore they are Bernoulli flows. In particular we produce examples of ergodic and non-Anosov geodesic flows. 

• Yingjian Liu (IMPA)