Poster session

Title: Generalized pseudo-Anosov Maps from Holomorphic Dynamics

Abstract: In this talk, I develop a new connection between the dynamics of quadratic polynomials on the complex plane and the dynamics of homeomorphisms of surfaces. In particular, given a quadratic polynomial, we look at its dynamics on the Hubbard tree, and we discuss whether one can construct an extension of it which is generalized pseudo-Anosov homeomorphism. We present a criterion to determine for which quadratic polynomials such an extension exists. 

Title: Exponential mixing implies Bernoulli.

Abstract: The strongest ergodic property that describes chaoticity is the Bernoulli

property i.e. isomorphic to a Bernoulli shift.

Ergodic ⊂ Weak Mixing ⊂ Mixing ⊂ Kolmogorov ⊂ Bernoulli

The aim of this poster is to comment about the work of Dolgopyat, Kanigowski,

Rodriguez Hertz in which exponential mixing implies Bernoulli for a smooth measure

in the C^{1+α} case.

Exponential mixing means that the rate of convergence of the correlation sequence

C_n(φ, ψ) is exponential, the Kolmogorov property implies mixing but do not give us

information about the rate of convergence.

To obtain Bernoullicity they use the Ornstein-Weiss argument, we will show this

tecnique and as a aplication we will talk about a result of Ponce, Tahzibi, Varao for

Derived of Anosov in which Kolmogorov implies Bernoulli.

Title: Topologically mixing suspension flows over shift spaces

Abstract: Bowen and Plante established a dichotomy for Anosov flows that states that a flow is either topologically mixing or is a constant time suspension flow.  We show that there are roof functions over shift spaces that are not cohomologous to a constant yet produce non-mixing suspension flows.  We also provide necessary and sufficient conditions guaranteeing topological mixing for suspension flows over a certain family of shift spaces.  This is a continuation of joint work with Todd Fisher.

Title: Construction of partially hyperbolic geodesic flows and robustly transitivity

Abstract: Here we address the problem of constructing partially hyperbolic geodesic flows that are not of Anosov type. This problem was firstly approached by Carneiro and Pujals. Their construction is based on deforming the Riemannian metric along a closed geodesic in order to break the hyperbolic behavior. Initially based on their work, we propose a new construction that has several advantages in order to analyze the remaining hyperbolic behavior. With the new construction, we can have finer estimates of the curvature and prove ergodicity in some cases. Besides that, we present a criterion for obtaining robustly transitive geodesic flow inside this class. For the robust transitivity criterion, we present a new notion of SH-saddle property. As a consequence of these results, we can observe new behaviors for the geodesic flow in manifolds with conjugate points.

Title: Smale Spaces and Groups of Intermediate Growth

Abstract: This poster exhibits a connection between Smale spaces and groups of intermediate growth, sketched as follows.

1.  Given a Smale space (Ω,f), we associate with it a contracting self-similar inverse semigroup $H$  (acting on a one-sided Markov shift space).

2. For a contracting self-similar inverse semigroup, its limit solenoid is a Smale orbispace under the shift map, denoted (S,s). Especially, if the inverse semigroup $H$ is obtained from 1., then (S,s) is a Smale space that is topologically conjugate to  (Ω,f).

3. In certain nice cases (e.g. bounded type with finite cycles), we can turn $H$ into a group $G_0$ and fragment $G_0$ to produce a group $G$ of intermediate growth.

Title: Unique measure of maximal entropy for Viana maps

Abstract: Viana maps are a robust class of non-uniformly expanding maps with a critical region. We examine the uniqueness of the measure of maximal entropy for Viana maps. By applying general techniques developed by Climenhaga and Thompson, we demonstrate that a weaker version of expansivity and specification is satisfied. These conditions ensure the existence of a unique measure of maximal entropy.

Title: On the spectrum of Sturmian Hamiltonians of bounded type in a small coupling regime

Abstract: We prove that the Hausdorff dimension of the spectrum of a discrete Schrödinger operator with Sturmian potential of bounded type tends to one as coupling tends to zero. The proof is based on trace map formalism.

Title: Non-stationary random dynamical systems

Abstract: The theory of random dynamical systems is rich and extensive. Most if not all results there were obtained under the assumption that the random transformations on each step are chosen with respect to the same distribution. Which of the results survive if we move to a non-stationary setting? In our talk, we study statistical properties of random dynamical systems in two different settings. The first one is smooth random dynamics on a manifold. In that setting, we establish various regularity results for stationary measures as well as their natural non-stationary analogs. In the second part we study actions of compact metrizable groups on compact metric spaces. Here we prove that trajectories of non-stationary random actions equidistribute as long as the "no deterministic images" condition is satisfied. We also prove ergodic theorem and a large deviation estimate result under the same condition and apply our results to study random walks on compact groups.