Title: Decay of mass near the equator for stationary measures
Abstract.
For products of random invertible matrices in arbitrary dimension, when a one-dimensional equator is present, we study the behavior of stationary measures on the projective space. By introducing a suitable Margulis function and using a recoupling technique, we obtain quantitative estimates on how much mass stationary measures assign to neighborhoods of the equator.
Título: Dinâmica e rigidez de métricas Finsler
Resumo:
Nesta palestra, faremos uma breve introdução às métricas de Finsler em variedades fechadas. Este tipo de métrica, que generaliza a noção clássica de métrica riemanniana, surge naturalmente em diversos contextos da geometria, análise e sistemas dinâmicos. Discutiremos as propriedades geométricas e dinâmicas que distinguem as métricas de Finsler das riemannianas, com ênfase no seu fluxo geodésico.
O foco principal será sobre problemas de rigidez. Mostraremos como certas condições geométricas (como propriedades na curvatura) ou dinâmicas (impostas ao fluxo geodésico) forçam uma métrica de Finsler a ser, na verdade, riemanniana. Apresentaremos resultados clássicos e recentes que ilustram esse fenômeno de rigidez.
Title: Measure of Maximal Entropy for a class of C1 endomorphisms with critical points
Abstract.
In the presentation, we discuss the existence and uniqueness of measures of maximal entropy for a certain class of endomorphisms introduced by Lizana-Ranter, which are homotopic to a linear volume-expanding endomorphism. This class of endomorphisms on T^2 are C^1-robustly transitive partially hyperbolic with persistence of critical points. This is an ongoing project joint with Y. Lima, C. Lizana.
Title: Measures of Maximal entropy for skew products over Anosov flows
Abstract.
In this work we prove the uniqueness of measure of maximal entropy for transitive skew products over Anosov flows that have a neutral center. As an application we prove the uniqueness of measure of maximal entropy for ergodic frame flows.
Title: On the Continuity of Fiber Lyapunov Exponents in Skew Product Systems
Abstract.
We study the relationship between fiber entropy and fiber Lyapunov exponents in skew product systems with two-dimensional fibers. Extending the work of Buzzi-Crovisier-Sarig on surface diffeomorphisms, we show that for smooth skew products, continuity of fiber entropy implies continuity of the fiber Lyapunov exponents. This is joint work with Marin and Poletti.
Title: On the Finiteness of Measures of Maximal Entropy for Certain Partially Hyperbolic Diffeomorphisms
Abstract.
In the study of dynamical systems, invariant measures that maximize entropy, called measures of maximal entropy (MMEs), capture the full “complexity” of the dynamics. A natural question is whether such measures exist, how many there are, and under which conditions finiteness can be guaranteed. In this talk, I will present recent results on the finiteness of MMEs for certain partially hyperbolic diffeomorphisms.
Title: A Sufficient Condition For Robustly transitive Diffeomorphisms
Abstract.
We give sufficient conditions under which partially hyperbolic skew product maps that admit a more general version of the notion Some Hyperbolicity, so called SH saddle property introduced by Piñeyrúa, are robustly topologically transitive. Furthermore, we present an example of a partially hyperbolic skew product maps in T^n ×S_2, where S_2 is the bi-torus, and one of the fiber is the diffeomorphism topologically conjugate to the Thurston’s Pseudo-Anosov map obtained by Gerber-Katok. We show that this map satisfies the hypotheses of the main result. This is an ongoing project joint with C. Lizana.
Title: Ornstein Isomorphism Theorem for n-to-1 Extended Bernoulli Transformations
Abstract.
Ornstein's theorem establishes metric entropy as a complete invariant for Bernoulli shifts. In this talk, we present how Ornstein’s ideas can be adapted to classify the locally invertible n-to-1 LM-Bernoulli transformations. We also establish a relation between folding and metric entropies for these dynamics.