Pablo D. Carrasco

Federal University of Minas Gerais | UFMG · Departamento de Matemática

We develop a general geometric method to establish the existence of positive Lyapunov exponents for a class of skew products. The technique is applied to show non-uniform hyperbolicity of some conservative partially hyperbolic diffeomorphisms having as center dynamics coupled products of standard maps, notably for skew-products whose fiber dynamics...

Bochi-Katok-Rodriguez Hertz proposed recently a program on the flexibility of Lyapunov exponents for conservative Anosov diffeomorphisms, and obtained partial results in this direction. For conservative Anosov diffeomorphisms with strong hyperbolic properties we establish extended flexibility results for their Lyapunov exponents. We give examples o...

We develop a geometric method to establish existence and uniqueness of equilibrium states associated to some Hölder potentials for center isometries (as are regular elements of Anosov actions), in particular the entropy maximizing measure and the SRB measure. It is also given a characterization of equilibrium states in terms of their disintegration...

Consider a three-dimensional partially hyperbolic diffeomorphism. It is proved that under some rigid hypothesis on the tangent bundle dynamics, the map is (modulo finite covers and iterates) an Anosov diffeomorphism, a (generalized) skew-product or the time-one map of an Anosov flow, thus recovering a well-known classification conjecture of the sec...

We consider weakly expansive holonomy pseudogroup foliations of compact manifolds. Our main results show the number of compact leaves is generally countable, and at most finite for codimension-one cases. We show examples of such foliations, demonstrating the results are sharp.

We consider the horocyclic flow corresponding to a (topologically mixing) Anosov flow or diffeomorphism, and establish the uniqueness of transverse quasi-invariant measures with Hölder Jacobians. In the same setting, we give a precise characterization of the equilibrium states of the hyperbolic system, showing that existence of a family of Radon me...

In this work we consider foliations of compact manifolds whose holonomy pseudo-group is expansive, and analyze their number of compact leaves. Our main result is that in the codimension-one case this number is at most finite, and we give examples of such foliations having one compact leaf.

We develop a geometric method to establish the existence and uniqueness of equilibrium states associated to some Hölder potentials for center isometries (as are regular elements of Anosov actions), in particular, the entropy maximizing measure and the SRB measure. A characterization of equilibrium states in terms of their disintegrations along stab...

We show that in nearly every homotopy class of any non-invertible endomorphism of the two-torus there exists a $\mathcal C^1$ open set of non-uniformly hyperbolic area preserving maps (one positive and one negative exponent at Lebesgue almost every point), without dominated splitting. Moreover, these maps are continuity points of the (averaged) Lya...

Bochi-Katok-Rodriguez Hertz proposed in [BKH21] a program on the flexibility of Lyapunov exponents for conservative Anosov diffeomorphisms, and obtained partial results in this direction. For conservative Anosov diffeomorphisms with strong hyperbolic properties we establish extended flexibility results for their Lyapunov exponents. We give examples...

In this note we report some advances in the study of ther- modynamic formalism for a class of partially hyperbolic systems—cen- ter isometries—that includes regular elements in Anosov actions. The techniques are of geometric flavor (in particular, not relying on sym- bolic dynamics) and even provide new information in the classical case. For such s...

I prepared these notes for lecturing Ergodic Theory (PhD level) in the Federal University of Minas Gerais (UFMG), during 2020. I haven't revised them completely so they most likely contain mistakes, but still they may have some useful parts: in any case, proceed with caution. The latest version can be found in my webpage.

In this note we report some advances in the study of thermodynamic formalism for a class of partially hyperbolic system -- center isometries, that includes regular elements in Anosov actions. The techniques are of geometric flavor (in particular, not relying in symbolic dynamics) and even provide new information in the classical case. For such syst...

We consider the horocyclic flow corresponding to a (topologically mixing) Anosov flow or diffeomorphism, and establish the uniqueness of transverse quasi-invariant measures with H\"older Jacobians. In the same setting, we give a precise characterization of the equilibrium states of the hyperbolic system, showing that existence of a family of Radon...

We prove the topological entropy remains constant inside the class of partially hyperbolic diffeomorphisms of Td with simple central bundle (that is, when it decomposes into one dimensional sub-bundles with controlled geometry) and such that their induced action on H1(Td) is hyperbolic. In absence of the simplicity condition we construct a robustly...

In this note we consider a symmetric random walk defined by a $(f,f^{-1})$ Kalikow type system, where $f$ is the time-one map of the geodesic flow corresponding to an hyperbolic manifold. We provide necessary and sufficient conditions for the existence of an stationary measure for the walk that is equivalent to the volume in the corresponding unit...

We prove the topological entropy remains constant inside the class of partially hyperbolic diffeomorphisms of $\mathbb{T}^d$ with simple central bundle (that is, when it decomposes into one dimensional sub-bundles with controlled geometry) and such that their induced action on $H_1(\mathbb{T}^d)$ is hyperbolic. In absence of the simplicity conditio...

We present an example of a $\mathcal{C}^1$-robustly transitive skew-product with non-trivial, non-hyperbolic action on homology. The example is conservative, ergodic, non-uniformly hyperbolic and its fiber directions cannot be decomposed into two dominated expanded/contracted bundles.

It is proved a classification of three dimensional partially hyperbolic diffeomorphisms assuming some rigid hypotheses on the tangent bundle dynamics.

Partial hyperbolicity appeared in the 1960s as a natural generalization of hyperbolicity. In the last 20 years, there has been great activity in this area. Here we survey the state of the art in some related topics, focusing especially on partial hyperbolicity in dimension three. The reason for this is not only that it is the smallest dimension in...

We develop a general geometric method to establish the existence of positive Lyapunov exponents for a class of skew products. The technique is applied to show non-uniform hyperbolicity of some conservative partially hyperbolic diffeomorphisms having as center dynamics coupled products of standard maps, notably for skew-products whose fiber dynamics...

We introduce some tools of symbolic dynamics to study the hyperbolic directions of partially hyperbolic diffeomorphisms, emulating the well known methods available for uniformly hyperbolic systems.

We prove that the system resulting of coupling the standard map with a fast hyperbolic system is robustly non-uniformly hyperbolic.

According to the work of Dennis Sullivan, there exists a smooth flow on the 5-sphere all of whose orbits are periodic although there is no uniform bound on their periods. The question addressed in this article is whether these type of examples can occur in the partially hyperbolic context. That is, if does there exist a partially hyperbolic diffeom...