Critical points for surface diffeomorphisms
Enrique R. Pujals Federico Rodriguez Hertz
Using the definition of dominated splitting, we introduce the notion of critical set for any dissipative surface diffeomorphism as an intrinsically well-defined object. We obtain a series of results related to this concept.
keywords: partially hyperbolic systems smooth mappings and diffeomorphisms homeomorphisms and diffeomorphisms of planes and surfaces dominated splitting. nonuniformly hyperbolic systems
Density of accessibility for partially hyperbolic diffeomorphisms with one-dimensional center
Keith Burns Federico Rodriguez Hertz María Alejandra Rodriguez Hertz Anna Talitskaya Raúl Ures
It is shown that stable accessibility property is $C^r$-dense among partially hyperbolic diffeomorphisms with one-dimensional center bundle, for $r \geq 2$, volume preserving or not. This establishes a conjecture by Pugh and Shub for these systems.
keywords: density of accessibility one dimensional center. partial hyperbolicity
Measure and cocycle rigidity for certain nonuniformly hyperbolic actions of higher-rank abelian groups
Anatole Katok Federico Rodriguez Hertz
We prove absolute continuity of "high-entropy'' hyperbolic invariant measures for smooth actions of higher-rank abelian groups assuming that there are no proportional Lyapunov exponents. For actions on tori and infranilmanifolds the existence of an absolutely continuous invariant measure of this kind is obtained for actions whose elements are homotopic to those of an action by hyperbolic automorphisms with no multiple or proportional Lyapunov exponents. In the latter case a form of rigidity is proved for certain natural classes of cocycles over the action.
keywords: entropy nonuniform hyperbolicity measure rigidity synchronizing time change. Lyapunov metric
Global rigidity of certain Abelian actions by toral automorphisms
Federico Rodriguez Hertz
We prove global rigidity results for some linear abelian actions on tori. The type of actions we deal with includes in particular maximal rank semisimple actions on $\mathbb T^N$.
keywords: smooth conjugacy. rigidity Anosov actions
Uniqueness of large invariant measures for $\mathbb{Z}^k$ actions with Cartan homotopy data
Anatole Katok Federico Rodriguez Hertz
Every $C^2$ action $\a$ of $\mathbb{Z}k$, $k\ge 2$, on the $(k+1)$-dimensional torus whose elements are homotopic to the corresponding elements of an action $\ao$ by hyperbolic linear maps has exactly one invariant measure that projects to Lebesgue measure under the semiconjugacy between $\a$ and $\a_0$. This measure is absolutely continuous and the semiconjugacy provides a measure-theoretic isomorphism. The semiconjugacy has certain monotonicity properties and preimages of all points are connected. There are many periodic points for $\a$ for which the eigenvalues for $\a$ and $\a_0$ coincide. We describe some nontrivial examples of actions of this kind.
keywords: $\mathbb{Z}^k$ actions. nonuniform hyperbolicity measure rigidity
Rigidity of real-analytic actions of $SL(n,\Z)$ on $\T^n$: A case of realization of Zimmer program
Anatole Katok Federico Rodriguez Hertz
We prove that any real-analytic action of $SL(n,\Z),n\ge 3$ with standard homotopy data that preserves an ergodic measure $\mu$ whose support is not contained in a ball, is analytically conjugate on an open invariant set to the standard linear action on the complement to a finite union of periodic orbits.
keywords: Global rigidity Zimmer program. real analytic
Cohomology free systems and the first Betti number
Federico Rodriguez Hertz Jana Rodriguez Hertz
We prove that a cohomology free flow on a manifold $M$ fibers over a diophantine translation on $\T^{\beta_1}$ where $\beta_1$ is the first Betti number of $M$.
keywords: Cohomolgy equation tori. diophantine vector field
Federico Rodriguez Hertz
This special issue presents some of the lecture notes of the courses held in the 2008 and 2011 Summer Institutes at the Mathematics Research and Conference Center of Polish Academy of Sciences at Będlewo, Poland. The school was structured as daily courses with a double lecture each, in two parts of 45-50 minutes with a break in between.

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Partial hyperbolicity and ergodicity in dimension three
Federico Rodriguez Hertz María Alejandra Rodriguez Hertz Raúl Ures
In [15] the authors proved the Pugh–Shub conjecture for partially hyperbolic diffeomorphisms with 1-dimensional center, i.e., stably ergodic diffeomorphisms are dense among the partially hyperbolic ones. In this work we address the issue of giving a more accurate description of this abundance of ergodicity. In particular, we give the first examples of manifolds in which all conservative partially hyperbolic diffeomorphisms are ergodic.
keywords: partial hyperbolicity laminations. accessibility property ergodicity
Invariant distributions for homogeneous flows and affine transformations
Livio Flaminio Giovanni Forni Federico Rodriguez Hertz
We prove that every homogeneous flow on a finite-volume homogeneous manifold has countably many independent invariant distributions unless it is conjugate to a linear flow on a torus. We also prove that the same conclusion holds for every affine transformation of a homogenous space which is not conjugate to a toral translation. As a part of the proof, we have that any smooth partially hyperbolic flow on any compact manifold has countably many distinct minimal sets, hence countably many distinct ergodic probability measures. As a consequence, the Katok and Greenfield-Wallach conjectures hold in all of the above cases.
keywords: Cohomological equations homogeneous flows.

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