JMD
Critical points for surface diffeomorphisms
Enrique R. Pujals Federico Rodriguez Hertz
Journal of Modern Dynamics 2007, 1(4): 615-648 doi: 10.3934/jmd.2007.1.615
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
DCDS
Density of accessibility for partially hyperbolic diffeomorphisms with one-dimensional center
Keith Burns Federico Rodriguez Hertz María Alejandra Rodriguez Hertz Anna Talitskaya Raúl Ures
Discrete & Continuous Dynamical Systems - A 2008, 22(1&2): 75-88 doi: 10.3934/dcds.2008.22.75
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
JMD
Measure and cocycle rigidity for certain nonuniformly hyperbolic actions of higher-rank abelian groups
Anatole Katok Federico Rodriguez Hertz
Journal of Modern Dynamics 2010, 4(3): 487-515 doi: 10.3934/jmd.2010.4.487
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
JMD
Global rigidity of certain Abelian actions by toral automorphisms
Federico Rodriguez Hertz
Journal of Modern Dynamics 2007, 1(3): 425-442 doi: 10.3934/jmd.2007.1.425
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
JMD
Uniqueness of large invariant measures for $\mathbb{Z}^k$ actions with Cartan homotopy data
Anatole Katok Federico Rodriguez Hertz
Journal of Modern Dynamics 2007, 1(2): 287-300 doi: 10.3934/jmd.2007.1.287
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
DCDS
Rigidity of real-analytic actions of $SL(n,\Z)$ on $\T^n$: A case of realization of Zimmer program
Anatole Katok Federico Rodriguez Hertz
Discrete & Continuous Dynamical Systems - A 2010, 27(2): 609-615 doi: 10.3934/dcds.2010.27.609
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
DCDS
Cohomology free systems and the first Betti number
Federico Rodriguez Hertz Jana Rodriguez Hertz
Discrete & Continuous Dynamical Systems - A 2006, 15(1): 193-196 doi: 10.3934/dcds.2006.15.193
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
JMD
Preface
Federico Rodriguez Hertz
Journal of Modern Dynamics 2014, 8(3&4): i-i doi: 10.3934/jmd.2014.8.3i
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.

For more information please click the “Full Text” above.
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JMD
Errata to "Measure rigidity beyond uniform hyperbolicity: Invariant measures for Cartan actions on tori" and "Uniqueness of large invariant measures for $\Zk$ actions with Cartan homotopy data"
Boris Kalinin Anatole Katok Federico Rodriguez Hertz
Journal of Modern Dynamics 2010, 4(1): 207-209 doi: 10.3934/jmd.2010.4.207
N/A.
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JMD
Partial hyperbolicity and ergodicity in dimension three
Federico Rodriguez Hertz María Alejandra Rodriguez Hertz Raúl Ures
Journal of Modern Dynamics 2008, 2(2): 187-208 doi: 10.3934/jmd.2008.2.187
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

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