Dominated splitting, partial hyperbolicity and positive entropy

Eleonora Catsigeras; Xueting  Tian

doi:10.3934/dcds.2016006

Article Contents

2016, Volume 36, Issue 9: 4739-4759. Doi: 10.3934/dcds.2016006

This issue Previous Article Classifying GL$(n,\mathbb{Z})$-orbits of points and rational subspaces Next Article On the microscopic spacetime convexity principle for fully nonlinear parabolic equations II: Spacetime quasiconcave solutions

Dominated splitting, partial hyperbolicity and positive entropy

Eleonora Catsigeras^1, and
Xueting Tian^2,

1.
Instituto de Matemática y Estadística Prof. Rafael Laguardia (IMERL), Facultad de Ingeniería, Universidad de la República
2.
School of Mathematical Sciences, Fudan University, Shanghai 200433

Received: July 2015

Revised: March 2016

Published: May 2017

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Abstract

Let $f:M\rightarrow M$ be a $C^1$ diffeomorphism with a dominated splitting on a compact Riemanian manifold $M$ without boundary. We state and prove several sufficient conditions for the topological entropy of $f$ to be positive. The conditions deal with the dynamical behaviour of the (non-necessarily invariant) Lebesgue measure. In particular, if the Lebesgue measure is $\delta$-recurrent then the entropy of $f$ is positive. We give counterexamples showing that these sufficient conditions are not necessary. Finally, in the case of partially hyperbolic diffeomorphisms, we give a positive lower bound for the entropy relating it with the dimension of the unstable and stable sub-bundles.

Keywords:

Dominated splitting and partial hyperbolicity,
positive topological entropy and positive metric entropy,
volume,
smooth,
SRB,
SRB-like measures.

Mathematics Subject Classification: Primary: 37D30, 37B40, 37D25; Secondary: 37A35.

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