Dominated Splitting, Partial Hyperbolicity and Positive Entropy

Abstract

Let $f:M\to M$ 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.

24pages

References in corpus (3)

• Pesin Entropy Formula for C1 Diffeomorphisms with Dominated Splitting
• Symbolic extensions for partially hyperbolic diffeomorphisms
• Partial hyperbolicity and specification

Cited by in corpus (1)

• Accessibility and centralizers for partially hyperbolic flows