Dominated Splitting, Partial Hyperbolicity and Positive Entropy

TL;DR

This paper establishes conditions under which a $C^1$ diffeomorphism with dominated splitting has positive topological entropy, linking entropy to measure recurrence and sub-bundle dimensions, with implications for partially hyperbolic systems.

Contribution

It provides new sufficient conditions for positive entropy based on Lebesgue measure behavior and offers lower bounds for entropy in partially hyperbolic diffeomorphisms.

Findings

Lebesgue measure $ ext{delta}$-recurrence implies positive entropy.

Counterexamples show conditions are not necessary.

Lower bounds for entropy relate to unstable and stable sub-bundle dimensions.

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.