Dominated Splitting, Partial Hyperbolicity and Positive Entropy

Eleonora Catsigeras, Xueting Tian

Published 2014-09-22Version 1

Let $f:M\rightarrow M$ be a $C^1$ diffeomorphism on a compact Riemanian manifold $M$ with a dominated splitting. We state and prove several sufficient conditions for the topological entropy of $f$ be positive. In particular, if $f$ preserves a smooth measure, or if $f$ does not preserve a smooth measure but the Lebesgue measure is still recurrent, then the entropy of $f$ is positive. By means of an example, we show that two conditions together are not necessary to have positive entropy. For such kind of examples we state and prove other sufficient conditions to have positive entropy. Finally, in the case of partial hyperbolicity we give a positive lower bound for the entropy, according to the maximum dimension of the unstable and stable sub-bundles.