Robust existence of nonhyperbolic ergodic measures with positive entropy and full support

TL;DR

This paper demonstrates that on certain manifolds, robustly transitive partially hyperbolic diffeomorphisms can have a dense set of ergodic, fully supported measures with positive entropy, constructed via an extended control technique.

Contribution

It introduces abstract conditions for constructing ergodic measures with positive entropy and full support in the setting of partially hyperbolic diffeomorphisms.

Findings

Existence of nonhyperbolic ergodic measures with positive entropy and full support.

Dense subset of such measures in the space of diffeomorphisms.

Extension of control at any scale with a long and sparse tail technique.

Abstract

We prove that for some manifolds $M$ the set of robustly transitive partially hyperbolic diffeomorphisms of $M$ with one-dimensional nonhyperbolic centre direction contains a $C^1$-open and dense subset of diffeomorphisms with nonhyperbolic measures which are ergodic, fully supported and have positive entropy. To do so, we formulate abstract conditions sufficient for the construction of an ergodic, fully supported measure $\mu$ which has positive entropy and is such that for a continuous function $\phi\colon X\to\mathbb{R}$ the integral $\int\phi\,d\mu$ vanishes. The criterion is an extended version of the control at any scale with a long and sparse tail technique coming from the previous works.