A criterion for zero averages and full support of ergodic measures

TL;DR

The paper introduces a new criterion ensuring that weak* limits of Birkhoff averages have measures with dense orbits and zero average of a continuous function, with applications to nonhyperbolic ergodic measures.

Contribution

It provides an abstract criterion called control at any scale with a long sparse tail, linking Birkhoff averages to dense orbits and zero averages, with applications to dynamical systems.

Findings

Nonhyperbolic ergodic measures are generic in certain diffeomorphisms.

The criterion guarantees dense orbits with zero Birkhoff average for measures.

Applications include nonhyperbolic homoclinic classes.

Abstract

Consider a homeomorphism $f$ defined on a compact metric space $X$ and a continuous map $\phi\colon X \to \mathbb{R}$. We provide an abstract criterion, called \emph{control at any scale with a long sparse tail} for a point $x\in X$ and the map $\phi$, that guarantees that any weak$\ast$ limit measure $\mu$ of the Birkhoff average of Dirac measures $\frac1n\sum_0^{n-1}\delta(f^i(x))$ is such that $\mu$-almost every point $y$ has a dense orbit in $X$ and the Birkhoff average of $\phi$ along the orbit of $y$ is zero. As an illustration of the strength of this criterion, we prove that the diffeomorphisms with nonhyperbolic ergodic measures form a $C^1$-open and dense subset of the set of robustly transitive partially hyperbolic diffeomorphisms with one dimensional nonhyperbolic central direction. We also obtain applications for nonhyperbolic homoclinic classes.