A criterion for zero averages and full support of ergodic measures

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 control at any scale with a long sparse tail for a point $x\in X$ and the map $\phi$, which guarantees that any weak* limit measure $\mu$ of the Birkhoff average of Dirac measures $\frac1n\sum_0^{n-1}\delta(f^i(x))$ s 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.
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https://hal.archives-ouvertes.fr/hal-01758598
Contributor : Imb - Université de Bourgogne <>
Submitted on : Wednesday, April 4, 2018 - 3:55:59 PM
Last modification on : Friday, June 8, 2018 - 2:50:07 PM

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  • HAL Id : hal-01758598, version 1
  • ARXIV : 1609.07764

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Christian Bonatti, Lorenzo J. Díaz, Jairo Bochi. A criterion for zero averages and full support of ergodic measures. Moscow mathematical journal, 2018, 18 (1), pp.15-61. ⟨http://www.mathjournals.org/mmj/2018-018-001/2018-018-001-002.html⟩. ⟨hal-01758598⟩

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