Abstract:
Consider a homeomorphism $f$ defined on a compact metric space $X$ and a continuous map $\varphi\colon X\to\mathbb R$. We provide an abstract criterion, called control at any scale with a long sparse tail for a point $x\in X$ and the map $\varphi$, which guarantees that any weak$^*$ 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 $\varphi$ 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.
Key words and phrases:Birkhoff average, ergodic measure, Lyapunov exponent, nonhyperbolic measure, partial hyperbolicity, transitivity.
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