A criterion for zero averages and full support of ergodic measures

Consider a homeomorphism $f$ defined on a compact metric space $X$ and a continuous map $ϕ\colon X \to \mathbb{R}$. 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 $ϕ$, that guarantees that any weak$\ast$ limit measure $μ$ of the Birkhoff average of Dirac measures $\frac1n\sum_0^{n-1}δ(f^i(x))$ is such that $μ$-almost every point $y$ has a dense orbit in $X$ and the Birkhoff average of $ϕ$ 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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