# Moscow Mathematical Journal

Volume 18, Issue 1, January–March 2018 pp. 15–61.

A Criterion for Zero Averages and Full Support of Ergodic Measures

**Authors**:
Christian Bonatti (1), Lorenzo J. Díaz (2), and Jairo Bochi (3)

**Author institution:**(1) Institut de Mathématiques de Bourgogne

(2) Departamento de Matemática, Pontifícia Universidade Católica do Rio de Janeiro

(3) Facultad de Matemáticas, Pontificia Universidad Católica de Chile

**Summary: **

Consider a homeomorphism *f* defined on a compact metric
space *X* and a continuous map φ: *X* → ℝ. We provide an abstract
criterion, called control at any scale with a long sparse tail for a point
*x* ∈ *X* and the map φ, which guarantees that
any weak* limit measure μ of the Birkhoff average of Dirac measures
(1/*n*) ∑_{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.

2010 Math. Subj. Class. 37D25, 37D35, 37D30, 28D99.

**Keywords:**Birkhoff average, ergodic measure, Lyapunov exponent, nonhyperbolic measure, partial hyperbolicity, transitivity.

Contents Full-Text PDF