A criterion for zero averages and full support of ergodic measures

Christian Bonatti, Lorenzo J. Díaz, Jairo Bochi

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Abstract

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 (Formula Presented) 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 C1-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.

Original languageEnglish (US)
Pages (from-to)15-61
Number of pages47
JournalMoscow Mathematical Journal
Volume18
Issue number1
DOIs
StatePublished - Jan 1 2018

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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