## 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 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.

Original language | English (US) |
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Pages (from-to) | 15-61 |

Number of pages | 47 |

Journal | Moscow Mathematical Journal |

Volume | 18 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2018 |

## All Science Journal Classification (ASJC) codes

- Mathematics(all)