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Weak* and entropy approximation of nonhyperbolic measures: a geometrical approach

Part of: Measure-theoretic ergodic theory Dynamical systems with hyperbolic behavior

Published online by Cambridge University Press:  25 July 2019

LORENZO J. DÍAZ ,
KATRIN GELFERT  and
BRUNO SANTIAGO
LORENZO J. DÍAZ
Affiliation:
Departamento de Matemática PUC-Rio, Marquês de São Vicente 225, Gávea, Rio de Janeiro 22451-900, Brazil. e-mail: lodiaz@mat.puc-rio.br
KATRIN GELFERT
Affiliation:
Instituto de Matemática, Universidade Federal do Rio de Janeiro Av. Athos da Silveira Ramos 149, Cidade Universitária - Ilha do Fundão, Rio de Janeiro 21945-909, Brazil. e-mail: gelfert@im.ufrj.br
BRUNO SANTIAGO
Affiliation:
Instituto de Matemática e Estatística, Universidade Federal Fluminense Rua Professor Marcos Waldemar de Freitas Reis, s/n, Bloco H - Campus do Gragoatá São Domingos, Niterói 24210-201, Brazil. e-mail: brunosantiago@id.uff.br
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Abstract

We study C1-robustly transitive and nonhyperbolic diffeomorphisms having a partially hyperbolic splitting with one-dimensional central bundle whose strong un-/stable foliations are both minimal. In dimension 3, an important class of examples of such systems is given by those with a simple closed periodic curve tangent to the central bundle. We prove that there is a C1-open and dense subset of such diffeomorphisms such that every nonhyperbolic ergodic measure (i.e. with zero central exponent) can be approximated in the weak* topology and in entropy by measures supported in basic sets with positive (negative) central Lyapunov exponent. Our method also allows to show how entropy changes across measures with central Lyapunov exponent close to zero. We also prove that any nonhyperbolic ergodic measure is in the intersection of the convex hulls of the measures with positive central exponent and with negative central exponent.

Keywords

Blender-horseshoeentropyergodic measuresLyapunov exponentsminimal foliationspartial hyperbolicitytransitivityweak* topology

MSC classification

Primary: 37D25: Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
Secondary: 37D30: Partially hyperbolic systems and dominated splittings 28D20: Entropy and other invariants 28D99: None of the above, but in this section
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 169 , Issue 3 , November 2020 , pp. 507 - 545
Copyright
© Cambridge Philosophical Society 2019

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