Topological and ergodic aspects of partially hyperbolic diffeomorphisms and nonhyperbolic step skew products


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Abstract

We review some ergodic and topological aspects of robustly transitive partially hyperbolic diffeomorphisms with one-dimensional center direction. We also discuss step skew-product maps whose fiber maps are defined on the circle which model such dynamics. These dynamics are genuinely nonhyperbolic and exhibit simultaneously ergodic measures with positive, negative, and zero exponents as well as intermingled horseshoes having different types of hyperbolicity. We discuss some recent advances concerning the topology of the space of invariant measures and properties of the spectrum of Lyapunov exponents.

About the authors

L. J. Díaz

Departamento de Matemática PUC-Rio

Author for correspondence.
Email: lodiaz@mat.puc-rio.br
Brazil, Marquês de S˜ao Vicente 225, Gávea, Rio de Janeiro, 22451-900

K. Gelfert

Instituto de Matemática Universidade Federal do Rio de Janeiro

Email: lodiaz@mat.puc-rio.br
Brazil, Av. Athos da Silveira Ramos 149, Cidade Universitária – Ilha do Fund˜ao, Rio de Janeiro, 21945-909

M. Rams

Institute of Mathematics

Email: lodiaz@mat.puc-rio.br
Poland, ul. Śniadeckich 8, Warszawa, 00-656

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