Duke Mathematical Journal

New criteria for ergodicity and nonuniform hyperbolicity

F. Rodriguez Hertz, M. A. Rodriguez Hertz, A. Tahzibi, and R. Ures

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In this work we obtain a new criterion to establish ergodicity and nonuniform hyperbolicity of smooth measures of diffeomorphisms of closed connected Riemannian manifolds. This method allows us to give a more accurate description of certain ergodic components. The use of this criterion in combination with topological devices such as blenders lets us obtain global ergodicity and abundance of nonzero Lyapunov exponents in some contexts.

In the partial hyperbolicity context, we obtain that stably ergodic diffeomorphisms are C1-dense among volume-preserving partially hyperbolic diffeomorphisms with 2-dimensional center bundle. This is motivated by a well-known conjecture of Pugh and Shub.

Article information

Duke Math. J., Volume 160, Number 3 (2011), 599-629.

First available in Project Euclid: 7 November 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37D25: Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
Secondary: 37D30: Partially hyperbolic systems and dominated splittings 37D35: Thermodynamic formalism, variational principles, equilibrium states


Rodriguez Hertz, F.; Rodriguez Hertz, M. A.; Tahzibi, A.; Ures, R. New criteria for ergodicity and nonuniform hyperbolicity. Duke Math. J. 160 (2011), no. 3, 599--629. doi:10.1215/00127094-1444314. https://projecteuclid.org/euclid.dmj/1320674938

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