New criteria for ergodicity and non-uniform hyperbolicity

Abstract

In this work we obtain a new criterion to establish ergodicity and non-uniform hyperbolicity of smooth measures of diffeomorphisms. 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 non-zero Lyapunov exponents in some contexts. In the partial hyperbolicity context, we obtain that stably ergodic diffeomorphisms are C^1-dense among volume preserving partially hyperbolic diffeomorphisms with two-dimensional center bundle. This is motivated by a well known conjecture of C. Pugh and M. Shub.

27 pages, 6 figures

References in corpus (3)

• Creation of blenders in the conservative setting
• On the regularization of conservative maps
• Nonuniform Center Bunching and the Genericity of Ergodicity among $C^1$ Partially Hyperbolic Symplectomorphisms

Cited by in corpus (13)

• Uniqueness of SRB measures for transitive diffeomorphisms on surfaces
• Creation of blenders in the conservative setting
• Geometrical and measure-theoretic structures of maps with mostly expanding center
• Partially hyperbolic dynamics in dimension 3
• On the stable ergodicity of diffeomorphisms with dominated splitting
• Ergodicity and partial hyperbolicity on the 3-torus
• Statistical stability for diffeomorphisms with mostly expanding and mostly contracting centers
• Stable local dynamics: expansion, quasi-conformality and ergodicity
• $C^r-$prevalence of stable ergodicity for a class of partially hyperbolic systems
• Blender-producing mechanisms and a dichotomy for local dynamics for heterodimensional cycles
• Minimality and stable Bernouliness in dimension 3
• Some advances on generic properties of the Oseledets splitting
• Stable weakly shadowable volume-preserving systems are volume-hyperbolic