$C^1$ density of stable ergodicity

Abstract

We prove a $C^1$ version of a conjecture by Pugh and Shub: among partially hyperbolic volume-preserving $C^r$ diffeomorphisms, $r>1$, the stably ergodic ones are $C^1$-dense. To establish these results, we develop new perturbation tools for the $C^1$ topology: linearization of horseshoes while preserving entropy, and creation of "superblenders" from hyperbolic sets with large entropy.

arXiv comment58 pages, 11 figures. The long version of "Diffeomorphisms with positive metric entropy" arXiv:1408.4252v3 has been split in two, and this paper corresponds to the second part. The first part is available as the current version of arXiv:1408.4252

Cited by in corpus (7)

• Historic behavior in non-hyperbolic homoclinic classes
• Topological entropy and Hausdorff dimension of irregular sets for non-hyperbolic dynamical systems
• Stable local dynamics: expansion, quasi-conformality and ergodicity
• The approximation of Lyapunov exponents by horseshoes for $C^1$-diffeomorphisms with dominated splitting
• Dimension Estimates for Non-conformal Repellers and Continuity of Sub-additive Topological Pressure
• Horseshoes and Lyapunov exponents for Banach cocycles over nonuniformly hyperbolic systems
• A criterion for transitivity of area preserving partially hyperbolic endomorphism on torus