C¹ density of stable ergodicity

2017/09/14 by A. Avila, Sylvain Crovisier, Avila, A. +3 · 5 citations
Mathematics · Physics and Astronomy · #Dynamical Systems (math.DS) #FOS: Mathematics #Geometry and complex manifolds #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.1709.04983

openalex publication_date 2017/09/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove a C¹ version of a conjecture by Pugh and Shub: among partially hyperbolic volume-preserving C^r diffeomorphisms, r>1, the stably ergodic ones are C¹-dense. To establish these results, we develop new perturbation tools for the C¹ topology: linearization of horseshoes while preserving entropy, and creation of "superblenders" from hyperbolic sets with large entropy.

Cited by

• Simple Lyapunov spectrum of partially hyperbolic diffeomorphisms
• On Kolmogorov-typical properties of symplectic dynamics
• Dimension Estimates for Non-conformal Repellers and Continuity of Sub-additive Topological Pressure
• Partially hyperbolic diffeomorphisms homotopic to the identity in dimension three
• The approximation of Lyapunov exponents by horseshoes for C¹-diffeomorphisms with dominated splitting

Related

• C^r-prevalence of stable ergodicity for a class of partially hyperbolic systems
• The local C¹-density of stable ergodicity
• Partial hyperbolicity and ergodicity in dimension three
• Finite Measures of Maximal Entropy for an Open Set of Partially Hyperbolic Diffeomorphisms
• Entropy of C¹ diffeomorphisms without a dominated splitting