Hyperbolicity versus non-hyperbolic ergodic measures inside homoclinic classes

Cheng Cheng , Sylvain Crovisier (1) , Shaobo Gan , Xiaodong Wang (2) , Dawei Yang (3)

1 LMO - Laboratoire de Mathématiques d'Orsay
2 CALTECH - California Institute of Technology
3 School of Mathematics

We prove that, for $C^1$-generic diffeomorphisms, if a homoclinic class is not hyperbolic, then there is a non-hyperbolic ergodic measure supported on it. This proves a conjecture by D\'iaz and Gorodetski [28]. We also discuss the conjectured existence of periodic points with different stable dimension in the class.

Domaines: Systèmes dynamiques [math.DS]

Cheng Cheng, Sylvain Crovisier, Shaobo Gan, Xiaodong Wang, Dawei Yang. Hyperbolicity versus non-hyperbolic ergodic measures inside homoclinic classes. Ergodic Theory and Dynamical Systems, 2019, 39, pp.1805-1823. ⟨10.1017/etds.2017.106⟩. ⟨hal-02394717⟩