Non-hyperbolic ergodic measures for non-hyperbolic homoclinic classes

TL;DR

This paper demonstrates that for generic $C^1$-diffeomorphisms, the presence of a homoclinic class with saddles of different indices guarantees an invariant ergodic measure with at least one zero Lyapunov exponent, indicating non-hyperbolic behavior.

Contribution

It establishes a link between heterogeneity in saddle indices within homoclinic classes and the existence of non-hyperbolic ergodic measures for generic $C^1$-diffeomorphisms.

Findings

Existence of non-hyperbolic ergodic measures in certain homoclinic classes.

Generic $C^1$-diffeomorphisms with heterogenous saddle indices have non-hyperbolic measures.

Zero Lyapunov exponent confirmed in these measures.

Abstract

We prove that for a generic $C^1$-diffeomorphism existence of a homoclinic class with periodic saddles of different indices (dimension of the unstable bundle) implies existence an invariant ergodic non-hyperbolic (one of the Lyapunov exponents is equal to zero) measure of $f$.