Non-hyperbolic ergodic measures with large support

TL;DR

This paper demonstrates that in a generic set of differentiable dynamical systems, certain complex invariant sets support ergodic measures with zero Lyapunov exponents, expanding understanding of non-hyperbolic dynamics.

Contribution

It establishes the existence of non-hyperbolic ergodic measures with large support on homoclinic classes in a residual subset of $ ext{Diff}^1(M)$, revealing new generic properties.

Findings

Residual subset of $ ext{Diff}^1(M)$ with specified properties

Homoclinic classes contain non-hyperbolic ergodic measures

Supports of these measures coincide with certain homoclinic classes

Abstract

We prove that there is a residual subset $\mathcal{S}$ in $\text{Diff}^1(M)$ such that, for every $f\in \mathcal{S}$, any homoclinic class of $f$ with invariant one dimensional central bundle containing saddles of different indices (i.e. with different dimensions of the stable invariant manifold) coincides with the support of some invariant ergodic non-hyperbolic (one of the Lyapunov exponents is equal to zero) measure of $f$.