Nonhyperbolicity of invariant measures on maximal attractor

TL;DR

This paper demonstrates that on high-dimensional manifolds, certain attractors can have invariant measures with zero Lyapunov exponents, highlighting nonhyperbolic behavior in dynamical systems.

Contribution

It establishes the existence of nonhyperbolic invariant measures on maximal attractors for a broad class of smooth diffeomorphisms on manifolds of dimension four or higher.

Findings

Existence of nonhyperbolic invariant measures on attractors

Construction of examples via skew products over horseshoe

Transfer of results to smooth diffeomorphisms

Abstract

The article states that for every compact manifold M of dimension 4 or higher there is an area U in a set of smooth diffeomorphisms over M such that every map f from U has local maximal partially hyperbolic attractor and nonatomic ergodic invariant measure on it where one of Lyapunov exponents vanish. The result is first proved for skew products over horseshoe and then transferred onto smooth diffeomorphisms.