Ergodic measures with multi-zero Lyapunov exponents inside homoclinic classes

TL;DR

This paper proves the existence of non-hyperbolic ergodic measures with zero Lyapunov exponents inside homoclinic classes of generic diffeomorphisms, revealing complex dynamical behavior.

Contribution

It establishes new conditions under which non-hyperbolic ergodic measures with zero Lyapunov exponents exist within homoclinic classes, expanding understanding of dynamical systems.

Findings

Existence of ergodic measures with zero Lyapunov exponents in certain homoclinic classes.

Construction of measures supported on entire homoclinic classes.

Identification of conditions involving dominated splittings and Jacobians for measure existence.

Abstract

We prove that for $C^1$ generic diffeomorphisms, if a homoclinic class $H(P)$ contains two hyperbolic periodic orbits of indices $i$ and $i+k$ respectively and $H(P)$ has no domination of index $j$ for any $j\in\{i+1,\cdots,i+k-1\}$, then there exists a non-hyperbolic ergodic measure whose $(i+l)^{th}$ Lyapunov exponent vanishes for any $l\in\{1,\cdots, k\}$, and whose support is the whole homoclinic class. We also prove that for $C^1$ generic diffeomorphisms, if a homoclinic class $H(P)$ has a dominated splitting of the form $E\oplus F\oplus G$, such that the center bundle $F$ has no finer dominated splitting, and $H(p)$ contains a hyperbolic periodic orbit $Q_1$ of index $\dim(E)$ and a hyperbolic periodic orbit $Q_2$ whose absolute Jacobian along the bundle $F$ is strictly less than $1$, then there exists a non-hyperbolic ergodic measure whose Lyapunov exponents along the center bundle $F$ all vanish and whose support is the whole homoclinic class.