Path connectedness and entropy density of the space of ergodic hyperbolic measures

TL;DR

This paper proves that the space of hyperbolic ergodic measures within an isolated homoclinic class is path connected and entropy dense under certain conditions, enhancing understanding of measure structure in dynamical systems.

Contribution

It establishes the path connectedness and entropy density of hyperbolic ergodic measures in homoclinic classes, a novel result in dynamical systems theory.

Findings

The space of hyperbolic ergodic measures is path connected.

This space is entropy dense within the set of measures.

The closure of this space remains path connected.

Abstract

We show that the space of hyperbolic ergodic measures of a given index supported on an isolated homoclinic class is path connected and entropy dense provided that any two hyperbolic periodic points in this class are homoclinically related. As a corollary we obtain that the closure of this space is also path connected.