Fundamental domain of invariant sets and applications

TL;DR

This paper constructs a fundamental domain for invariant sets with finite peaks in dynamical systems and explores their properties under partially hyperbolic diffeomorphisms, revealing volume-related dichotomies.

Contribution

It introduces a method to construct fundamental domains for finite-peak sets and applies this to partially hyperbolic diffeomorphisms to establish volume dichotomies.

Findings

Finite-peak sets can be covered by a fundamental domain.

In accessible partially hyperbolic systems, finite-peak sets are either full volume or transitive points have positive volume.

Abstract

Let $X$ be a compact metric space and $f:X\to X$ a homeomorphism on $X$. We construct a fundamental domain for the set with finite peaks for each cocycle induced by $\phi\in C(X,R)$. In particular we prove that if a partially hyperbolic diffeomorphism is accessible, then either the set with finite peaks for the Jacobian cocycle is of full volume, or the set of transitive points is of positive volume.