Typical path components in tent map inverse limits

TL;DR

This paper investigates the structure of path components in the inverse limit space of a tent map, revealing measure-theoretic and topological dichotomies based on the critical orbit's density.

Contribution

It characterizes the measure and topological properties of path components in tent map inverse limits, highlighting a dichotomy depending on the critical orbit's density.

Findings

Set of points with bi-infinite, bi-dense path components has full measure.

Topological dichotomy: dense G_delta set in one case, its complement dense G_delta in the other.

Behavior depends on whether the critical orbit of the tent map is dense or not.

Abstract

In the inverse limit ${\hat{I}}_s$ of a tent map $f_s$ restricted to its core, the set $\mathcal{GR}$ of points whose path components are bi-infinite and bi-dense has full measure with respect to the measure induced on $\hat{I}_s$ by the unique absolutely continuous invariant measure of $f_s$. With respect to topology, there is a dichotomy. When the parameter $s$ is such that the critical orbit of $f_s$ is not dense, $\mathcal{GR}$ contains a dense $G_\delta$ set. In contrast, when the critical orbit of $f_s$ is dense, the complement of $\mathcal{GR}$ contains a dense $G_\delta$ set.