Typical path components in tent map inverse limits
Volume 250 / 2020
Fundamenta Mathematicae 250 (2020), 301-318
MSC: Primary 37B45; Secondary 37E05.
DOI: 10.4064/fm810-1-2020
Published online: 20 March 2020
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.
