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arXiv · 1712.00739

Typical path components in tent map inverse limits

Philip Boyland·André de Carvalho·Toby Hall
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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.

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Philip Boyland, André de Carvalho, Toby Hall. 2017-12-03. Typical path components in tent map inverse limits. https://arxiv.org/abs/1712.00739

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