Typical path components in tent map inverse limits

In the inverse limit ${\widehat{I}}_s$ of a tent map $f_s$ restricted to its core, the set $\mathcal{G}\mathcal{R}$ of points whose path components are bi-infinite and bi-dense has full measure with respect to the measure induced on ${\widehat{I}}_s$ 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, $\mathcal{G}\mathcal{R}$ contains a dense $G_{\delta }$ set. In contrast, when the critical orbit of $f_s$ is dense, the complement of $\mathcal{G}\mathcal{R}$ contains a dense $G_{\delta }$ set.