  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.
