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Abstract

We investigate generalizations of Ingram's Conjecture involving maps on trees. We show that for a class of tentlike maps on the $k$-star with periodic critical orbit, different maps in the class have distinct inverse limit spaces. We do this by showing that such maps satisfy the conclusion of the Pseudo-isotopy Conjecture, i.e., if $h$ is a homeomorphism of the inverse limit space, then there is an integer $N$ such that $h$ and $\widehat\sigma^N$ switch composants in the same way, where $\widehat\sigma$ is the standard shift map of the inverse limit space.