Explosion of smoothness from a point to everywhere for
conjugacies between Markov families
Abstract
For uniformly asymptotically affine (uaa) Markov maps on train
tracks, we prove the following type of rigidity result: if a
topological conjugacy between them is (uaa) at a point in the
train track then the conjugacy is (uaa) everywhere. In
particular, our methods apply to the case in which the domains
of the Markov maps are Cantor sets. We also present similar
statements for (uaa) and Cr expanding
circle maps with r>1.