  We show that symbolic finite-to-one extensions of the type constructed by O.
Sarig for surface diffeomorphisms induce H\"older-continuous conjugacies on
large sets. We deduce this from their Bowen property. This notion, introduced
in a joint work with M. Boyle, generalizes a fact first observed by R.\ Bowen
for Markov partitions. We rely on the notion of degree from finite equivalence
theory and magic word isomorphisms. As an application, we give lower bounds on
the number of periodic points first for surface diffeomorphisms (improving a
result of Sarig) and for Sinai billiards maps (building on a result of Baladi
and Demers). Finally we characterize surface diffeomorphisms admitting a
H\"older-continuous coding of all their aperiodic hyperbolic measures and give
a slightly weaker construction preserving local compactness.

Dynamical Systems