  This paper investigates dynamics that persist under isotopy in classes of
orientation-preserving homeomorphisms of orientable surfaces. The persistence
of periodic points with respect to periodic and strong Nielsen equivalence is
studied. The existence of a dynamically minimal representative with respect to
these relations is proved and necessary and sufficient conditions for the
isotopy stability of an equivalence class are given. It is also shown that most
the dynamics of the minimal representative persist under isotopy in the sense
that any isotopic map has an invariant set that is semiconjugate to it.

Dynamical Systems;Geometric Topology