Abstract

We study the minimal set of (Lefschetz) periods of the $C^1$ Morse-Smale diffeomorphisms on a (orientable or not) compact surface without boundary inside its class of homology. In fact our study extends to the $C^1$ diffeomorphisms on these surfaces having finitely many periodic orbits all of them hyperbolic and with the same action on the homology as the Morse-Smale diffeomorphisms.

We mainly have two kind of results. First we completely characterize the minimal sets of periods for the $C^1$ Morse-Smale diffeomorphisms on a compact surface without boundary of genus $g=0,1,...,3$ for orientable surfaces and genus $g=1,2,...,9$ for non-orientable ones. But the proof of these results provides an algorithm for characterizing these minimal sets of periods for the $C^1$ Morse-Smale diffeomorphisms on compact surfaces without boundary of arbitrary genus.

Second we study what kind of subsets of positive integers can be minimal sets of periods of the $C^1$ Morse-Smale diffeomorphisms on a compact surface without boundary.