In this paper will consider diffeomorphisms f : M → M of a compact surface M , isotopic to the identity map, and study the growth of their periodic points. An important approach to studying such properties of diffeomorphisms is via the rotation set ρ(f) and the associated rotation vectors. For the particular case of homeomorphisms of tori Franks has shown that if ρ(f) has non-empty interior the...

Let (X, d) be a nonempty metric space, and let (2X , Hd) be the hyperspace of all nonempty compact subsets of X with the Hausdorff metric. Let F :X → 2X be an ε-contractive map. A general condition is given that guarantees the existence of a periodic point of F (the theorem extends a result of Edelstein to multi-valued maps). The condition holds when X is compact; hence, F has a periodic point ...

We show that S. Saito’s fixed point formula serves as a powerful tool for counting the number of isolated periodic points of an area-preserving surface map admitting periodic curves. His notion of periodic curves of types I and II plays a central role in our discussion. We establish a Shub-Sullivan type result on the stability of local indices under iterations of the map, the finiteness of the ...