Growth of Periodic Points and Rotation Vectors on Surfaces

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 then every rational point in int(ρ(f)) is represented by a periodic orbit [6]. In a recent paper, Sharp showed that (for diffeomorphisms) under this hypothesis the number of periodic points with prescribed rational rotation vector has exponential growth [18]. Subsequently, Hayakawa studied the analogous problem for compact surfaces of genus at least 2 and obtained a partial generalization of Franks’ result [10]. In this paper we shall improve on this result. Furthermore, we shall extend Sharp’s result on the exponential growth of periodic points to this setting. Before we state our main result we briefly describe the idea of the rotation set associated to a point, and the rotation vector associated to a periodic point, for a homeomorphsism f :M →M isotopic to the identity map on a compact surfaceM . The rotation set ρx(f) ⊂ H1(M,R) associated to a point x ∈M can be heuristically interpreted as describing the “asymptotic drift” in homology of the orbit of x. In the special case that f(x) = x is a periodic point then ρx(f) is a single vector. A precise definition is given in the next section.