The asymptotic behavior of least pseudo-Anosov dilatations

Let S = Sg,n be an orientable surface with genus g and n marked points. The mapping class group of S is defined to be the group of homotopy classes of orientation preserving homeomorphisms of S. We denote it by Mod(S). Given a pseudo-Anosov element f ∈ Mod(S), let λ(f ) denote the dilatation of f (see section 2.1). We define L(Sg,n) := {log λ(f )|f ∈ Mod(Sg,n) pseudo-Anosov}. This is precisely the length spectrum of the moduli space Mg,n of Riemann surfaces of genus g with n marked points with respect to the Teichmuller metric; see [Iva88]. There is a shortest closed geodesic and we denote its length