Counting conjugacy classes of fully irreducibles: double exponential growth

Abstract

Inspired by results of Eskin and Mirzakhani (J Mod Dyn 5(1):71–105, 2011) counting closed geodesics of length ≤L in the moduli space of a fixed closed surface, we consider a similar question in the Out(Fr) setting. The Eskin-Mirzakhani result can be equivalently stated in terms of counting the number of conjugacy classes (within the mapping class group) of pseudo-Anosovs whose dilatations have natural logarithm ≤L. Let Nr(L) denote the number of Out(Fr)-conjugacy classes of fully irreducibles satisfying that the natural logarithm of their dilatation is ≤L. We prove for r≥3 that as L→∞, the number Nr(L) has double exponential (in L) lower and upper bounds. These bounds reveal behavior not present in the surface setting or in classical hyperbolic dynamical systems.