Counting conjugacy classes of fully irreducibles: double exponential growth

Inspired by results of Eskin and Mirzakhani (J Mod Dyn 5(1):71–105, 2011) counting closed geodesics of length \(\le L\) in the moduli space of a fixed closed surface, we consider a similar question in the \(Out (F_r)\) 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 \(\le L\). Let \({\mathfrak {N}}_r(L)\) denote the number of \(Out (F_r)\)-conjugacy classes of fully irreducibles satisfying that the natural logarithm of their dilatation is \(\le L\). We prove for \(r\ge 3\) that as \(L\rightarrow \infty \), the number \({\mathfrak {N}}_r(L)\) 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.