The problem of counting closed geodesics of bounded length, originally in the setting of negatively curved manifolds, goes back to the classic work of Margulis in 1960s about the dynamics of the geodesic flow. Since then Margulis' results have been generalized to many other contexts where some whiff of hyperbolicity is present.
Thus a 2011 result of Eskin and Mirzakhani shows that for a closed hyperbolic surface S of genus $g\ge 2$, the number $N(L)$ of closed Teichmuller geodesics of length $\le L$ in the moduli space of $S$ grows as $e^{hL}/(hL)$ where $h=6g-6$. The number $N(L)$ is also equal to the number of conjugacy classes of pseudo-Anosov elements $\phi$ in the mapping class group $MCG(S)$ with $\log\lambda(\phi)\le L$, where $\lambda(\phi)>1$ is the ``dilatation" or ``stretch factor" of $\phi$.

We consider an analogous problem in the $Out(F_r)$ setting, for the action of the outer automorphism group $Out(F_r)$ of the free group $F_r$ of rank $r$ on a ``cousin" of the Teichmuller space, called the Culler-Vogtmann Outer space. In this context being a ``fully irreducible" element of $Out(F_r)$ serves as a natural counterpart of being pseudo-Anosov. Every fully irreducible $\phi\in Out(F_r)$ acts on the Outer space as a loxodromic isometry with translation length $\log\lambda(\phi)$, where again $\lambda(\phi)$ is the stretch factor of $\phi$. We estimate the number $N_r(L)$ of fully irreducible elements $\phi\in Out(F_r)$ with $\log\lambda(\phi)\le L$. The number $N_r(L)$ can also be interpreted as the number of homotopy classes of closed “contracting" geodesics in the moduli space of the Outer space. We prove, for $r\ge 3$, that $N_r(L)$ grows \emph{doubly exponentially} in $L$ as $L\to\infty$, in terms of both lower and upper bounds. This result reveals new behavior not present in classic hyperbolic dynamical systems. The talk is based on a joint paper with Catherine Pfaff.