The frequency space of a free group

Abstract

We analyze the structure of the \emph{frequency space} $Q(F)$ of a nonabelian free group $F=F(a_1,...,a_k)$ consisting of all shift-invariant Borel probability measures on $\partial F$ and construct a natural action of $Out(F)$ on $Q(F)$. In particular we prove that for any outer automorphism $\phi$ of $F$ the \emph{conjugacy distortion spectrum} of $\phi$, consisting of all numbers $||\phi(w)||/||w||$, where $w$ is a nontrivial conjugacy class, is the intersection of $\mathbb Q$ and a closed subinterval of $\mathbb R$ with rational endpoints. We also provide an algorithm for detecting strict hyperbolicity of an automorphism of $F$.