Geometric entropy of geodesic currents on free groups

Abstract

A \emph{geodesic current} on a free group $F$ is an $F$-invariant measure on the set $\partial^2 F$ of pairs of distinct points of $\partial F$. The space of geodesic currents on $F$ is a natural companion of Culler-Vogtmann's Outer space $cv(F)$ and studying them together yields new information about both spaces as well as about the group $Out(F)$. The main aim of this paper is to introduce and study the notion of {\it geometric entropy} $h_T(\mu)$ of a geodesic current $\mu$ with respect to a point $T$ of $cv(F)$, which can be viewed as a length function on $F$. The geometric entropy is defined as the slowest rate of exponential decay of $\mu$-measures of bi-infinite cylinders in $F$, as the $T$-length of the word defining such a cylinder goes to infinity. We obtain an explicit formula for $h_{T'}(\mu_T)$, where $T,T'$ are arbitrary points in $cv(F)$ and where $\mu_T$ denotes a Patterson-Sullivan current corresponding to $T$. It involves the volume entropy $h(T)$ and the extremal distortion of distances in $T$ with respect to distances in $T'$. It follows that, given $T$ in the projectivized outer space $CV(F)$, $h_{T'}(\mu_T)$ as function of $T'\in CV(F)$ achieves a strict global maximum at $T'=T$. We also show that for any $T\in cv(F)$ and any geodesic current $\mu$ on $F$, $h_T(\mu)\le h(T)$, where the equality is realized when $\mu=\mu_T$. For points $T\in cv(F)$ with simplicial metric (where all edges have length one), we relate the geometric entropy of a current and the measure-theoretic entropy.