Title

# Geometric entropy of geodesic currents on free groups

Authors
Kapovich, Ilya
Published in Kolyada, S. Proceedings of the Special Semester "Dynamical Numbers", MPI Bonn. Basel: Birkhäuser. 2010
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 \$Tin 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 \$Tin cv(F)\$ with simplicial metric (where all edges have length one), we relate the geometric entropy of a current and the measure-theoretic entropy.
Note Updated version, incorporating the referee's comments
Stable URL http://archive-ouverte.unige.ch/unige:11892
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