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Geometric entropy of geodesic currents on free groups

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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Deposited on : 2010-09-24

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