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Entropy for automorphisms of free groups

Authors:	Marie Choda

Institution:	Department of Mathematics, Osaka Kyoiku University, Asahigaoka, Kashiwara 582-8582, Japan

Abstract:	Let $\sigma$ be the automorphism of the free group $F_\infty$ which is arising from a permutation of the free generators of $F_\infty.$ The $\sigma$ naturally induces the automorphism $\hat \sigma$ of the reduced -algebra $C^*_r(F_\infty),$ and also the automorphism $\bar{\hat \sigma}$ of the group factor $L(F_\infty).$ We show that the Brown-Germain entropy $ha(\sigma )$ is zero. This implies that the Brown-Voiculescu topological entropy $ht(\hat \sigma ),$ the Connes-Narnhofer-Thirring dynamical entropy $h_\phi (\hat \sigma )$ and the Connes-Størmer entropy $H(\bar{\hat \sigma} )$ are all zero.

Keywords:	Entropy free group $C^*$-algebra amenability

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