Entropy for automorphisms of free groups

Choda, Marie

doi:10.1090/S0002-9939-06-08318-3

Entropy for automorphisms of free groups
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by Marie Choda

Proc. Amer. Math. Soc. 134 (2006), 2905-2911

DOI: https://doi.org/10.1090/S0002-9939-06-08318-3

Published electronically: April 11, 2006

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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 $C^*$-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.

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