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Expansive algebraic actions of discrete residually finite amenable groups and their entropy

Christopher Deninger·Klaus Schmidt
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Abstract

We prove an entropy formula for certain expansive actions of a countable discrete residually finite group $Γ$ by automorphisms of compact abelian groups in terms of Fuglede-Kadison determinants. This extends an earlier result proved by the first author under somewhat more restrictive conditions. The main tools for this generalization are a representation of the $Γ$-action by means of a `fundamental homoclinic point', and the description of entropy in terms of the renormalized logarithmic growth-rate of the set of $Γ_n$-fixed points, where $(Γ_n, n\ge1)$ is a decreasing sequence of finite index normal subgroups of $Γ$ with trivial intersection.

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Christopher Deninger, Klaus Schmidt. 2006-10-09. Expansive algebraic actions of discrete residually finite amenable groups and their entropy. https://arxiv.org/abs/math/0605723

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