Expansive algebraic actions of discrete residually finite amenable groups and their entropy
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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.
Using a recent result of Y. Choi the discussion of expansiveness was strengthened and shortened. The paper will appear in: Ergodic Theory and Dynamical Systems
Using a recent result of Y. Choi the discussion of expansiveness was strengthened and shortened. The paper will appear in: Ergodic Theory and Dynamical Systems