Abstract
This paper deals with the problem of finding the range of entropy values resulting from actions of discrete amenable groups by automorphisms of compact abelian groups. When the acting group G is locally normal, we obtain an entropy formula and show that the full range of entropy values [0, infinity] occurs for actions of G. We consider related entropy range problems, give sufficient conditions for zero entropy and, as a consequence, verify the known relationship between completely positive entropy and mixing for these actions.
Highlights
Let G be a countably infinite discrete amenable group and X a compact metrizable abelian group with Borel σ-algebra B(X) and normalized Haar measure μX
Suppose α is a G-action by μX -preserving transformations αg of X and consider the metric entropy h(α) with respect to μX
For a given class of discrete amenable groups G, a natural problem is to determine the range of possible entropy values, H(G) = {h(α) : α is an algebraic G-action for some G ∈ G}
Summary
Let G be a countably infinite discrete amenable group and X a compact metrizable abelian group with Borel σ-algebra B(X) and normalized Haar measure μX. Theorem 3.1(1) shows that to calculate the entropy of an algebraic action of a locally normal group G on a compact finite-dimensional abelian group X, it is enough to consider the submodule M of M = X consisting of all elements of finite order.
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