Some remarks on the entropy for algebraic actions of amenable groups

TL;DR

This paper investigates the entropy of algebraic actions of amenable groups, analyzing its possible values, decomposing it into prime-based components, and extending prior work on p-adic entropy with elementary proofs.

Contribution

It introduces a natural decomposition of entropy into prime and infinity components for algebraic actions of amenable groups, extending Lind and Ward's p-adic entropy results.

Findings

Entropy values are characterized for algebraic actions of amenable groups.

A decomposition of entropy into prime-based summands and an infinity component is established.

Elementary proofs reprove fundamental results about classes of amenable groups.

Abstract

In this short note we study the entropy for algebraic actions of certain amenable groups. The possible values for this entropy are studied. Various fundamental results about certain classes of amenable groups are reproved using elementary arguments and the entropy invariant. We provide a natural decomposition of the entropy into summands contributed by individual primes and a summand corresponding to infinity. These results extend previous work by Lind and Ward on p-adic entropy.