On an entropy of $Z_+^k$-actions

TL;DR

This paper explores a generalized entropy concept for multi-parameter actions, analyzing its properties and calculating bounds for actions on circles and tori generated by expanding endomorphisms.

Contribution

It studies a novel entropy definition for $bZ_+^k$-actions, including properties, explicit calculations, and bounds for actions on circles and tori.

Findings

Entropy can be nonzero even when generators have zero entropy.

Explicit entropy values are computed for actions on circles.

An upper bound for entropy on tori is established using preimage entropies.

Abstract

In this paper, a definition of entropy for $\mathbb{Z}_+^k(k\geq2)$-actions due to S. Friedland \cite{Friedland} is studied. Unlike the traditional definition, it may take a nonzero value for actions whose generators have finite (even zero) entropy as single transformations. Some basic properties are investigated and its value for the $\mathbb{Z}_+^k$-actions on circles generated by expanding endomorphisms is given. Moreover, an upper bound of this entropy for the $\mathbb{Z}_+^k$-actions on tori generated by expanding endomorphisms is obtained via the preimage entropies, which are entropy-like invariants depending on the "inverse orbits" structure of the system.