$k$-type entropy of $\mathbb{Z}^d$ actions

TL;DR

This paper introduces the concept of $k$-type entropy for $bZ^d$-actions on compact metric spaces, explores its properties, and computes it for certain $bZ^2$-actions on a torus, linking it to classical entropy.

Contribution

It defines and analyzes a new $k$-type entropy concept for $bZ^d$-actions, connecting it with existing dynamical measures and calculating it for specific examples.

Findings

$k$-type entropy is well-defined and shares properties with classical entropy.

Connections established between $k$-type entropy and other dynamical notions.

Explicit calculations of $k$-type entropy for some $bZ^2$-actions on a torus.

Abstract

We introduce the concept of \(k\)-type entropy for dynamical systems generated by \(\mathbb{Z}^d\)-actions on compact metric spaces. We investigate its fundamental properties and establish connections with classical entropy and other \(k\)-type dynamical notions. The $k$-type entropy of some $\mathbb{Z}^2$-actions on a two dimensional torus is also calculated.