Algebraic entropy of shift endomorphisms on abelian groups

TL;DR

This paper studies the algebraic entropy of generalized shift endomorphisms on abelian groups, revealing its dependence on the underlying map and providing examples for various entropy behaviors.

Contribution

It introduces a method to compute algebraic entropy for generalized shifts, highlighting the influence of the map and offering a versatile tool for counterexamples.

Findings

Algebraic entropy depends mainly on the map mbda.

Provides explicit formulas for entropy in various cases.

Demonstrates the use of generalized shifts as a universal counterexample tool.

Abstract

For every finite-to-one map $\lambda:\Gamma\to\Gamma$ and for every abelian group $K$, the generalized shift $\sigma_\lambda$ of the direct sum $\bigoplus_\Gamma K$ is the endomorphism defined by $(x_i)_{i\in\Gamma}\mapsto(x_{\lambda(i)})_{i\in\Gamma}$. In this paper we analyze and compute the algebraic entropy of a generalized shift, which turns out to depend on the cardinality of $K$, but mainly on the function $\lambda$. We give many examples showing that the generalized shifts provide a very useful universal tool for producing counter-examples.