Strings of group endomorphisms

TL;DR

This paper introduces two new types of strings and their string numbers for group endomorphisms, revealing a dichotomy where these numbers are only zero or infinity, impacting the understanding of algebraic entropy.

Contribution

It defines two special string types and demonstrates a dichotomy in their string numbers, advancing the study of algebraic entropy in group endomorphisms.

Findings

String numbers are only zero or infinity for all group endomorphisms.

Dichotomy applies to the newly introduced string types.

Implications for algebraic entropy computation.

Abstract

Recently the strings and the string number of self-maps were used in the computation of the algebraic entropy of special group endomorphisms. We introduce two special kinds of strings, and their relative string numbers. We show that a dichotomy holds for all these three string numbers; in fact, they admit only zero and infinity as values on group endomorphisms.