Euclidean-type algorithm for functions with same or similar origin

TL;DR

This paper introduces an Euclidean-type algorithm to determine when two maps are kin, i.e., iterates of the same map up to a commuting homeomorphism, and analyzes their topological entropy.

Contribution

It characterizes kin maps and provides a novel algorithm to test kinship, along with entropy computations for related diagonal maps.

Findings

The algorithm effectively tests kinship between maps.

Characterization of kin maps in terms of their iterates.

Topological entropy formulas for diagonal maps in kin diagrams.

Abstract

Maps $f,g\colon X\to X$ are called kin if they are forward iterates of the same map $\varphi\colon X\to X$, up to a composition with a commuting homeomorphism. Kin form an important class of commuting maps on $X$. In this paper, we characterize kin, and give an Euclidean-type algorithm which tests when two maps $f,g\colon X\to X$ are kin. Furthermore, we compute the topological entropy of diagonal maps induced by commuting diagonal kin diagrams.