Euclidean-type algorithm for functions with same or similar origin

Maps $f,g\text{\hspace{0.17em}⁣}:X\to X$ are called kin if they are forward iterates of the same map $\phi \text{\hspace{0.17em}⁣}:X\to X$, 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 $f,g\text{\hspace{0.17em}⁣}:X\to X$ are kin. Furthermore, we compute the topological entropy of diagonal maps induced by commuting diagonal kin diagrams.