Self semi conjugations of Ulam's Tent-map

Abstract

We study the self-semiconjugations of the Tent-map $f:\, x\mapsto 1-|2x-1|$ for $x\in [0,\, 1]$. We prove that each of these semi-conjugations $ξ$ is piecewise linear. For any $n\in \mathbb{N}$ we denote $A_n = f^{-n}(0)$ and describe the maps $ψ:\, A_n\rightarrow [0,\, 1]$ such that $ψ\circ f = f\circ ψ$. Also we describe all possible restrictions, of self-semiconjugations of the Tent-map onto $A_n$ and prove that for any $α\in A_n\setminus A_{n-1}$ a restriction is completely determined by its value at $α$.