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 $\xi$ is piecewise linear. For any $n\in \mathbb{N}$ we denote $A_n = f^{-n}(0)$ and describe the maps $\psi:\, A_n\rightarrow [0,\, 1]$ such that $\psi\circ f = f\circ \psi$. Also we describe all possible restrictions, of self-semiconjugations of the Tent-map onto $A_n$ and prove that for any $\alpha\in A_n\setminus A_{n-1}$ a restriction is completely determined by its value at $\alpha$.