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 $\psi :A_n\to [0,1]$ 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 $\alpha$.

arXiv comment20 pages