Allowed patterns of symmetric tent maps via commuter functions

Abstract

We introduce a new technique to study pattern avoidance in dynamical systems, namely the use of a commuter function between non-conjugate dynamical systems. We investigate the properties of such a commuter function, specifically $h:[0,1]\to [0,1]$ satisfying $T_1\circ h=h\circ T_{\mu }$, where $T_{\mu }$ denotes a symmetric tent map of height $\mu$. We make use of this commuter function to prove strict inclusion of the set of allowed patterns of $T_{\mu }$ in the set of allowed patterns of $T_1$.