Allowed patterns of symmetric tent maps via commuter functions

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 $T_1$.