Rational functions with identical measure of maximal entropy

Abstract

We discuss when two rational functions $f$ and $g$ can have the same measure of maximal entropy. The polynomial case was completed by (Beardon, Levin, Baker-Eremenko,Schmidt-Steinmetz, etc., 1980s-90s), and we address the rational case following Levin-Przytycki (1997). We show: $\mu_f = \mu_g$ implies that $f$ and $g$ share an iterate ($f^n = g^m$ for some $n$ and $m$) for general $f$ with degree $d \geq 3$. And for generic $f\in \Rat_{d\geq 3}$, $\mu_f = \mu_g$ implies $g=f^n$ for some $n \geq 1$. For generic $f\in \Rat_2$, $\mu_f = \mu_g$ implies that $g= f^n$ or $\sigma_f\circ f^n$ for some $n\geq 1$, where $\sigma_f\in PSL_2(\C)$ permutes two points in each fiber of $f$. Finally, we construct examples of $f$ and $g$ with $\mu_f = \mu_g$ such that $f^n \neq \sigma\circ g^m$ for any $\sigma \in PSL_2(\C)$ and $m,n\geq 1$.