Rational functions with identical measure of maximal entropy

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–1990s, and we address the rational case following Levin and Przytycki (1997). We show: _μf=_μg${\mu }_f={\mu }_g$ implies that f and g   share an iterate (fⁿ=g^m$f^n=g^m$ for some n and m) for general f   with degree d≥3$d\ge 3$. And for generic f∈_Ratd≥3$f\in {\mathrm{Rat}}_{d\ge 3}$, _μf=_μg${\mu }_f={\mu }_g$ implies g=fⁿ$g=f^n$ for some n≥1$n\ge 1$. For generic f∈_Rat2$f\in {\mathrm{Rat}}_2$, _μf=_μg${\mu }_f={\mu }_g$ implies that g=fⁿ$g=f^n$ or _σf°fⁿ${\sigma }_f°f^n$ for some n≥1$n\ge 1$, where _σf∈_PSL2(C)${\sigma }_f\in {\mathit{PSL}}_2(\mathbb{C})$ permutes two points in each fiber of f. Finally, we construct examples of f and g   with _μf=_μg${\mu }_f={\mu }_g$ such that fⁿ≠σ°g^m$f^n\ne \sigma °g^m$ for any σ∈_PSL2(C)$\sigma \in {\mathit{PSL}}_2(\mathbb{C})$ and m,n≥1$m,n\ge 1$.