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 implies that f and g share an iterate (fn=gm for some n and m) for general f with degree d≥3. And for generic f∈Ratd≥3, μf=μg implies g=fn for some n≥1. For generic f∈Rat2, μf=μg implies that g=fn or σf∘fn for some n≥1, where σf∈PSL2(C) permutes two points in each fiber of f. Finally, we construct examples of f and g with μf=μg such that fn≠σ∘gm for any σ∈PSL2(C) and m,n≥1.