Equal Compositions of Rational Functions

We study the following questions: (1) What are all solutions to f ◦ f̂ = g ◦ ĝ in complex rational functions f, g ∈ C(X) and meromorphic functions f̂ , ĝ on the complex plane? (2) For which rational functions f(X) and g(X) with coefficients in an algebraic number field K does the equation f(a) = g(b) have infinitely many solutions with a, b ∈ K? We utilize various algebraic, geometric and analytic results in order to resolve both questions in the case that the numerator of f(X) − g(Y ) is an irreducible polynomial in C[X,Y ] of sufficiently large degree. Our work answers a 1973 question of Fried in all but finitely many cases, and makes significant progress towards answering a 1924 question of Ritt and a 1997 question of Lyubich and Minsky. Date: December 31, 2015.