On semiconjugate rational functions

Let A and B be rational functions on the Riemann sphere. The function B is
said to be semiconjugate to the function A if there exists a non-constant rational
function X such that A\circ X= X\circ B  (*).

The semiconjugacy condition generalises both the classical conjugacy relation and
the commutativity condition. In the talk we present a description of solutions of
functional equaton (*) in terms of orbifolds of non-negative Euler characteristic on
the Riemann sphere, and discuss numerous relations of this equation with complex
dynamics and number theory.