On semi-conjugate rational functions
Let $A$ and $B$ be rational functions on the Riemann sphere. The function $B$ is said to be semi-conjugate to the function $A$ if
there exists a non-constant rational function $X$ such that
A\circ X=X\circ B. (*)
The semi-conjugacy condition generalises both the classical conjugacy relation and the commutativity condition. In the talk we present a description of solutions of functional equation (*) 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.