Let A be a rational function of degree at least two on the Riemann sphere. We say that A is tame if the algebraic curve A(x)−A(y)=0 has no factors of genus zero or one distinct from the diagonal. In this paper, we show that if tame rational functions A and B have orbits with infinite intersection, then A and B have a common iterate. We also show that for a tame rational function A decompositions of its iterates A∘d, d≥1, into compositions of rational functions can be obtained from decompositions of a single iterate A∘N for N big enough.
|Journal||Journal of the European Mathematical Society|
|State||Published - 2021|