Tame rational functions: Decompositions of iterates and orbit intersections

Let A be a rational function of degree at least 2 on the Riemann sphere. We say that A is tame if the algebraic curve A(x) - A(y) D 0 has no factors of genus 0 or 1 distinct from the diagonal. In this paper, we show that if tame rational functions A and B have some 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 large enough.