Tame rational functions: Decompositions of iterates and orbit intersections

Source abstract

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)=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^{\circ d}, d\geq 1, into compositions of rational functions can be obtained from decompositions of a single iterate A^{\circ N} for N large enough.