Computing conjugating sets and automorphism groups of rational functions

Let ϕ and ψ be endomorphisms of the projective line of degree at least 2, defined over a field F. From a dynamical perspective, a significant question is to determine whether ϕ and ψ are conjugate (or to answer the related question of whether a given rational function ϕ has a nontrivial automorphism). We construct efficient algorithms for computing the set of conjugating maps (resp., the group of automorphisms), with an emphasis on the case where F is a finite field or a number field. Each of our algorithms takes advantage of different dynamical structures, so context (e.g., field of definition and degree of the map) determines the preferred algorithm.