Orbit length generating functions of automorphisms of a rooted regular binary tree

Abstract

To every automorphism w of an infinite rooted regular binary tree we associate a two variable generating function \Phi_w that encodes information on the orbit structure of w. We prove that this is a rational function if w can be described by finitely many recursion relations of a particular form. We show that this condition is satisfied for all elements of the discrete iterated monodromy group \Gamma associated to a postcritically finite quadratic polynomial over C. For such \Gamma we also prove that there are only finitely many possibilities for the denominator of \Phi_w, and we describe a procedure to determine their lowest common denominator.