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

2014/03/31 by Richard Pink, Pink, Richard
Mathematics · #20E08 (37B20 #37D40) #FOS: Mathematics #Group Theory (math.GR) #math.GR #msc:20E08

paper · pdf · doi:10.48550/arxiv.1403.8019

arxiv created 2014/03/31 · arxiv updated 2014/04/01

Abstract

To every automorphism w of an infinite rooted regular binary tree we associate a two variable generating function Φ_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 Γassociated to a postcritically finite quadratic polynomial over C. For such Γwe also prove that there are only finitely many possibilities for the denominator of Φ_w, and we describe a procedure to determine their lowest common denominator.

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