Enumerating pseudo-Anosov foliations

Abstract

Let M be a closed oriented surface of genus at least two, and let SF denote the space of all projective classes of measured foliations on M. The authors have previously given a criterion in terms of certain combinatorial words for an element of &^ to be left invariant by a pseudo-Anosov map of M: such foliations are characterized by the fact that the associated word is eventually periodic. The current work derives an estimate which says roughly that the dilatation of the corresponding pseudo-Anosov map is large if the periodic part of the word is long. This estimate is then used to bound the number of distinct conjugacy classes of foliations invariant under pseudo-Anosov maps of M in terms of a specified bound on the dilatations. 1. Introduction. This paper is a sequel to our paper [5] in which we described a method of representing measured foliations carried by a train track by means of semi-infinite combinatorial words in a finite alphabet. We showed that pseudo-Anosov foliations (i.e., those that are preserved by pseudo-Anosov maps) are characterized by being representable by eventually periodic convergent words. In this paper, we establish an inequality relating the length of the repeating part of the word corresponding to a pseudo-Anosov foliation and the dilatation factor of a pseudo-Anosov map preserving that foliation. As a by-product, for each real number R, we shall describe a finite set of foliations, whose cardinality is bounded in terms of R and which contains a representative of each conjugacy class of pseudo-Anosov foliation which is preserved by a pseudo-Anosov map of dilatation at most R. Throughout this paper, M will denote a closed, oriented, smooth surface of negative Euler characteristic. Moreover, unless otherwise stated, we will tacitly assume that each train track considered is trivalent, i.e., there are exactly three half-branches incident on each switch. For the basic facts about measured foliations and train tracks, we refer the reader to [1], [3], [4], [5], or [6]. It is necessary to further refine some of the notions developed in [5] (whose terminology we will adopt), and this is the purpose of the next paragraph.