§2. Definitions and Preliminaries

2.1. Except in §2.6, Theorems 3.5 and 3.6, and the last

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part of §11, where we deal with the disk D , we will work with

a compact orientable C surface M without boundary. As in

[G-K], the non-orientable case can be handled in a completely

analogous manner, once the generalization of the topological

theory mentioned in §1 of [F-L-P] is developed.

2.2. We will use widely the definitions and notations in

[G-K]. For the convenience of the reader, we will review them

here.

2.3. Measured Foliations.

A measured foliation of M is a foliation 7 which is

smooth outside a finite set of singular points x.,.. . ,x ,

together with a transverse measure u- . At x. a finite

number p = p(i) 3 of leaves of 7 meet. We say that x.

is a p-prong singularity of J. (See [G-K], paragraph 2.1.)

If, in addition, we allow one-prong singularities (setting

p = 1 in [G-K] paragraph 2.1),at which a single leaf of 7

terminates in a point, then (J,p.) is called a measured foliation

with "spines".

2.4. Pseudo-Anosov and Generalized Pseudo-Anosov Maps.

For each p 0, let D = {z = t1+it2 : tj+t^ p2}.

A pseudo-Anosov map f, as defined by Thurston ([F-L-P], [T]),

has the following form. There are two transversal measured folia-

tions (Js,nS) and (3FU,M.U) and a constant X 1 such that

(2.1)

f(*s,ns)

=

(*S,i nS)

and

f(*U,nU)

=

(*U,\nU).

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