§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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