12 MARLIES GERBER
where H(s..,s ) = (In \)s-.s95 as before. Since this
Hamiltonian function is real analytic in a neighborhood of
3D , we see that g~ is a ° diffeomorphism of D which is
real analytic in a neighborhood of
dD2.
2.7. Markov partitions.
Both the proof of the topological conjugacy between g
and f in [G-K]and the conditional stability results for g
(Theorems 3,1 and 3.5) rely on the use of Markov partitions.
By the construction in [F-L-P], §9, for any pseudo-Anosov
or generalized pseudo-Anosov map f, there is a Markov partition
{R.,...,RN) of M, as described in [G-K], paragraph 2.3.
Let A = (a..),-. .^M be the transition matrix for
13 li,jN
{Rl5...,RN}, i.e.
( 1, if Int f(R±) f l Int R. i 4
a.. \
1^ I 0, otherwise.
The union of the boundaries of the R, is the union of
segments L. . and L? . of singular leaves extending P. .
and P? . , respectively. Let L = u L. . and
x^ i=l,...,m 1,D
j=0,...,p(i)-l
let
LS
be defined similarly. Assume that
a1
in paragraph 2.4
u
is such that fL 0
c-1
p (D
.)cP(i)"1
U P . . and
? " 1 = 0 1,:3
-1Ls -1 P(i)-1 s
f f l p/(llat) c U P T . for i = 1,. . ,,m.
1 a
j=0
1,:
In [G-K] it is shown that the L^ . and L? . are
segments of unstable and stable leaves for g as well. More-
over, gR, = fR, for k = 1,...,N, and the boundaries of the
R, form the boundaries of a Markov partition for g with the
same transition matrix A ([G-K],§7).
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