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 C° ° 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).