CONDITIONAL STABILITY, REAL ANALYTIC PSEUDO-ANOSOV MAPS 9

(2.10) (X-l

) M # l o

^_^!l_j

in

u±,

if p(i)=1

islf...fB.

(2.11) p*(dt dt ) at those points of U., i m, which do

not correspond to singular coordinates (s,,s2) with

s + * .

Sl

Ss

2 -

rm

0

An easy computation shows that themeasures in (2.9) and (2.10)

are equal to constant multiples of (p*(dt,dt?) in some

neighborhood of x. and consequently v is a smooth measure.

(For (2.9), this computation is carried out in [G-K], §4.)

In the smooth (t., ,t2) coordinate system, in some neighbor-

hood of x., the vector field generating g (i.e., (#T • )^v ,

- 1

l 5 3

p(i-)

if p(i) 3; (*

i0

) * x * V Y i f P ( i ) = 1 a n d x i s reg*rded

as given in the (s..,s2) coordinate system) is Hamiltonian with

respect to a constant multiple of dt dt with Hamiltonian function

(2.12) H o i . . = const • Im(t +it0)p(l), if p(i) 3

1,3 i z

(2.13) H o

x"1

o $ = const • t^t^+t*), if p(i) = 1,

where H(s-,s2 = (In X)s1s is the Hamiltonian function for

v with respect to ds,ds2 and the constants are all positive.

(See [G-K], §4.) Since these Hamiltonian functions are C°°,it

follows that g is a C°° diffeomorphism of M. Note, however,

that g is not

C00

with respect to the (t,,t2) coordinates in

all of U., i = l,...,m, because of the "piecing together" required

in the definitions of T and X.

As shown in [G-K], §§5-8, g is Bernoulli with respect to

v and is topologically conjugate to f through a homeomorphism