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
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