CONDITIONAL STABILITY, REAL ANALYTIC PSEUDO-ANOSOV MAPS 11
to singular coordinates (s,,s2) in U,, IL or U3 with
2
s. . +
s2
^
r2.
n
The measure v is smooth except at x^, where
its density function goes to infinity.
Now consider the map t from a deleted neighborhood of
2
xu to a neighborhood of the boundary of D
given in the smooth coordinate system z = t-. + it = pe (as
in (2.2)) by
.
(
iT. [. 2trp IT
\|r(pe ) = y i - £_ e
v1(S2)
Then an easy computation shows that \| ^ maps the normalized
vl—
measure on a deleted neighborhood of x to the
V 1 ( S } dudu
normalized Lebesgue measure in a neighborhood of dD .
Clearly, \|/ - can be extended to a C°° diffeomorphism (still
2 2
denoted \|r ) from S - {x } to int D . Since, by
Moser!s
oo
9
theorem [M],there is a C diffeomorphism of D equal to
the identity in a neighborhood of
dD2
which sends
vl \ duldu2
K*\ o~7 » we may assume, by composing with such
v(SzY n
1
2
diffeomorphism of D if necessary, that
* (
V l
\v,(Sshv
du1du2
2 - 1 2
Define a map g2 on D by g2 = ty gjty on int D and
2
extending to 3D by continuity. Then r r preserves du,du2.
2
Again, an easy computation shows that in a neighborhood of dD",
the vector field \|rC^j
A
" n^*v (with v as in (2.7)), which
' duxdu2
generates g9, is Hamiltonian with respect to with
Hamiltonian function
1 2 2 u2
H ° *! + +" = const (l-u1-u2
/ 2X 2
/ul+U2
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