CONDITIONAL STABILITY, REAL ANALYTIC PSEUDO-ANOSOV MAPS
map of the vector field v given by
(2.7)
Un \)sn
s2 = -Un\)s2
An invariant measure \i for f i s given by cpv§? -(ds., ds
0
)
1
ljj -L
l
in S? . , i = 1,...,m, j = Q,...,p(i)-l,and by p*(dt,dt0)
in U., i m. Note that \i is smooth except at the singular
points. The density function for [i vanishes at x., if
p(i) 3, and goes to infinity at x. , if p(i) = 1. As
shown in §10 of [F-L-P] , f is Bernoulli with respect to |JL.
2.5. Smooth models of pseudo-Anosov and generalized
pseudo-Anosov maps.
The smooth models g of [G-K]are constructed by "slowing
down" the vector field v in a neighborhood of each singular
point. In the neighborhood c p . ( B ,) of a singular point
i a
x., i = l,...,m, with p = p(i) {1,3,4,5,...}, we let gQ
be the time-one map of the new vector field vv given by
(2.8)
Un \)s.Y (s?+s^)
1 p 1 2
-Un X)s0Y (s?+si:) ,
2 p 1 2
where Y is a function defined on [0,°°) such that
(2.8a)
(i) For u
(ii)
f
2=1
Y (u) = \
P
u1/3,
if
if
Y is C except at
(iii)
Yf(u)
0 for u 0
P
P * 3
(iv) Y (u)
P
for u r.
Previous Page Next Page