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.