CONDITIONAL STABILITY, REAL ANALYTIC PSEUDO-ANOSOV MAPS 5

s s u u

If (J ,[i ) and (7

9

[L ) are measured foliations with

spines and f has the above form (with p = 1 at the one-

prong singularities), then f is said to be a generalized

pseudo-Anosov map.

At each singular point x., with p = p(i), we define the

stable and unstable prongs

rs -1f ix . ^ 2j+l , .

n

P.. ± = P - {z = pe : 0pa, T = -J— IT}, 3 = 0,l,...,p-l,

J- 5 J 1 P

and

p. . = p. {z = pe : 0 p a, x = —*-

TT

}, 3 = 0,1,.. . ,p-l,

1 5 J 1 P

and the stable and unstable sectors

S? . = (pT^z = pe1T : 0 p a , ^ T ^ ff - T - H r ^ tr}, j = 0 , 1 , . . . , p-1 ,

1

9 J 1 p p

and

s" . = cp^lz = pe1T : 0 p a, ^f I T T ^f^ " 3 = 0 , 1 , . . . , p - 1 .

13

J 1 P P

(Then s! _ = S? = U., if p(i) = 1.)

1,0 i,n 0 1 *

A singular leaf of 7 [7 ] is defined to be a singularity

x# together with the extension along J 17 ] (away from x.)

1 1

of a stable [unstable] prong at x. .

While, in general, f may permute the singular leaves, we

will assume here, as in [G-K], that f(x.) = x. and

'

1 1

f(P? .)c P* , f-X(P^ c P" i = l,...,m,

j = 0,...,p(i)-l. Here again, the arguments in the general

case are the same, but require more cumbersome notation.

Consider the mapping

(2.3a) $. . : p.S? . -• {w : Re w 0} given by

1 1 1

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