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
!»3
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