isotopic to the identity. Another proof of this result is con-
tained, as a special case, in the proofs of Theorems 3.1 and 3.5
of the present paper.
2.6. Disk case.
We now describe a construction of a C diffeomorphism
2 2 2
g2 of D = {(ul5u2) : u + u 1} for which we will obtain a
conditional stability type result (Theorem 3.5) similar to
that for g (Theorem 3.1). This will be used to prove the
existence of real analytic Bernoulli diffeomorphisms on D
(Corollary 3.7).
We begin with a generalized pseudo-Anosov map f on S
which has four singular points x,, x , x , x with p(i) = 1,
i = l,...,if. (For a concrete example of such a map f, first
2 2 2
take a linear hyperbolic automorphism fn of T = 3 R / Z
having positive eigenvalues which leaves the points x,! = (0,0),
x2 = %,0) x3 = ^0,"2)' xi+= (9"'2* * fixed- Then factor through
2 2
the map J:T - * T defined by J(v ,v ) = (l-v^l-v ). The
2 2
factor space T L is homeomorphic to S , and with the
appropriate choice of the homeomorphism, the factorization of
fn copied over to S is a generalized pseudo-Anosov map with
1-prong singularities at the points corresponding to x', xl,
x£, x^. See Paragraph 4 of [Kl],§2.)
Next we let g. , be the map on S obtained from f by
"slowing down" and "blowing up" near the points x,, x^, x3
exactly as in the construction of g. Outside the neighborhoods
2 2 2
of x-., x^, x3 corresponding to s, + s2 - rn we
Zi =
Then g, is a homeomorphism which is a C diffeomorphism
except at x^. Also, g. . has an invariant measure v given by
(2.10) in tL, i = 1,2,3; by (p*#* Q(ds1ds2) in U^, and by
cp|(dt dt ) at those points of U. , i H, which do not correspond
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