10 MARLIES GERBER

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

2

existence of real analytic Bernoulli diffeomorphisms on D

(Corollary 3.7).

2

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

2

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.)

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

^e^

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