2 MARLIES GERBER
The obvious obstruction to real analyticity for the Bernoulli
diffeomorphisms in [G-K], as well as those in [Kl],is the use of
bump functions in the smoothing process. However, in this
paper we obtain a conditional structural stability result for the
C* models in [G-K] which leads to real analytic Bernoulli models
of pseudo-Anosov maps. As conjectured by M. Herman, we prove that
a smooth model g from [G-K] is topologically stable under small
perturbations which preserve a certain finite number of jets of
g at each singularity of f. (See Theorem 3.1 for the precise
statement.) The class of such perturbations includes a real
analytic diffeomorphism (Theorem 3.2) and we thus obtain real
analytic analogues (Corollary 3.3) of the results in [G-K].
The conditional structural stability result is also valid
for the smooth models of generalized pseudo-Anosov maps. (See the
definition in [F-L-P], §13.) In particular, this allows us to
work on S2, where only the generalized type of pseudo-Anosov
map exists. This leads to the main corollary in this paper: the
existence of real analytic diffeomorphisms of S2 and of D2
which are Bernoulli with respect to Lebesgue measure.
Smooth ergodic diffeomorphisms of D2 were first constructed
by D.V. Anosov and A.B. Katok [A-K] in 1970. Topologically
transitive mappings of
were constructed much earlier by
L.G. Snirel'man [Sn] and A.S. Besicovitch [Bl], [B2],and a C00
version of their construction was later obtained by E.A. Sidorov [Si].
However, in the real analytic setting even topologically transitive
examples on D2 were apparently not previously known.
I am grateful to A. Katok for suggesting to me the problems
dealt with here and for many helpful conversations throughout the
preparation of this paper.