Does there exists a real analytic area preserving ergodic diffeomorphism on $S^2$?
(Possibly by perturbing a rotation in the real-analytic topology?)
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$\begingroup$ One can show such examples exists in the $C^\infty$ category by perturbing a rotation. But the construction does not carry over to the real-analytic category. Also a construction is possible for $S^3$ in the real-analytic category using the fact that it is $SU(2)$. I am curious to know if any work is known for $S^2$ or even dimensional spheres in general. $\endgroup$– user42388Commented Feb 18, 2014 at 14:32
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In $S^2$ it is possible to construct such an example by a quite different method (not as a limit of rotations) which is indeed Bernoulli with respect to Lebesgue.
The idea is to quotient the cat map of the torus by $\pm Id$ to get a homeomorphism of the sphere which is Bernoulli and then notice that the map is "stable" in the sense that perturbations which preserve the structure of the singularities are essentially the same. In such a way, a real analytic perturbation preserving the structure of singularities can be found and one can prove that indeed the map is Bernoulli.
I might be wrong since I recall reading this a long time ago in this paper of Lewowicz and Lima de Saa which I cannot access right now. I recall also that Gerber had some paper in the same lines (see here). I am sure this is done for usual pseudo-Anosov maps, and I cannot see a reason for this not to work in the $S^2$ example mentioned above though I might be missing something.
In higher dimensions this won't definitely work, I think the discussion in this paper might be of interest to you.
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$\begingroup$ Does the perturbation remove the singularity of the $S^2$? $\endgroup$– PengfeiCommented Feb 20, 2014 at 2:27