Does there exists a real analytic area preserving ergodic diffeomorphism on $S^2$? 


(Possibly by perturbing a rotation in the real-analytic topology?)



Real analytic ergodic diffeomorphisms of the two sphere
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](http://journals.cambridge.org/action/displayAbstract;jsessionid=6D90B16B80CB2B4202E574DC4F274677.journals?fromPage=online&aid=2140120) 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](http://books.google.com.uy/books?id=Z40ZNVcgOFIC&pg=PA1&lpg=PA1&dq=gerber%20analytic%20models%20pseudo%20anosov&source=bl&ots=I5luAED0PJ&sig=f86Feeo8wTFgZQ_XcOx3ZhK1PBI&hl=es&sa=X&ei=NjUFU-D3FpS1sATCn4GACA&ved=0CEUQ6AEwAg#v=onepage&q=gerber%20analytic%20models%20pseudo%20anosov&f=false)). 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](http://arxiv.org/pdf/1309.3137.pdf) might be of interest to you. 


