Problem 50

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Is there a transitive/ergodic diffeomophism on $\mathbb S^2, \mathbb D^2$?



See [1], [2] for a 2004 survey.


  1. [katok1970new] Katok AB.  1970.  New examples in smooth ergodic theory. Ergodic diffeomorphisms. Trans. Mosc. Math. Soc.. 23
  2. [fayad2004constructions] Fayad B, Katok A.  2004.  Constructions in elliptic dynamics. Ergodic Theory and Dynamical Systems. 24:1477–1520.

I am a bit surprised by this question because I think the answer was known at that time. Following the work of Plykin, we can start from an hyperbolic toral automorphism and map the torus to the sphere using a Jacobi elliptic function. The result is a pseudo-Anosov diffeomorphism of the sphere that inherits the properties of the toral automorphism with respect to a measure absolutely continuous with respect to Lebesgue. Then we can project the sphere on the plane stereographically and send the plane in a disk. Actually the procedure goes back to Schroeder (1871) when starting from multiplication by 2 on a 2-dimensional torus, and gives a transitive rational transformation of the Riemann sphere.

Or do I miss something?

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