PermalinkSubmitted by Yves Coudene on Thu, 12/01/2016 - 08:06
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.
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FL
See [1], [2] for a 2004 survey.
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I am a bit surprised by this
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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