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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