Cannon--Thurston curves do not equidistribute

Abstract: Thurston showed that a fibered three-manifold is hyperbolic if and only if it is a mapping torus of a surface by a pseudo-Anosov map. For such a three-manifold, the inclusion of the fibre (at the level of the universal covers) gives an exponentially distorted copy of a hyperbolic plane in hyperbolic three-space. Nonetheless, Cannon--Thurston showed that one still obtains a continuous map from the circle at infinity (for the hyperbolic plane) to the sphere at infinity (for hyperbolic three-space), resulting in a space filling curve. This fits in a much broader context of Cannon--Thurston maps as for example studied by Mahan and others. We show that in contrast to classical Peano type curves, the Cannon--Thurston curves do not equidistribute: a large class of measures on the circle pushes forward to singular measures on the sphere.

This is joint work with Maher, Pfaff, and Uyanik.

algebraic topologydifferential geometrydynamical systemsgroup theorygeometric topologysymplectic geometry

Audience: researchers in the topic

( paper )

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