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.