Vaibhav Gadre

Cannon–Thurston curves do not equidistribute

Thurston showed that a fibered 3-manifold is hyperbolic if and only if it is a mapping torus of a surface by a pseudo-Anosov map. For such a 3-manifold, the inclusion of the fiber (at the level of the universal covers) gives an exponentially distorted copy of a hyperbolic plane in hyperbolic 3-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 3-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. With Maher, Pfaff, Uyanik, 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.