Cohomology fractals, Cannon-Thurston maps, and the geodesic flow

TL;DR

This paper introduces cohomology fractals in hyperbolic three-manifolds, explores their relation to Cannon-Thurston maps, and analyzes their limiting distribution as the visual radius grows, revealing they form a distribution rather than a function.

Contribution

It establishes a correspondence between cohomology fractals and Cannon-Thurston maps for fibrations, and proves the limiting behavior of these fractals as a distribution on the sphere at infinity.

Findings

Cohomology fractals are normally distributed with diverging standard deviations.

The limit of cohomology fractals is a distribution, not a function.

The correspondence between cohomology fractals and Cannon-Thurston maps is proven for fibrations.

Abstract

Cohomology fractals are images naturally associated to cohomology classes in hyperbolic three-manifolds. We generate these images for cusped, incomplete, and closed hyperbolic three-manifolds in real-time by ray-tracing to a fixed visual radius. We discovered cohomology fractals while attempting to illustrate Cannon-Thurston maps without using vector graphics; we prove a correspondence between these two, when the cohomology class is dual to a fibration. This allows us to verify our implementations by comparing our images of cohomology fractals to existing pictures of Cannon-Thurston maps. In a sequence of experiments, we explore the limiting behaviour of cohomology fractals as the visual radius increases. Motivated by these experiments, we prove that the values of the cohomology fractals are normally distributed, but with diverging standard deviations. In fact, the cohomology fractals do not converge to a function in the limit. Instead, we show that the limit is a distribution on the sphere at infinity, only depending on the manifold and cohomology class.