Weaving Geodesics and New Phenomena in Horocyclic Dynamics

TL;DR

This paper constructs new examples of hyperbolic surfaces with unique horocyclic dynamics, including minimal orbit closures and singular invariant measures, revealing complex recurrence phenomena in horocyclic flows.

Contribution

It introduces the first examples of non-trivial minimal horocyclic orbit closures and singular invariant measures on geometrically infinite hyperbolic surfaces.

Findings

Existence of non-trivial minimal horocyclic orbit closures.

Construction of infinite locally-finite conservative horocyclic measures that are singular.

Surfaces supporting horocyclic orbit closures with arbitrary Hausdorff dimension in (1,2).

Abstract

We construct geometrically infinite hyperbolic surfaces supporting horocycles with tailored recurrence properties. In particular, we obtain the first examples of non-trivial minimal horocyclic orbit closures and of infinite locally-finite conservative horocyclic invariant measures which are singular with respect to the geodesic flow. Other examples include surfaces supporting horocyclic orbit closures of arbitrary Hausdorff dimension in $(1,2)$.