Weaving Geodesics and New Phenomena in Horocyclic Dynamics
Françoise Dal'bo, James Farre, Or Landesberg, Yair Minsky
TL;DR
This work analyzes horospherical dynamics on geometrically infinite hyperbolic surfaces by introducing loom surfaces built in the band model of $\mathbb{H}^2$ and a slack/weaving framework that prescribes horocycle recurrence. The authors construct the first nontrivial minimal horocyclic orbit closures and prove the existence of locally finite $N$-invariant ergodic measures supported on these closures, which are singular to the geodesic flow. They then realize horocycle orbit closures with prescribed fractional Hausdorff dimensions in $(1,2)$ via distal loom surfaces and accumulation data $E=\mathrm{accum}(\mathscr{S}_+(\eta_k^\pm))$, enabling fixed and locally varying fractal dimensions and even discrete sub-invariance. The results yield counterexamples to horospherical infinite-measure rigidity under relaxed tameness conditions, illustrating highly irregular and flexible horocyclic dynamics on geometrically infinite surfaces. Overall, the paper broadens the understanding of horocycle flows beyond finite- and geometrically finite settings by explicitly constructing minimal sets, exotic invariant measures, and fractal-dimension orbit closures.
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 $(1,2)$.