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Horocyclic trajectories in hyperbolic solenoidal surfaces of finite type

Fernando Alcalde Cuesta, Álvaro Carballido Costas, Matilde Martínez, Alberto Verjovsky

TL;DR

This work develops a topological dynamical theory for laminated horocycle flows on unit tangent bundles $T^1\mathcal{X}$ of hyperbolic solenoidal surfaces of finite type by extending Hedlund-style dichotomies to inverse-limit constructions. It establishes a cohesive framework of solenoidal surfaces (including McCord solenoids), analyzes leafwise hyperbolic dynamics, and proves a Hedlund-type theorem for towers of finite-area coverings. A detailed classification of horocycle-invariant minimal sets is provided via a trichotomy and explicit McCord-solenoid examples, revealing how cusp lifts and transverse Cantor dynamics shape orbit closures. Furthermore, the paper proves a convergence theorem: under positive time, minimal sets approach cuspidal ends in the end-compactified boundary, while under negative time they converge to the full end-compactification, highlighting the link between end structure and laminated dynamics. Collectively, these results illuminate how transverse Cantor structures and cuspidal ends govern laminated horocycle dynamics, bridging hyperbolic geometry, inverse limits, and Cantor dynamics in finite-type solenoids.

Abstract

We study the dynamical properties of the laminated horocycle flow on the unit tangent bundles of 2-dimensional smooth solenoidal manifolds of finite type. These laminations are the analog of complete hyperbolic surfaces of finite area.

Horocyclic trajectories in hyperbolic solenoidal surfaces of finite type

TL;DR

This work develops a topological dynamical theory for laminated horocycle flows on unit tangent bundles of hyperbolic solenoidal surfaces of finite type by extending Hedlund-style dichotomies to inverse-limit constructions. It establishes a cohesive framework of solenoidal surfaces (including McCord solenoids), analyzes leafwise hyperbolic dynamics, and proves a Hedlund-type theorem for towers of finite-area coverings. A detailed classification of horocycle-invariant minimal sets is provided via a trichotomy and explicit McCord-solenoid examples, revealing how cusp lifts and transverse Cantor dynamics shape orbit closures. Furthermore, the paper proves a convergence theorem: under positive time, minimal sets approach cuspidal ends in the end-compactified boundary, while under negative time they converge to the full end-compactification, highlighting the link between end structure and laminated dynamics. Collectively, these results illuminate how transverse Cantor structures and cuspidal ends govern laminated horocycle dynamics, bridging hyperbolic geometry, inverse limits, and Cantor dynamics in finite-type solenoids.

Abstract

We study the dynamical properties of the laminated horocycle flow on the unit tangent bundles of 2-dimensional smooth solenoidal manifolds of finite type. These laminations are the analog of complete hyperbolic surfaces of finite area.

Paper Structure

This paper contains 18 sections, 15 theorems, 91 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Solenoidal surfaces
  3. Definition of solenoidal manifolds
  4. The universal solenoid
  5. The universal hyperbolic solenoid
  6. Hyperbolic solenoidal surfaces
  7. Hyperbolic solenoidal surfaces of finite type
  8. Examples of hyperbolic solenoidal surfaces of finite type
  9. Solenoidal cusps
  10. Examples
  11. Inverse limits and McCord solenoids
  12. Hedlund's Theorem for hyperbolic solenoidal surfaces of finite type
  13. Minimal sets for the horocycle flow
  14. A trichotomy for inverse limits of surface coverings
  15. Examples of McCord solenoids of different class
  16. ...and 3 more sections

Key Result

Theorem 1

Let $\mathcal{X}$ be the inverse limit of a tower of finite coverings based on a hyperbolic surface $S$ of finite area. Then any orbit of the horocycle flow on the unit tangent bundle $\mathcal{Y} = T^1 \mathcal{X}$ of $\mathcal{X}$ is either dense or projects onto a periodic orbit of the horocycle

Figures (2)

  • Figure 1: 2-to-1 covering that keeps the number of cusps
  • Figure 2: 2-to-1 covering that duplicates the number of cusps

Theorems & Definitions (60)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Definition 1
  • Definition 2
  • Remark 1
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • ...and 50 more