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Inducing recurrent flows by twisting on infinite surfaces with unbounded cuffs

Hrant Hakobyan, Michael Pandazis, Dragomir Saric

Abstract

A Riemann surface $X$ is parabolic if and only if the geodesic flow (for the hyperbolic metric) on the unit tangent bundle of $X$ is ergodic. Consider a Riemann surface $X$ with a single topological end and a sequence $α_n$ of pairwise disjoint, simple closed geodesics converging to the end, called {\it cuffs}. Basmajian, the first and the third author, proved that when the lengths $\ell (α_n)$ of cuffs are at most $2\log n$, the surface $X$ is parabolic. One could expect that having arbitrary large cuff lengths $\ell (α_n)$ (think of $\ell (α_n)=n!^{n!}$) would allow the geodesic flow to escape to infinity, thus making $X$ not parabolic. Contrary to this and motivated by their proof of the Surface Subgroup Theorem, Kahn and Marković conjectured that for every choice of lengths $\ell (α_n)$, there is a choice of twists that would make $X$ parabolic. We show that their conjecture is essentially true. Namely, for any sequence of positive numbers $\{ a_n\}$, there is a choice of lengths $\ell (α_n)\geq a_n$ such that the (relative) twists by $1/2$ make $X$ parabolic. This result extends to the surfaces with countably many ends while it does not hold for uncountably many ends.

Inducing recurrent flows by twisting on infinite surfaces with unbounded cuffs

Abstract

A Riemann surface is parabolic if and only if the geodesic flow (for the hyperbolic metric) on the unit tangent bundle of is ergodic. Consider a Riemann surface with a single topological end and a sequence of pairwise disjoint, simple closed geodesics converging to the end, called {\it cuffs}. Basmajian, the first and the third author, proved that when the lengths of cuffs are at most , the surface is parabolic. One could expect that having arbitrary large cuff lengths (think of ) would allow the geodesic flow to escape to infinity, thus making not parabolic. Contrary to this and motivated by their proof of the Surface Subgroup Theorem, Kahn and Marković conjectured that for every choice of lengths , there is a choice of twists that would make parabolic. We show that their conjecture is essentially true. Namely, for any sequence of positive numbers , there is a choice of lengths such that the (relative) twists by make parabolic. This result extends to the surfaces with countably many ends while it does not hold for uncountably many ends.

Paper Structure

This paper contains 8 sections, 13 theorems, 40 equations, 11 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Tight flute surfaces
  3. Non-planar surfaces
  4. Overview and open questions
  5. Preliminaries
  6. A reduction from countably many to a single end
  7. Parabolic flute surfaces with arbitrary large cuffs
  8. Loch-Ness monster surfaces and basic end surfaces

Key Result

Theorem 1.2

For every non-decreasing sequence $\ell_n$ there is a sequence $\ell_n'\geq\ell_n$ such that the half-twist flute surface $X(\{\ell_n'\},\{1/2\})$ is parabolic.

Figures (11)

  • Figure 1: A flute surface
  • Figure 2: The infinite Loch-Ness monster surface
  • Figure 3: A surface with finitely many non-planar ends.
  • Figure 4: A decomposition into subsurfaces.
  • Figure 5: A decomposition into subsurfaces.
  • ...and 6 more figures

Theorems & Definitions (23)

  • Conjecture 1.1: Kahn-Marković
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Conjecture 1.8: Basmajian, Hakobyan, Pandazis, Šarić
  • Definition 3.1
  • Theorem 3.2
  • proof
  • ...and 13 more