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Closed geodesics in dilation surfaces

Adrien Boulanger, Selim Ghazouani, Guillaume Tahar

Abstract

We prove that directions of closed geodesics in every dilation surface form a dense subset of the circle. The proof draws on a study of the degenerations of the Delaunay triangulation of dilation surfaces under the action of Teichmüller flow in the moduli space.

Closed geodesics in dilation surfaces

Abstract

We prove that directions of closed geodesics in every dilation surface form a dense subset of the circle. The proof draws on a study of the degenerations of the Delaunay triangulation of dilation surfaces under the action of Teichmüller flow in the moduli space.

Paper Structure

This paper contains 29 sections, 26 theorems, 6 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Affine structures on surfaces
  3. Dilation surfaces vs general affine surfaces
  4. Action of $SL(2,\mathbb{R})$ and strategy of proof.
  5. An important shortcoming and an open problem
  6. Acknowledgements.
  7. Dilation surfaces
  8. Dilation cones
  9. Generalities
  10. Cylinders
  11. Moduli of cylinders
  12. Pencils
  13. Non-polygonable surfaces
  14. Action of $SL(2,\mathbb{R})$
  15. Delaunay polygonations
  16. ...and 14 more sections

Key Result

Theorem 1.1

For any closed dilation surface $\Sigma$, there is a dense set of directions $\theta$ such that the directional foliation $\mathcal{F}_{\theta}$ has a periodic leaf. Equivalently, the set of directions covered by a cylinder is dense in $\mathbb{RP}^{1}$.

Figures (16)

  • Figure 1: The sides of the two polygons are glued according to their names. Topologically, the resulting surface has genus two and has only one singularity which corresponds to the extremal points of these two polygons.
  • Figure 2: On the left a flat cylinder with a closed geodesic in dashed (corresponding to the only direction in which there is a closed geodesic). On the right a dilation cylinder with two closed geodesics of two different directions.
  • Figure 3: A fundamental domain for the action of $z \mapsto 2 z$ on a cone of angle $\theta > \pi$. Any trajectory entering the cylinder is trapped within it forever regardless of the direction of the trajectory, as the one represented here. This property prevents a polygonation to "connect" both sides of the cylinder.
  • Figure 4: A Delaunay disk with four boundary singularities. Their convex hull in the disk is a face of the Delaunay polygonation.
  • Figure 5: The three first polygons of a degenerating sequence of type 1 whose vertices all converge toward $s_{\infty}$.
  • ...and 11 more figures

Theorems & Definitions (61)

  • Theorem 1.1
  • Corollary 1.2
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Corollary 2.6
  • ...and 51 more