More Like this

1. Outer billiards in the spaces of oriented geodesics of the three‐dimensional space forms

   https://doi.org/10.1112/jlms.12922  

   Godoy, Yamile ; Harrison, Michael ; Salvai, Marcos ( June 2024 , Journal of the London Mathematical Society)

   Let be the three‐dimensional space form of constant curvature , that is, Euclidean space , the sphere , or hyperbolic space . Let be a smooth, closed, strictly convex surface in . We define an outer billiard map on the four‐dimensional space of oriented complete geodesics of , for which the billiard table is the subset of consisting of all oriented geodesics not intersecting . We show that is a diffeomorphism when is quadratically convex. For , has a Kähler structure associated with the Killing form of . We prove that is a symplectomorphism with respect to its fundamental form and that can be obtained as an analogue to the construction of Tabachnikov of the outer billiard in defined in terms of the standard symplectic structure. We show that does not preserve the fundamental symplectic form on associated with the cross product on , for . We initiate the dynamical study of this outer billiard in the hyperbolic case by introducing and discussing a notion of holonomy for periodic points.

    

   more » « less
2. Teichmüller dynamics and unique ergodicity via currents and Hodge theory

   https://doi.org/10.1515/crelle-2019-0037  

   McMullen, Curtis T. ( December 2019 , Journal für die reine und angewandte Mathematik (Crelles Journal))

   null (Ed.)

   Abstract We present a cohomological proof that recurrence of suitable Teichmüller geodesics impliesunique ergodicity of their terminal foliations.This approach also yields concrete estimates for periodic foliations andnew results for polygonal billiards. 

   more » « less
3. Hodge and Teichmüller

   https://doi.org/10.3934/jmd.2022007  

   Kahn, Jeremy ; Wright, Alex ( January 2022 , Journal of Modern Dynamics)

   We consider the derivative \begin{document}$ D\pi $\end{document} of the projection \begin{document}$ \pi $\end{document} from a stratum of Abelian or quadratic differentials to Teichmüller space. A closed one-form \begin{document}$ \eta $\end{document} determines a relative cohomology class \begin{document}$ [\eta]_\Sigma $\end{document}, which is a tangent vector to the stratum. We give an integral formula for the pairing of \begin{document}$ D\pi([\eta]_\Sigma) $\end{document} with a cotangent vector to Teichmüller space (a quadratic differential). We derive from this a comparison between Hodge and Teichmüller norms, which has been used in the work of Arana-Herrera on effective dynamics of mapping class groups, and which may clarify the relationship between dynamical and geometric hyperbolicity results in Teichmüller theory.

    

   more » « less
4. COARSE AND FINE GEOMETRY OF THE THURSTON METRIC

   https://doi.org/10.1017/fms.2020.3  

   DUMAS, DAVID ; LENZHEN, ANNA ; RAFI, KASRA ; TAO, JING ( January 2020 , Forum of Mathematics, Sigma)

   We study the geometry of the Thurston metric on the Teichmüller space of hyperbolic structures on a surface $S$ . Some of our results on the coarse geometry of this metric apply to arbitrary surfaces $S$ of finite type; however, we focus particular attention on the case where the surface is a once-punctured torus. In that case, our results provide a detailed picture of the infinitesimal, local, and global behavior of the geodesics of the Thurston metric, as well as an analogue of Royden’s theorem. 

   more » « less
5. Limit sets of Weil–Petersson geodesics with nonminimal ending laminations

   https://doi.org/10.1142/S1793525319500456  

   Brock, Jeffrey ; Leininger, Christopher ; Modami, Babak ; Rafi, Kasra ( March 2020 , Journal of Topology and Analysis)

   null (Ed.)

   In this paper, we construct examples of Weil–Petersson geodesics with nonminimal ending laminations which have [Formula: see text]-dimensional limit sets in the Thurston compactification of Teichmüller space. 

   more » « less