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Introducing sub-Riemannian and sub-Finsler billiards

  • *Corresponding author: Álvaro del Pino

    *Corresponding author: Álvaro del Pino

The first author is supported by Deutsche Forschungsgemeinschaft (DFG) under Germany's Excellence Strategy EXC-2181/1 - 390900948 (the Heidelberg STRUCTURES Excellence Cluster), as well as by SFB/TRR 191 "Symplectic Structures in Geometry, Algebra and Dynamics" funded by the DFG. The second author was supported by the NWO 016.Veni.192.013 grant for the duration of the project

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  • We define billiards in the context of sub-Finsler Geometry. We provide symplectic and variational (or rather, control theoretical) descriptions of the problem and show that they coincide. We then discuss several phenomena in this setting, including the failure of the reflection law to be well-defined at singular points of the boundary distribution, the appearance of gliding and creeping orbits, and the behavior of reflections at wavefronts.

    We then study some concrete tables in $ 3 $-dimensional euclidean space endowed with the standard contact structure. These can be interpreted as planar magnetic billiards, of varying magnetic strength, for which the magnetic strength may change under reflection. For each table we provide various results regarding periodic trajectories, gliding orbits, and creeping orbits.

    Mathematics Subject Classification: Primary: 53C17.Secondary: 53D25, 37C83.


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  • Figure 1.  The $ n $-gon from Lemma 5.10, with $ n = 6 $, and an arc connecting two of the boundary points (in this case, adjacent also in the hexagon, so $ m = 1 $). The lightly shaded region must have the same area as the sum of the dark regions

    Figure 2.  The situation described in Subsection 5.5.3. A projection of a half-line that reflects at a point in $ H $, and its continuation after the reflection point

    Figure 3.  The bigon from Lemma 5.16 under the assumption that $ q_1 $ appears in the arc before the tangency with respect to the origin. The lightly shaded area is positive and the dark area is negative. In this case, the latter is larger and therefore the trajectory hits the bottom boundary; that is, $ \psi $ is not in the range described in the Lemma

    Figure 4.  The bigon from Lemma 5.16 under the assumption that $ q_1 $ lies beyond the tangency point from the origin. Here the positive area is larger than the negative area, so we obtain one of the trajectories described in the Lemma

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