# American Institute of Mathematical Sciences

July  2022, 42(7): 3187-3232. doi: 10.3934/dcds.2022014

## Introducing sub-Riemannian and sub-Finsler billiards

 1 Universität Heidelberg, Mathematisches Institut, Mathematikon, Im Neuenheimer Feld 205, 69120 Heidelberg, Germany 2 Utrecht University, Department of Mathematics, Budapestlaan 6, 3584 Utrecht, The Netherlands

*Corresponding author: Álvaro del Pino

Received  December 2020 Revised  November 2021 Published  July 2022 Early access  March 2022

Fund Project: 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

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.

Citation: Lucas Dahinden, Álvaro del Pino. Introducing sub-Riemannian and sub-Finsler billiards. Discrete and Continuous Dynamical Systems, 2022, 42 (7) : 3187-3232. doi: 10.3934/dcds.2022014
##### References:
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##### References:
 [1] A. Agrachev, Compactness for sub-Riemannian length minimizers and subanalyticity, Rend. Sem. Mat. Univ. Pol. Torino, 56 (1998), 1–12 (2001). [2] A. A. Agrachev and Y. L. Sachkov, Control Theory from the Geometric Viewpoint, Encyclopaedia of Mathematical Sciences, 87, Springer-Verlag Berlin Heidelberg, 2004. doi: 10.1007/978-3-662-06404-7. [3] A. A. Agrachev and A. V. Sarychev, Abnormal sub-Riemannian geodesics: Morse index and rigidity, Ann. Inst. Henri Poincaré Anal. Non Linéaire, 13 (1996), 635-690.  doi: 10.1016/s0294-1449(16)30118-4. [4] S. Artstein-Avidan, R. Karasev and Y. Ostrover, From symplectic measurements to the Mahler conjecture, Duke Math. J., 163 (2014), 2003-2022.  doi: 10.1215/00127094-2794999. [5] S. Artstein-Avidan and Y. Ostrover, Bounds for Minkowski billiard trajectories in convex bodies, Int. Mat. Res. Not., 2014 (2014), 165-193.  doi: 10.1093/imrn/rns216. [6] D. Barilari, U. Boscain and D. Cannarsa, On the induced geometry on surfaces in 3D contact sub-Riemannian manifolds, ESAIM: Control, Optimisation and Calculus of Variations, 28 (2022), Paper No. 9, 28 pp. doi: 10.1051/cocv/2021104. [7] D. Barilari, U. Boscain, D. Cannarsa and K. Habermann, Stochastic processes on surfaces in three-dimensional contact sub-Riemannian manifolds, Annales de l'Institut Henri Poincaré Probabilités et Statistiques, 57 (2021), 1388–1410. doi: 10.1214/20-aihp1124. [8] A. Belotto da Silva, A. Figalli, A. Parusinski and L. Rifford, Strong Sard Conjecture and regularity of singular minimizing geodesics for analytic sub-Riemannian structures in dimension 3, preprint, arXiv: 1810.03347. [9] Y. Colin de Verdière, L. Hillairet and E. Trélat, Spectral asymptotics for sub-Riemannian Laplacians, I: Quantum ergodicity and quantum limits in the 3-dimensional contact case, Duke Math. J., 167 (2018), 109-174.  doi: 10.1215/00127094-2017-0037. [10] H. Geiges, An Introduction to Contact Topology, Cambridge Studies in Advanced Mathematics, 109, Cambridge University Press, 2008. doi: 10.1017/CBO9780511611438. [11] E. Giroux, Convexitè en topologie de contact, Comment. Math. Helv., 66 (1991), 637-677.  doi: 10.1007/BF02566670. [12] L. Hsu, Calculus of variations via the Griffiths formalism, J. Differential Geom., 36 (1992), 551-589.  doi: 10.4310/jdg/1214453181. [13] B. Khesin and S. Tabachnikov, Pseudo-Riemannian geodesics and billiards, Advances in Mathematics, 221 (2009), 1364-1396.  doi: 10.1016/j.aim.2009.02.010. [14] R. Montgomery, Abnormal minimizers, SIAM Journal on Control and Optimization, 32 (1994), 1605-1620.  doi: 10.1137/S0363012993244945. [15] R. Montgomery, A Tour of Subriemannian Geometries, their Geodesics and Applications, Mathematical Surveys and Monographs, 91, American Mathematical Society, Providence, RI, (2002). doi: 10.1090/surv/091. [16] P. Pansu, Submanifolds and differential forms on Carnot manifolds, after M. Gromov and M. Rumin, preprint, arXiv: 1604.06333. [17] L. Rizzi and T. Rossi, Heat content asymptotics for sub-Riemannian manifolds, J. Math. Pures Appl., 148 (2021), 267–307, preprint, arXiv: 2005.01666. doi: 10.1016/j.matpur.2020.12.004. [18] N. Savale, A Gutzwiller type trace formula for the magnetic Dirac operator, Geom. Funct. Anal., 28 (2018), 1420-1486.  doi: 10.1007/s00039-018-0462-y. [19] I. Zelenko, Nonregular abnormal extremals of 2-distributions: Existence, second variation and rigidity, J. Dynamical and Control systems, 5 (1999), 347–383. doi: 10.1023/A:1021766616913.
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
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
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
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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