On length spectrum rigidity of dispersing billiard systems

Otto Vaughn Osterman

doi:10.3934/jmd.2023025

Article Contents

2023, Volume 19: 847-878. Doi: 10.3934/jmd.2023025

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On length spectrum rigidity of dispersing billiard systems

Otto Vaughn Osterman

Department of Mathematics, University of Maryland, 4176 Campus Dr, College Park, MD 20742, USA

Received: September 14, 2022

Revised: June 12, 2023

Published online: November 15, 2023

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Abstract

We consider the class of dispersing billiard systems in the plane formed by removing three convex analytic scatterers satisfying the non-eclipse condition. The collision map in this system is conjugated to a subshift, providing a natural labeling of periodic points. We study the problem of marked length spectrum rigidity for this class of systems. We show that two such systems have the same marked length spectrum if and only if their collision maps are analytically conjugate to each other near a homoclinic orbit and that two scatterers and the marked length spectrum together uniquely determine the third scatterer. To do so, we conjugate the system to a Birkhoff normal form and show that the length spectral data of a certain class of periodic orbits can be expressed as a type of asymptotic power series expansion. We relate this asymptotic series to the power series of two analytic functions describing the dynamics of the normalized system and show that we can recover the full power series expansions of these functions.

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Mathematics Subject Classification: Primary: 37D99; Secondary: 37C83, 37J46.

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Figure 1. The homoclinic orbit $ \left\{ {{x_k^\infty}} \right\}_{k \in \mathbb{Z}} $ with $ x_k^\infty = \left( {{i_k^\infty, s_k^\infty, r_k^\infty}} \right) $, and the period-$ 2 $ orbit $ \left\{ {{(1,0,0), (2,0,0)}} \right\} $

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Figure 2. The conjugacy between a cyclicity-$ 2 $ orbit of the collision map and the dynamics of the orbit in Birkhoff coordinates

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Figure 3. Commutative diagram used in the proof of Proposition 20

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Figure 4. The process of determining the collision point $ \gamma_3(s_0^n) $ from the point $ \gamma_1(s_1^n) $, the unit normal $ \vec{\nu} $ to $ D_1 $, and the angle of incidence $ \varphi_1^n $

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References

[1]	K. G. Andersson and R. B. Melrose, The propagation of singularities along gliding rays, Invent. Math., 41 (1977), 197-232. doi: 10.1007/BF01403048.
[2]	V. I. Arnold, Mathematical Methods in Classical Mechanics, 2nd ed., Graduate Texts in Mathematics, 60, Springer-Verlag, 1989. doi: 10.1007/978-1-4757-2063-1.
[3]	P. Bálint, J. De Simoi, V. Kaloshin and M. Leguil, Marked length spectrum, homoclinic orbits and the geometry of open dispersing billiards, Comm. Math. Phys., 374 (2020), 1531-1575. doi: 10.1007/s00220-019-03448-x.
[4]	G. D. Birkhoff, Surface transformations and their dynamical applications, Acta Math., 43 (1922), 1-119. doi: 10.1007/BF02401754.
[5]	Y. Colin de Verdiere, Sur les longueurs des trajectories périodiques d'un billard, Séminare de Théorie Spectrale de Géometrétrie, 1 (1982-83), 1-18.
[6]	J. de Simoi, V. Kaloshin and M. Leguil, Marked length spectral determination of analytic chaotic billiards with axial symmetries, preprint, arXiv: 1905.00890, 2019.
[7]	J. de Simoi, V. Kaloshin and Q. Wei, Dynamical spectral rigidity among $Z^2$-symmetric strictly convex domains close to a circle, Ann. of Math. (2), 186 (2017), 277-314. doi: 10.4007/annals.2017.186.1.7.
[8]	I. M. Gessel, A combinatorial proof of the multivariable Lagrange inversion formula, J. Combin. Theory Ser. A, 45 (1987), 178-195. doi: 10.1016/0097-3165(87)90013-6.
[9]	A. Iantchenko, J. Sjöstrand and M. Zworski, Birkhoff normal forms in semi-classical inverse problems, Math. Res. Lett., 9 (2002), 337-362. doi: 10.4310/MRL.2002.v9.n3.a9.
[10]	M. Ikawa, Decay of solutions of the wave equation in the exterior of several convex bodies, Ann. Inst. Fourier, 38 (1988), 113-146. doi: 10.5802/aif.1137.
[11]	M. Kac, Can one hear the shape of a drum?, Amer. Math. Monthly, 73 (1966), 1-23. doi: 10.2307/2313748.
[12]	V. Kaloshin, M. Leguil and K. Zhang, Marked length spectrum determines a generic convex analytic domain, manuscript, 2021.
[13]	A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Mathematics and its Applications, 54, Cambridge University Press, 1995. doi: 10.1017/CBO9780511809187.
[14]	R. B. Melrose and J. Sjöstrand, Singularities in boundary value problems. II, Comm. Pure Appl. Math., 35 (1982), 129-168. doi: 10.1002/cpa.3160350202.
[15]	J. Moser, The analytic invariants of an area-preserving mapping near a hyperbolic fixed point, Comm. Pure Appl. Math., 9 (1956), 673-692. doi: 10.1002/cpa.3160090404.
[16]	V. Petkov and L. Stojanov, Periods of multiple reflecting geodesics and inverse spectral results, Amer. J. Math., 109 (1987), 619-668. doi: 10.2307/2374608.
[17]	V. M. Petkov and L. N. Stoyanov, Geometry of the Generalized Geodesic Flow and Inverse Spectral Problems, 2nd ed, Wiley, 2017. doi: 10.1002/9781119107682.
[18]	S. Zelditch, Inverse spectral problem for analytic domains I: Balian-Bloch trace formula, Comm. Math. Phys., 248 (2004), 357-407. doi: 10.1007/s00220-004-1074-y.
[19]	S. Zelditch, Inverse spectral problem for analytic domains II: $Z_2$-symmetric domains, Ann. of Math. (2), 170 (2009), 205-269. doi: 10.4007/annals.2009.170.205.
[20]	S. Zelditch, Inverse resonance problem for $Z_2$-symmetric analytic obstacles in the plane, in Geometric Methods in Inverse Problems and PDE Control, 289–321, IMA Vol. Math. Appl., 137, Springer-Verlag, 2004. doi: 10.1007/978-1-4684-9375-7_11.