American Institute of Mathematical Sciences

October  2015, 35(10): 5055-5082. doi: 10.3934/dcds.2015.35.5055

Computing Mather's $\beta$-function for Birkhoff billiards

 1 Department of Mathematics, Università degli Studi Roma Tor Vergata, Viale della Ricerca Scientifica 1, 00133 Rome, Italy

Received  September 2013 Revised  February 2014 Published  April 2015

This article is concerned with the study of Mather's $\beta$-function associated to Birkhoff billiards. This function corresponds to the minimal average action of orbits with a prescribed rotation number and, from a different perspective, it can be related to the maximal perimeter of periodic orbits with a given rotation number, the so-called Marked length spectrum. After having recalled its main properties and its relevance to the study of the billiard dynamics, we stress its connections to some intriguing open questions: Birkhoff conjecture and the isospectral rigidity of convex billiards. Both these problems, in fact, can be conveniently translated into questions on this function. This motivates our investigation aiming at understanding its main features and properties. In particular, we provide an explicit representation of the coefficients of its (formal) Taylor expansion at zero, only in terms of the curvature of the boundary. In the case of integrable billiards, this result provides a representation formula for the $\beta$-function near $0$. Moreover, we apply and check these results in the case of circular and elliptic billiards.
Citation: Alfonso Sorrentino. Computing Mather's $\beta$-function for Birkhoff billiards. Discrete and Continuous Dynamical Systems, 2015, 35 (10) : 5055-5082. doi: 10.3934/dcds.2015.35.5055
References:
 [1] E. Y. Amiran, Caustics and evolutes for convex planar domains, J. Differential Geom., 28 (1988), 345-357. [2] M. Bialy, Convex billiards and a theorem by E. Hopf, Math. Z., 214 (1993), 147-154. doi: 10.1007/BF02572397. [3] G. D. Birkhoff, Dynamical systems with two degrees of freedom, Trans. Amer. Math. Soc., 18 (1917), 199-300. doi: 10.1090/S0002-9947-1917-1501070-3. [4] G. D. Birkhoff, On the periodic motions of dynamical systems, Acta Math., 50 (1927), 359-379. doi: 10.1007/BF02421325. [5] A. Delshams and R. Ramírez-Ros, Poincaré-Melnikov-Arnold method for analytic planar maps, Nonlinearity, 9 (1996), 1-26. doi: 10.1088/0951-7715/9/1/001. [6] V. Guillemin and R. Melrose, A cohomological invariant of discrete dynamical systems, in E. B. Christoffel (Aachen/Monschau, 1979), Birkhäuser, Basel-Boston, Mass., 1981, 672-679. [7] B. Halpern, Strange billiard tables, Trans. Amer. Math. Soc., 232 (1977), 297-305. doi: 10.1090/S0002-9947-1977-0451308-7. [8] V. Kovachev and G. Popov, Invariant tori for the billiard ball map, Trans. Amer. Math. Soc., 317 (1990), 45-81. doi: 10.1090/S0002-9947-1990-0989578-5. [9] V. F. Lazutkin, Existence of caustics for the billiard problem in a convex domain (in Russian), Izv. Akad. Nauk SSSR Ser. Mat., 37 (1973), 186-216. [10] V. F. Lazutkin, KAM Theory and Semiclassical Approximations to Eigenfunctions, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Vol. 24, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-642-76247-5. [11] S. Marvizi and R. Melrose, Spectral invariants of convex planar regions, J. Differential Geom., 17 (1982), 475-502. [12] D. Massart and A. Sorrentino, Differentiability of Mather's average action and integrability on closed surfaces, Nonlinearity, 24 (2011), 1777-1793. doi: 10.1088/0951-7715/24/6/005. [13] J. N. Mather, Glancing billiards, Ergodic Theory Dynam. Systems, 2 (1982), 397-403. doi: 10.1017/S0143385700001681. [14] J. N. Mather, Differentiability of the minimal average action as a function of the rotation number, Bol. Soc. Brasil. Mat. (N.S.), 21 (1990), 59-70. doi: 10.1007/BF01236280. [15] J. N. Mather and G. Forni, Action minimizing orbits in Hamiltonian systems, in Transition to Chaos in Classical and Quantum Mechanics (Montecatini Terme, 1991), Lecture Notes in Math., 1589, Springer, Berlin, 1994, 92-186. doi: 10.1007/BFb0074076. [16] G. Popov, Invariants of the length spectrum and spectral invariants of planar convex domains, Commun. Math. Phys., 161 (1994), 335-364. doi: 10.1007/BF02099782. [17] J. Pöschel, Integrability of hamiltonian systems on cantor sets, Comm. Pure Appl. Math., 35 (1982), 653-696. doi: 10.1002/cpa.3160350504. [18] G. Sapiro and A. Tannenbaum, On affine place curve evolution, J. Funct. Anal., 119 (1994), 79-120. doi: 10.1006/jfan.1994.1004. [19] K. F. Siburg, The Principle of Least Action in Geometry and Dynamics, Lecture Notes in Mathematics, Vol. 1844, Springer-Verlag, 2004. doi: 10.1007/b97327. [20] A. Sorrentino, Lecture notes on Mather's theory for Lagrangian systems, preprint, arXiv:1011.0590. [21] A. Sorrentino and C. Viterbo, Action minimizing properties and distances on the group of Hamiltonian diffeomorphisms, Geom. Topol., 14 (2010), 2383-2403. doi: 10.2140/gt.2010.14.2383. [22] S. Tabachnikov, Geometry and Billiards, Student Mathematical Library, Vol. 30, American Mathematical Society, 2005. [23] M. B. Tabanov, New ellipsoidal confocal coordinates and geodesics on an ellipsoid, J. Math. Sci., 82 (1996), 3851-3858. doi: 10.1007/BF02362647. [24] M. P. Wojtkowski, Two applications of Jacobi fields to the billiard ball problem, J. Differential Geom., 40 (1994), 155-164.

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References:
 [1] E. Y. Amiran, Caustics and evolutes for convex planar domains, J. Differential Geom., 28 (1988), 345-357. [2] M. Bialy, Convex billiards and a theorem by E. Hopf, Math. Z., 214 (1993), 147-154. doi: 10.1007/BF02572397. [3] G. D. Birkhoff, Dynamical systems with two degrees of freedom, Trans. Amer. Math. Soc., 18 (1917), 199-300. doi: 10.1090/S0002-9947-1917-1501070-3. [4] G. D. Birkhoff, On the periodic motions of dynamical systems, Acta Math., 50 (1927), 359-379. doi: 10.1007/BF02421325. [5] A. Delshams and R. Ramírez-Ros, Poincaré-Melnikov-Arnold method for analytic planar maps, Nonlinearity, 9 (1996), 1-26. doi: 10.1088/0951-7715/9/1/001. [6] V. Guillemin and R. Melrose, A cohomological invariant of discrete dynamical systems, in E. B. Christoffel (Aachen/Monschau, 1979), Birkhäuser, Basel-Boston, Mass., 1981, 672-679. [7] B. Halpern, Strange billiard tables, Trans. Amer. Math. Soc., 232 (1977), 297-305. doi: 10.1090/S0002-9947-1977-0451308-7. [8] V. Kovachev and G. Popov, Invariant tori for the billiard ball map, Trans. Amer. Math. Soc., 317 (1990), 45-81. doi: 10.1090/S0002-9947-1990-0989578-5. [9] V. F. Lazutkin, Existence of caustics for the billiard problem in a convex domain (in Russian), Izv. Akad. Nauk SSSR Ser. Mat., 37 (1973), 186-216. [10] V. F. Lazutkin, KAM Theory and Semiclassical Approximations to Eigenfunctions, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Vol. 24, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-642-76247-5. [11] S. Marvizi and R. Melrose, Spectral invariants of convex planar regions, J. Differential Geom., 17 (1982), 475-502. [12] D. Massart and A. Sorrentino, Differentiability of Mather's average action and integrability on closed surfaces, Nonlinearity, 24 (2011), 1777-1793. doi: 10.1088/0951-7715/24/6/005. [13] J. N. Mather, Glancing billiards, Ergodic Theory Dynam. Systems, 2 (1982), 397-403. doi: 10.1017/S0143385700001681. [14] J. N. Mather, Differentiability of the minimal average action as a function of the rotation number, Bol. Soc. Brasil. Mat. (N.S.), 21 (1990), 59-70. doi: 10.1007/BF01236280. [15] J. N. Mather and G. Forni, Action minimizing orbits in Hamiltonian systems, in Transition to Chaos in Classical and Quantum Mechanics (Montecatini Terme, 1991), Lecture Notes in Math., 1589, Springer, Berlin, 1994, 92-186. doi: 10.1007/BFb0074076. [16] G. Popov, Invariants of the length spectrum and spectral invariants of planar convex domains, Commun. Math. Phys., 161 (1994), 335-364. doi: 10.1007/BF02099782. [17] J. Pöschel, Integrability of hamiltonian systems on cantor sets, Comm. Pure Appl. Math., 35 (1982), 653-696. doi: 10.1002/cpa.3160350504. [18] G. Sapiro and A. Tannenbaum, On affine place curve evolution, J. Funct. Anal., 119 (1994), 79-120. doi: 10.1006/jfan.1994.1004. [19] K. F. Siburg, The Principle of Least Action in Geometry and Dynamics, Lecture Notes in Mathematics, Vol. 1844, Springer-Verlag, 2004. doi: 10.1007/b97327. [20] A. Sorrentino, Lecture notes on Mather's theory for Lagrangian systems, preprint, arXiv:1011.0590. [21] A. Sorrentino and C. Viterbo, Action minimizing properties and distances on the group of Hamiltonian diffeomorphisms, Geom. Topol., 14 (2010), 2383-2403. doi: 10.2140/gt.2010.14.2383. [22] S. Tabachnikov, Geometry and Billiards, Student Mathematical Library, Vol. 30, American Mathematical Society, 2005. [23] M. B. Tabanov, New ellipsoidal confocal coordinates and geodesics on an ellipsoid, J. Math. Sci., 82 (1996), 3851-3858. doi: 10.1007/BF02362647. [24] M. P. Wojtkowski, Two applications of Jacobi fields to the billiard ball problem, J. Differential Geom., 40 (1994), 155-164.
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