Dynamical zeta functions for billiards

Chaubet, Yann; Petkov, Vesselin

doi:10.5802/aif.3743

Dynamical zeta functions for billiards
[Fonctions zêta dynamiques pour les billards]

Chaubet, Yann ¹ ; Petkov, Vesselin ²

¹ Université de Nantes, Laboratoire de mathématiques Jean Leray (UMR CNRS 6629) Département de mathématiques, 2 rue de la Houssinière, 44322 Nantes Cédex 3, France
² Université de Bordeaux, Institut de Mathématiques de Bordeaux, 351, Cours de la Libération, 33405 Talence, France

Annales de l'Institut Fourier, Online first, 56 p.

Abstract (VO)
Résumé

Let $D \subset ℝ^{d}, d ⩾ 2,$ be the union of a finite collection of pairwise disjoint strictly convex compact obstacles. Let $μ_{j} \in ℂ, Im μ_{j} > 0$ be the resonances of the Laplacian in the exterior of $D$ with Neumann or Dirichlet boundary condition on $\partial D$ . For $d$ odd, $u (t) = \sum_{j} e^{i | t | μ_{j}}$ is a distribution in $𝒟^{'} (ℝ ∖ {0})$ and the Laplace transforms of the leading singularities of $u (t)$ yield the dynamical zeta functions $η_{N}, η_{D}$ for Neumann and Dirichlet boundary conditions, respectively. These zeta functions play a crucial role in the analysis of the distribution of the resonances. Under a non-eclipse condition, for every $d ⩾ 2$ we show that $η_{N}$ and $η_{D}$ admit a meromorphic continuation to the whole complex plane. In the particular case when the boundary $\partial D$ is real analytic, by using a result of Fried [17], we prove that the function $η_{D}$ cannot be entire. Following Ikawa [29], this implies the existence of a strip ${z \in ℂ : 0 < Im z ⩽ α}$ containing an infinite number of resonances $μ_{j}$ for the Dirichlet problem. Moreover, for $α ≫ 1$ we obtain a lower bound for the resonances lying in this strip.

Soit $d ⩾ 2$ , et $D \subset ℝ^{d}$ une union finie d’obstacles strictement convexes, compacts et deux à deux disjoints. Soient $μ_{j} \in ℂ, Im μ_{j} > 0,$ les résonances du Laplacien à l’extérieur de $D$ avec conditions aux limites de Neumann ou de Dirichlet sur $\partial D$ . Pour $d$ impair, la formule $u (t) = \sum_{j} e^{i | t | μ_{j}}$ définit une distribution de $𝒟^{'} (ℝ ∖ {0})$ . Les transformées de Laplace des singularités principales de $u (t)$ s’expriment comme des fonctions zêta dynamiques $η_{N}$ et $η_{D}$ , associées aux conditions aux limites de Neumann et Dirichlet, respectivement. Ces fonctions zêta jouent un rôle crucial dans l’analyse de la distribution des résonances. Sous une condition de non-éclipse, pour $d ⩾ 2$ quelconque, nous montrons que $η_{N}$ et $η_{D}$ admettent un prolongement méromorphe à tout le plan complexe. Dans le cas particulier où la frontière $\partial D$ est analytique réelle, en utilisant un résultat de Fried [17], nous prouvons que la fonction $η_{D}$ ne peut pas être entière. Ceci implique, d’après un résultat de Ikawa [29], l’existence d’une bande ${z \in ℂ : 0 < Im z ⩽ α}$ contenant un nombre infini de résonances $μ_{j}$ pour le problème de Dirichlet. De plus, pour $α ≫ 1$ , nous obtenons une borne inférieure sur le nombre de résonances se trouvant dans cette bande.

Reçu le : 2022-08-30
Révisé le : 2023-10-19
Accepté le : 2024-04-12
Première publication : 2025-09-22

DOI : 10.5802/aif.3743

Classification : 37D20, 37D05, 37D40, 81U24
Keywords: Dynamical zeta function, Billiard flow, Scattering Resonances, Modified Lax–Phillips conjecture
Mots-clés : Fonction zêta dynamique, Flot de billard, Résonances, Conjecture de Lax–Phillips modifiée

Affiliations des auteurs :

Chaubet, Yann ¹ ; Petkov, Vesselin ²

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Chaubet, Yann; Petkov, Vesselin. Dynamical zeta functions for billiards. Annales de l'Institut Fourier, Online first, 56 p.

Bibliographie
Cité par

[1] Baladi, Viviane Dynamical zeta functions and dynamical determinants for hyperbolic maps, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, 68, Springer, 2018 | MR | Zbl | DOI

[2] Bálint, Péter; Chernov, Nikolai; Szász, Domokos; Tóth, Imre Péter Geometry of multi-dimensional dispersing billiards, Geometric methods in dynamics. I (Astérisque), Volume 286, Société Mathématique de France, 2003, pp. 119-150 | Numdam | Zbl

[3] Bardos, Claude; Guillot, Jean-Claude; Ralston, James La relation de Poisson pour l’équation des ondes dans un ouvert non borné. Application à la théorie de la diffusion, Commun. Partial Differ. Equations, Volume 7 (1982) no. 8, pp. 905-958 | DOI | MR | Zbl

[4] Barkhofen, Sonja; Weich, Tobias; Potzuweit, Alexander; Stöckmann, H.-J.; Kuhl, Ulrich; Zworski, Maciej Experimental observation of the spectral gap in microwave n-disk systems, Phys. Rev. Lett., Volume 110 (2013) no. 16, 164102

[5] Bernstein, Vladimir Leçons sur les progrès récents de la théorie de séries de Dirichlet, professées au Collège de France, Gauthier-Villars, 1933 | Zbl

[6] Bonthonneau, Yannick Guedes; Jézéquel, Malo FBI Transform in Gevrey classes and Anosov flows, Astérisque, 456, Société Mathématique de France, 2025, 233 pages | Zbl

[7] Bowen, Rufus; Ruelle, David The ergodic theory of Axiom A flows, Invent. Math., Volume 29 (1975) no. 3, pp. 181-202 | DOI | MR | Zbl

[8] Burq, Nicolas Contrôle de l’équation des plaques en présence d’obstacles strictement convexes, Mémoire de la Société Mathématique de France, Nouvelle Série, 55, Société Mathématique de France, 1993, 126 pages | Numdam | MR

[9] Chernov, Nikolai; Markarian, Roberto Chaotic billiards, Mathematical Surveys and Monographs, 127, American Mathematical Society, 2006 | DOI | MR | Zbl

[10] Conley, Charles; Easton, Robert Isolated invariant sets and isolating blocks, Trans. Am. Math. Soc., Volume 158 (1971) no. 1, pp. 35-61 | MR | Zbl | DOI

[11] Cvitanović, Predrag; Vattay, Gábor; Wirzba, Andreas Quantum fluids and classical determinants, Classical, Semiclassical and Quantum Dynamics in Atoms (Lecture Notes in Physics), Volume 485, Springer, 1997, pp. 29-62 | DOI | Zbl

[12] Delarue, Benjamin; Schütte, Philipp; Weich, Tobias Resonances and weighted zeta functions for obstacle scattering via smooth models, Ann. Henri Poincaré, Volume 25 (2024) no. 2, pp. 1607-1656 | DOI | MR | Zbl

[13] Dyatlov, Semyon; Guillarmou, Colin Pollicott–Ruelle resonances for open systems, Ann. Henri Poincaré, Volume 17 (2016) no. 11, pp. 3089-3146 | DOI | MR | Zbl

[14] Dyatlov, Semyon; Zworski, Maciej Dynamical zeta functions for Anosov flows via microlocal analysis, Ann. Sci. Éc. Norm. Supér., Volume 49 (2016) no. 3, pp. 543-577 | DOI | MR | Numdam | Zbl

[15] Dyatlov, Semyon; Zworski, Maciej Mathematical Theory of Scattering Resonances, Graduate Studies in Mathematics, 200, American Mathematical Society, 2019 | DOI | MR | Zbl

[16] Faure, Frédéric; Tsujii, Masato The semiclassical zeta function for geodesic flows on negatively curved manifolds, Invent. Math., Volume 208 (2017) no. 3, pp. 851-998 | DOI | MR | Zbl

[17] Fried, David Meromorphic zeta functions for analytic flows, Commun. Math. Phys., Volume 174 (1995) no. 1, pp. 161-190 | DOI | MR | Zbl

[18] Giulietti, Paolo; Liverani, Carlangelo; Pollicott, Mark Anosov flows and dynamical zeta functions, Ann. Math., Volume 178 (2013) no. 2, pp. 687-773 | DOI | Zbl

[19] Gouëzel, Sébastien; Liverani, Carlangelo Compact locally maximal hyperbolic sets for smooth maps: fine statistical properties, J. Differ. Geom., Volume 79 (2008) no. 3, pp. 433-477 | MR | Zbl

[20] Guillarmou, Colin; Mazzucchelli, Marco; Tzou, Leo Boundary and lens rigidity for non-convex manifolds, Am. J. Math., Volume 143 (2021) no. 2, pp. 533-575 | DOI | Zbl

[21] Guillemin, Victor Lectures on spectral theory of elliptic operators, Duke Math. J., Volume 44 (1977) no. 3, pp. 485-517 | Zbl | DOI

[22] Guillemin, Victor; Melrose, Richard The Poisson summation formula for manifolds with boundary, Adv. Math., Volume 32 (1979) no. 3, pp. 204-232 | Zbl | DOI | MR

[23] Hasselblatt, Boris Hyperbolic dynamical systems, North-Holland, 2002, pp. 239-319

[24] Hayman, Walter. K. Meromorphic functions, Oxford Mathematical Monographs, Clarendon Press, 1964, xiv+191 pages | MR | Zbl

[25] Hörmander, Lars The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis, Grundlehren der Mathematischen Wissenschaften, 256, Springer, 1983, ix+391 pages | MR | Zbl

[26] Ikawa, Mitsuru On the poles of the scattering matrix for two strictly convex obstacles, J. Math. Kyoto Univ., Volume 23 (1983) no. 1, pp. 127-194 | MR | Zbl

[27] Ikawa, Mitsuru Trapping obstacles with a sequence of poles of the scattering matrix converging to the real axis, Osaka J. Math., Volume 22 (1985) no. 4, pp. 657-689 | Zbl | MR

[28] Ikawa, Mitsuru Decay of solutions of the wave equation in the exterior of several convex bodies, Ann. Inst. Fourier, Volume 38 (1988) no. 2, pp. 113-146 | DOI | Numdam | Zbl | MR

[29] Ikawa, Mitsuru On the existence of poles of the scattering matrix for several convex bodies, Proc. Japan Acad., Ser. A, Volume 64 (1988) no. 4, pp. 91-93 | MR | Zbl

[30] Ikawa, Mitsuru On the poles of the scattering matrix for several convex bodies, Algebraic analysis, Vol. I, Academic Press Inc., 1988, pp. 243-251 | Zbl | MR

[31] Ikawa, Mitsuru On the distribution of poles of the scattering matrix for several convex bodies, Functional-analytic methods for partial differential equations (Tokyo, 1989) (Lecture Notes in Mathematics), Volume 1450, Springer, 1990, pp. 210-225 | Zbl | DOI | MR

[32] Ikawa, Mitsuru Singular perturbation of symbolic flows and poles of the zeta functions, Osaka J. Math., Volume 27 (1990) no. 2, pp. 281-300 | MR | Zbl

[33] Ikawa, Mitsuru Singular perturbation of symbolic flows and poles of the zeta functions, Addentum, Osaka J. Math., Volume 29 (1992), pp. 161-174 | Zbl

[34] Jin, Long; Tao, Zhongkai Counting Policott–Ruelle resonances for Axiom A flows, Commun. Math. Phys., Volume 406 (2025) no. 2, 26 | Zbl

[35] Jin, Long; Zworski, Maciej A local trace formula for Anosov flows, Ann. Henri Poincaré, Volume 18 (2017) no. 1, pp. 1-35 (with appendices by Frédéric Naud) | Zbl | DOI | MR

[36] Kovachev, Valery Smoothness of the billiard ball map for strictly convex domains near the boundary, Proc. Am. Math. Soc., Volume 103 (1988) no. 3, pp. 856-860 | Zbl | DOI | MR

[37] Lax, Peter D.; Phillips, Ralph S. Scattering theory, Pure and Applied Mathematics, 26, Academic Press Inc., 1989, xii+309 pages (with appendices by Cathleen S. Morawetz and Georg Schmidt) | Zbl | MR

[38] Lin, Kevin K; Zworski, Maciej Quantum resonances in chaotic scattering, Chem. Phys. Lett., Volume 355 (2002) no. 1-2, pp. 201-205

[39] Liverani, Carlangelo; Wojtkowski, Maciej P. Generalization of the Hilbert metric to the space of positive definite matrices, Pac. J. Math., Volume 166 (1994) no. 2, pp. 339-355 | DOI | MR | Zbl

[40] Melrose, Richard Scattering theory and the trace of the wave group, J. Funct. Anal., Volume 45 (1982) no. 1, pp. 29-40 | DOI | MR | Zbl

[41] Morita, Takehiko The symbolic representation of billiards without boundary condition, Trans. Am. Math. Soc., Volume 325 (1991) no. 2, pp. 819-828 | DOI | MR | Zbl

[42] Parry, William; Pollicott, Mark Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque, 187-188, Société Mathématique de France, 1990, 268 pages | Numdam | Zbl

[43] Perron, Oskar Zur Theorie der Dirichletschen Reihen, J. Reine Angew. Math., Volume 134 (1908), pp. 95-143 | DOI | MR | Zbl

[44] Petkov, Vesselin Analytic singularities of the dynamical zeta function, Nonlinearity, Volume 12 (1999) no. 6, pp. 1663-1681 | DOI | MR | Zbl

[45] Petkov, Vesselin Lower bounds on the number of scattering poles for several strictly convex obstacles, Asymptotic Anal., Volume 30 (2002) no. 1, pp. 81-91 | DOI | Zbl | MR

[46] Petkov, Vesselin Dynamical zeta function for several strictly convex obstacles, Can. Math. Bull., Volume 51 (2008) no. 1, pp. 100-113 | DOI | MR | Zbl

[47] Petkov, Vesselin M.; Stoyanov, Luchezar N. Geometry of the generalized geodesic flow and inverse spectral problems, John Wiley & Sons, 2017, xv+410 pages | Zbl | DOI | MR

[48] Potzuweit, Alexander; Weich, Tobias; Barkhofen, Sonja; Kuhl, Ulrich; Stöckmann, H.-J.; Zworski, Maciej Weyl asymptotics: from closed to open systems, Phys. Rev. E, Volume 86 (2012) no. 6, 066205

[49] Schütte, Philipp; Weich, Tobias; Barkhofen, Sonja Meromorphic continuation of weighted zeta functions on open hyperbolic systems, Commun. Math. Phys., Volume 398 (2023) no. 2, pp. 655-678 | DOI | MR | Zbl

[50] Sjöstrand, Johannes A trace formula and review of some estimates for resonances, Microlocal analysis and spectral theory (Lucca, 1996) (NATO ASI Series. Series C. Mathematical and Physical Sciences), Volume 490, Kluwer Academic Publishers, 1997, pp. 377-437 | DOI | Zbl | MR

[51] Stoyanov, Luchezar Spectrum of the Ruelle operator and exponential decay of correlations for open billiard flows, Am. J. Math., Volume 123 (2001) no. 4, pp. 715-759 | DOI | MR | Zbl

[52] Stoyanov, Luchezar Scattering resonances for several small convex bodies and the Lax–Phillips conjecture, Memoirs of the American Mathematical Society, 199, American Mathematical Society, 2009 no. 933, vi+76 pages | Zbl

[53] Stoyanov, Luchezar Non-integrability of open billiard flows and Dolgopyat-type estimates, Ergodic Theory Dyn. Syst., Volume 32 (2012) no. 1, pp. 295-313 | DOI | MR | Zbl

[54] Vodev, Georgi Sharp bounds on the number of scattering poles in even-dimensional spaces, Duke Math. J., Volume 74 (1994) no. 1, pp. 1-17 | Zbl | DOI | MR

[55] Vodev, Georgi Sharp bounds on the number of scattering poles in the two-dimensional case, Math. Nachr., Volume 170 (1994), pp. 287-297 | Zbl | DOI | MR

[56] Wirzba, Andreas Quantum mechanics and semiclassics of hyperbolic n-disk scattering systems, Phys. Rep., Volume 309 (1999) no. 1-2, pp. 1-116 | DOI | MR

[57] Zworski, Maciej Poisson formulae for resonances, Séminaire sur les Équations aux Dérivées Partielles, 1996–1997, Éditions de l’École polytechnique, 1997 (Exp. No. XIII, 14p.) | Numdam | Zbl | MR

[58] Zworski, Maciej Poisson formula for resonances in even dimensions, Asian J. Math., Volume 2 (1998) no. 3, pp. 609-617 | Zbl | DOI | MR

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