Abstract
We give a simple proof of ergodicity of eigenfunctions of the Laplacian with Dirichlet boundary conditions on compact Riemannian manifolds with piecewise smooth boundaries and ergodic billiards. Examples include the “Bunimovich stadium”, the “Sinai billiard” and the generic polygonal billiard tables of Kerckhoff, Masur and Smillie.
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Bunimovich, L.A.: On the ergodic properties of nowhere dispersive billiards. Commun. Math. Phys.65, 295–312 (1979)
Chazarain, J.: Construction de la paramétrix du problème mixte hyperbolique pour l'équation des ondes. C.R. Acad. Sci. Paris276, 1213–1215 (1973)
Colin de Verdière, Y.: Ergodicité et fonctions propres du laplacien. Commun. Math. Phys.102, 497–502 (1985)
Cornfeld, I.P., Fomin, S.V., Sinai, Ya.G.: Ergodic Theory. Grundlehren Math. Wiss.245, Berlin: Springer, 1982
Dodziuk, J.: Eigenvalues of the Laplacian and the heat equation. Am. Math. Monthly88, 686–695 (1981)
Gérard, P., Leichtnam, E.: Ergodic properties of eigenfunctions for the Dirichlet problem. Duke Math. J.71(2), 559–607 (1993)
Farris, M.: Egorov's theorem on a manifold with diffractive boundary. Comm. P.D.E.6(6), 651–687 (1981)
Guillemin, V., Melrose, R.B.: A cohomological invariant of discrete dynamical systems. In: Christoffel Centennial Volume, P.I. Putzer, F. Feher, eds. Basel: Birkhäser, 1981, pp. 672–679
Hörmander, L.: The Analysis of Linear Partial Differential Operators. Vols.3 and4, Grundlehren Math. Wiss.274 and275, Berlin, Heidelberg, New York: Springer, 1986
Kenig, C., Pipher, J.: Theh-path distribution of the lifetime of conditioned Brownian motion for non-smooth domains. Probability Th. Rel. Fields82, 615–623 (1989)
Kerckhoff, S., Masur, H., Smillie, J.: Ergodicity of billiard flows and quadratic differentials. Ann. Math.124, 293–311 (1986)
Melrose, R.B., Sjöstrand, J.: Singularities in boundary value problems I. Comm. Pure Appl. Math.31, 593–617 (1978)
Petkov, V., Stoyanov, L.N.: Geometry of Reflecting Rays and Inverse Spectral Problems. New York: John Wiley and Sons, 1992
Schnirelman, A.I.: Ergodic properties of eigenfunctions. Usp. Math. Nauk29, 181–182 (1974)
Sinai, Ya.G.: Dynamical systems with elastic reflections. Ergodic properties of dispersive billiards. Usp. Mat. Nauk25, 141–192 (1970), Russ. Math. Surv.25, 137–189 (1970)
Walters, P.: An Introduction to Ergodic Theory. Grad. Texts in Math.79, Berlin, Heidelberg, New York: Springer, 1982
Zelditch, S.: Uniform distribution of eigenfunctions on compact hyperbolic surfaces. Duke Math. J.55, 919–941 (1987)
Zelditch, S.: Quantum ergodicity ofC * dynamical systems, to appear in: Commun. Math. Phys.
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Communicated by Ya.G. Sinai
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Zelditch, S., Zworski, M. Ergodicity of eigenfunctions for ergodic billiards. Commun.Math. Phys. 175, 673–682 (1996). https://doi.org/10.1007/BF02099513
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DOI: https://doi.org/10.1007/BF02099513