Communications in Mathematical Physics

, Volume 294, Issue 2, pp 303–342 | Cite as

Entropic Bounds on Semiclassical Measures for Quantized One-Dimensional Maps



Quantum ergodicity asserts that almost all infinite sequences of eigenstates of quantized ergodic Hamiltonian systems are equidistributed in phase space. This, however, does not prohibit existence of exceptional sequences which might converge to different (non-Liouville) classical invariant measures. It has been recently shown by N. Anantharaman and S. Nonnenmacher in [20,21] (with H. Koch) that for Anosov geodesic flows the metric entropy of any semiclassical measure μ must satisfy a certain bound. This remarkable result seems to be optimal for manifolds of constant negative curvature, but not in the general case, where it might become even trivial if the (negative) curvature of the Riemannian manifold varies a lot. It has been conjectured by the same authors, that in fact, a stronger bound (valid in the general case) should hold.

In the present work we consider such entropic bounds using the model of quantized piecewise linear one-dimensional maps. For a certain class of maps with non-uniform expansion rates we prove the Anantharaman-Nonnenmacher conjecture. Furthermore, for these maps we are able to construct some explicit sequences of eigenstates which saturate the bound. This demonstrates that the conjectured bound is actually optimal in that case.


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  1. 1.
    Bohigas, O.: Random matrix theory and chaotic dynamics. In: Giannoni, M.J., Voros, A., Zinn-Justin, J., eds., Chaos et physique quantique, (École d’été des Houches, Session LII, 1989), Amsterdam: North Holland, 1991Google Scholar
  2. 2.
    Berry M.V.: Regular and irregular semiclassical wave functions. J. Phys. A 10, 2083–2091 (1977)ADSGoogle Scholar
  3. 3.
    Voros, A.: Semiclassical ergodicity of quantum eigenstates in the Wigner representation. In: Stochastic Behavior in Classical and Quantum Hamiltonian Systems, Casati, G., Ford, J., eds., Proceedings of the Volta Memorial Conference, Como, Italy, 1977, Lecture Notes in Phys. 93, Berlin: Springer, 1979, pp. 326–333Google Scholar
  4. 4.
    Lazutkin, V.F.: Semiclassical asymptotics of eigenfunctions. In: Partial Differential Equations V, Berlin: Springer, 1999Google Scholar
  5. 5.
    Schnirelman, A.I.: Ergodic properties of eigenfunctions. Usp. Mat. Nauk 29, no. 6 (180), 181–182 (1974)Google Scholar
  6. 6.
    Zelditch S.: Uniform distribution of the eigenfunctions on compact hyperbolic surfaces. Duke Math. J. 55, 919–941 (1987)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Colinde Verdière Y.: Ergodicité et fonctions propres du laplacien. Commun. Math. Phys. 102, 497–502 (1985)CrossRefGoogle Scholar
  8. 8.
    Gérard P., Leichtnam É.: Ergodic properties of eigenfunctions for the Dirichlet problem. Duke Math. J. 71(2), 559–607 (1993)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Zworski M., Zelditch S.: Ergodicity of eigenfunctions for ergodic billiards. Commun. Math. Phys. 175, 673–682 (1996)MATHCrossRefMathSciNetADSGoogle Scholar
  10. 10.
    Bouzouina A., De Bièvre S.: Equipartition of the eigenfunctions of quantized ergodic maps on the torus. Commun. Math. Phys. 178, 83–105 (1996)MATHCrossRefADSGoogle Scholar
  11. 11.
    Helffer B., Martinez A., Robert D.: Ergodicité et limite semi-classique. Commun. Math. Phys. 109, 313–326 (1987)MATHCrossRefMathSciNetADSGoogle Scholar
  12. 12.
    Rudnick Z., Sarnak P.: The behavior of eigenstates of arithmetic hyperbolic manifolds. Commun. Math. Phys. 161, 195–213 (1994)MATHCrossRefMathSciNetADSGoogle Scholar
  13. 13.
    Lindenstrauss E.: Invariant measures and arithmetic quantum unique ergodicity. Ann. Math. 163, 165–219 (2006)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Hassell, A.: Ergodic billiards that are not quantum unique ergodic, with an appendix by A. Hassell, L. Hillairet. Preprint (2008)[math,AP], 2008, to appear in Ann. of Math
  15. 15.
    Faure F., Nonnenmacher S., De Bièvre S.: Scarred eigenstates for quantum cat maps of minimal periods. Commun. Math. Phys. 239, 449–492 (2003)MATHCrossRefADSGoogle Scholar
  16. 16.
    Faure F., Nonnenmacher S.: On the maximal scarring for quantum cat map eigenstates. Commun. Math. Phys. 245, 201–214 (2004)MATHCrossRefMathSciNetADSGoogle Scholar
  17. 17.
    Anantharaman N., Nonnenmacher S.: Entropy of semiclassical measures of the Walsh-quantized baker’s map. Ann. H. Poincaré 8, 37–74 (2007)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Kelmer, D.: Arithmetic quantum unique ergodicity for symplectic linear maps of the multidimensional torus. Preprint (2005),, 2007, to appear in Ann. of Math.
  19. 19.
    Anantharaman N.: Entropy and the localization of eigenfunctions. Ann. of Math. 168(2), 435–475 (2008)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Anantharaman N., Nonnenmacher S.: Half–delocalization of eigenfunctions of the Laplacian on an Anosov manifold. Ann. de l’Inst. Fourier 57(7), 2465–2523 (2007)MATHMathSciNetGoogle Scholar
  21. 21.
    Anantharaman, N., Nonnenmacher, S., Koch, H.: Entropy of eigenfunctions.[math-ph], 2007
  22. 22.
    Pakoński P., Życzkowski K., Kuś M.: Classical 1D maps, quantum graphs and ensembles of unitary matrices. J. Phys. A 34(43), 9303–9317 (2001)MATHCrossRefMathSciNetADSGoogle Scholar
  23. 23.
    Berkolaiko G., Keating J.K., Smilansky U.: Quantum Ergodicity for Graphs Related to Interval Maps. Commun. Math. Phys. 273, 137–159 (2007)MATHCrossRefMathSciNetADSGoogle Scholar
  24. 24.
    Zyczkowski K., Kuś M., Słomczyński W., Sommers H.-J.: Random unistochastic matrices. J. Phys. A 36(12), 3425–3450 (2003)MATHCrossRefMathSciNetADSGoogle Scholar
  25. 25.
    Keller, G.: Equilibrium States in Ergodic Theory. London Mathematical Society Student Texts 42 Cambridge: Cambridge University Press, 1998Google Scholar
  26. 26.
    De Bièvre, S.: Quantum chaos: a brief first visit. In: Second Summer School in Analysis and Mathematical Physics (Cuernavaca, 2000), Vol. 289 of Contemp. Math., Providence, RI: Amer. Math. Soc., 2001, pp. 161–218Google Scholar
  27. 27.
    Deutsch D.: Uncertainty in quantum measurements. Phys. Rev. Lett. 50, 631–633 (1983)CrossRefMathSciNetADSGoogle Scholar
  28. 28.
    Kraus K.: Complementary observables and uncertainty relations. Phys. Rev. D 35, 3070–3075 (1987)CrossRefMathSciNetADSGoogle Scholar
  29. 29.
    Maassen H., Uffink J.B.M.: Generalized entropic uncertainty relations. Phys. Rev. Lett. 60, 1103–1106 (1988)CrossRefMathSciNetADSGoogle Scholar
  30. 30.
    Baladi, V.: Positive Transfer Operators and Decay of Correlations. Advanced Series in Nonlinear Dynamics, Vol 16, Singapore: World Scientific, 2000Google Scholar
  31. 31.
    Luzzatto, S.: Stochastic-like behavior in non-uniformly expanding maps. In: Handbook of Dynamical Systems, Vol. 1B, B. Hasselblatt and A. Katok (eds.), London: Elsevier, 2006, pp. 265–326Google Scholar
  32. 32.
    Denker M., Holzmann H.: Markov partitions for fibre expanding systems. Colloq. Math. 110, 485–492 (2008)MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Gutkin, B.: Quantum towers and entropic bounds on semiclassical measures. In preparationGoogle Scholar
  34. 34.
    Riviere, G.: Entropy of semiclassical measures in dimension 2.[math-ph], 2008
  35. 35.
    Nonnenmacher S., Rubin M.: Resonant eigenstates for a quantized chaotic system. Nonlinearity 20, 1387–1420 (2007)MATHCrossRefMathSciNetADSGoogle Scholar

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© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Fachbereich PhysikUniversität Duisburg-EssenDuisburgGermany

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