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Singularity of Projections of 2-Dimensional Measures Invariant Under the Geodesic Flow

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Abstract

We show that on any compact Riemann surface with variable negative curvature there exists a measure which is invariant and ergodic under the geodesic flow and whose projection to the base manifold is 2-dimensional and singular with respect to the 2-dimensional Lebesgue measure.

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References

  1. Abramov L.: On the entropy of a flow (Russian). Dokl. Akad. Nauk SSSR 128, 873–875 (1959)

    MathSciNet  MATH  Google Scholar 

  2. Anantharaman N., Nonnenmacher S.: Half-delocalization of eigenfunctions for the Laplacian on an Anosov manifold. Ann. Inst. Fourier 57, 2465–2723 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Anosov D.: Geodesic flows on closed Riemann manifolds with negative curvature. Proc. Steklov Inst. Math. 90, 3–210 (1967)

    Google Scholar 

  4. Bowen R.: Symbolic dynamics for hyperbolic flows. Amer. J. Math. 95, 429–460 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  5. Holland M., Melbourne I.: Central limit theorems and invariance principles for Lorenz attractors. J. London Math. Soc. 76(2), 345–364 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Katok, A., Hasselblatt, B.: Introduction to the modern theory of dynamical systems, Encyclopedia of Mathematics and its Applications, 54, Cambridge: Cambridge University Press, 1995

  7. Ledrappier F., Lindenstrauss E.: On the projections of measures invariant under the geodesic flow. Int. Math. Res. Not. 9, 511–526 (2003)

    Article  MathSciNet  Google Scholar 

  8. Lindenstrauss E.: Invariant measures and quantum unique ergodicity. Ann. Math. 163, 165–219 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. Livšic A.: Homology properties of y-Systems. Math. Notes 10, 754–757 (1971)

    Google Scholar 

  10. Mattila P.: Geometry of Sets and Measures in Euclidean Spaces Fractals and rectifiability. Cambridge University Press, Cambridge (1995)

    MATH  Google Scholar 

  11. Melbourne I., Nicol M.: Almost sure invariance principle for nonuniformly hyperbolic systems. Comm. Math. Phys. 260, 131–146 (2005)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  12. Melbourne I., Nicol M.: A vector-valued almost sure invariance principle for hyperbolic dynamical systems. Ann. Probab. 37, 478–505 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Mörters P., Peres Y.: Brownian motion, Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge (2010)

    Google Scholar 

  14. Pesin Ya., Sadovskaya V.: Multifractal analysis of conformal Axiom A flows. Comm. Math. Phys. 216, 277–312 (2001)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  15. Przytycki F., Urbański M., Zdunik A.: Gibbs and Hausdorff measures on repellers for holomorphic maps I. Ann. Math. 130, 1–40 (1989)

    Article  MATH  Google Scholar 

  16. Ratner M.: Markov decomposition for an u-flow on a 3-dimensional manifold. Math. Notes 6, 880–887 (1969)

    MathSciNet  Google Scholar 

  17. Ratner M.: Markov partitions for Anosov flows on n-dimensional manifolds. Israel J. Math. 15, 92–114 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  18. Rivière G.: Entropy of semiclassical measures in dimension 2. Duke Math. J. 155, 271–335 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ruelle D.: Thermodynamic Formalism Second edition. University Press, Cambridge (2004)

    Book  Google Scholar 

  20. Series C.: Symbolic dynamics for geodesic flows. Acta Math. 146, 103–128 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  21. Walters, P.: An introduction to ergodic theory. Graduate Texts in Math. 79. Berlin-Heidelberg-New York: Springer-Verlag, 1982

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Helsinki, P. O. Box 68, 00014, Helsinki, Finland

    Risto Hovila

  2. Department of Mathematical Sciences, University of Oulu, P. O. Box 3000, 90014, Oulu, Finland

    Esa Järvenpää & Maarit Järvenpää

  3. LPMA, UMR 7599, Université Paris 6, 4, Place Jussieu, 75252, Paris Cedex 05, France

    François Ledrappier

Authors
  1. Risto Hovila
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  2. Esa Järvenpää
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  3. Maarit Järvenpää
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  4. François Ledrappier
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Corresponding author

Correspondence to Risto Hovila.

Additional information

Communicated by G. Gallavotti

We acknowledge the Centre of Excellence in Analysis and Dynamics Research supported by the Academy of Finland. RH acknowledges the support from Jenny and Antti Wihuri Foundation. FL acknowledges the partial support from NSF Grant DMS-0801127.

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Hovila, R., Järvenpää, E., Järvenpää, M. et al. Singularity of Projections of 2-Dimensional Measures Invariant Under the Geodesic Flow. Commun. Math. Phys. 312, 127–136 (2012). https://doi.org/10.1007/s00220-011-1387-6

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  • DOI: https://doi.org/10.1007/s00220-011-1387-6

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