Advertisement

Communications in Mathematical Physics

, Volume 254, Issue 3, pp 695–717 | Cite as

(Non)regularity of Projections of Measures Invariant Under Geodesic Flow

  • Esa  JärvenpääEmail author
  • Maarit Järvenpää
  • Mika Leikas
Article

Abstract

We show that, unlike in the 2-dimensional case [LL], the Hausdorff dimension of a measure invariant under the geodesic flow is not necessarily preserved under the projection from the unit tangent bundle onto the base manifold if the base manifold is at least 3-dimensional. In the 2-dimensional case we reprove the preservation theorem due to Ledrappier and Lindenstrauss [LL] using the general projection formalism of Peres and Schlag [PS]. The novelty of our proof is that it illustrates the reason behind the failure of the preservation in the higher dimensional case. Finally, we show that the projected measure has fractional derivatives of order γ for all γ<(α−2)/2 provided that the invariant measure has finite α-energy for some α>2 and the base manifold has dimension 2.

Keywords

Neural Network Manifold Statistical Physic Complex System Peris 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bonetto, F., Kupiainen, A., Lebowitz, J.L.: Absolute continuity of projected SRB measures of coupled Arnold cat map lattices. To appear in Ergodic Theory Dynam. Systems, http://arxiv.org/abs/nlin/0310009, 2003
  2. 2.
    Bricmont, J., Kupiainen, A.: Coupled analytic maps. Nonlinearity 8, 379–396 (1995)CrossRefGoogle Scholar
  3. 3.
    Bricmont, J., Kupiainen, A.: High temperature expansions and dynamical systems. Commun. Math. Phys. 178, 703–732 (1996)Google Scholar
  4. 4.
    Bunimovich, L.A., Sinai, Ya. G.: Spacetime chaos in coupled map lattices. Nonlinearity 1, 491–516 (1988)CrossRefGoogle Scholar
  5. 5.
    Falconer, K.: Techniques in Fractal Geometry. Chichester: John Wiley & Sons, Ltd., 1997Google Scholar
  6. 6.
    Falconer, K., Mattila, P.: The packing dimension of projections and sections of measures. Math. Proc. Cambridge Philos. Soc. 119, 695–713 (1996)Google Scholar
  7. 7.
    Haase, H.: On the dimension of product measures. Mathematika 37, 316–323 (1990)Google Scholar
  8. 8.
    Hu, X., Taylor, J.: Fractal properties of products and projections of measures in ℝn. Math. Proc. Cambridge Philos. Soc. 115, 527–544 (1994)Google Scholar
  9. 9.
    Hunt, B.R., Kaloshin, Yu., V.: How projections affect the dimension spectrum of fractal measures?. Nonlinearity 10, 1031–1046 (1997)CrossRefGoogle Scholar
  10. 10.
    Hunt, B.R., Kaloshin, Yu., V.: Regularity of embeddings of infinite-dimensional fractal sets into finite-dimensional spaces. Nonlinearity 12, 1263–1275 (1999)CrossRefGoogle Scholar
  11. 11.
    Järvenpää, E.: SRB-measures for coupled map lattices. To appear in Lecture Notes in Phys., SpringerGoogle Scholar
  12. 12.
    Järvenpää, E., Järvenpää, M.: On the definition of SRB-measures for coupled map lattices. Commun. Math. Phys. 220, 1–12 (2001)CrossRefGoogle Scholar
  13. 13.
    Järvenpää, E., Järvenpää, M., Llorente, M.: Local dimensions of sliced measures and stability of packing dimensions of sections of sets. Adv. Math. 183, 127–154 (2004)CrossRefGoogle Scholar
  14. 14.
    Kaufman, R.: On Hausdorff dimension of projections. Mathematika 15, 153–155 (1968)Google Scholar
  15. 15.
    Ledrappier, F., Lindenstrauss, E.: On the projections of measures invariant under the geodesic flow. Int. Math. Res. Not. 9, 511–526 (2003)CrossRefGoogle Scholar
  16. 16.
    Marstrand, M.: Some fundamental geometrical properties of plane sets of fractional dimension. Proc. London Math. Soc. 4, 257–302 (1954)Google Scholar
  17. 17.
    Mattila, P.: Hausdorff dimension, orthogonal projections and intersections with planes. Ann. Acad. Sci. Fenn. Math. 1, 227–244 (1975)Google Scholar
  18. 18.
    Mattila, P.: Orthogonal projections, Riesz capacities and Minkowski content. Indiana Univ. Math. J. 39, 185–198 (1990)Google Scholar
  19. 19.
    Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces: Fractals and rectifiability. Cambridge: Cambridge University Press, 1995Google Scholar
  20. 20.
    Mattila, P.: Hausdorff dimension, projections, and Fourier transform. Publ. Mat. 48, 3–48 (2004)Google Scholar
  21. 21.
    Peres, Y., Schlag, W.: Smoothness of projections, Bernoulli convolutions, and the dimensions of exceptions. Duke Math. J. 102, 193–251 (2000)CrossRefGoogle Scholar
  22. 22.
    Sauer, T. D., Yorke, J. A.: Are dimensions of a set and its image equal for typical smooth functions?. Ergodic Theory Dynam. Systems 17, 941–956 (1997)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Esa  Järvenpää
    • 1
    Email author
  • Maarit Järvenpää
    • 1
  • Mika Leikas
    • 1
  1. 1.Department of Mathematics and StatisticsUniversity of JyväskyläFinland

Personalised recommendations