Advertisement

Communications in Mathematical Physics

, Volume 176, Issue 2, pp 307–320 | Cite as

Dimensions of invariant sets of expanding maps

Article

Abstract

We consider a compact invariant set Λ of an expanding map of a manifoldM and give upper and lowerbounds for the Hausdorff Dimension dim H (Λ), and box dimensionsdim B (Λ) and dim B (Λ). These bounds are given in terms of the topological entropy, topological pressure, and uniform Lyapunov exponents of the map.

A measure-theoretic version of these results is also included.

Keywords

Entropy Neural Network Statistical Physic Complex System Nonlinear Dynamics 
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. [B] Bedford, T.: Crinkly curves, Markov partitions and box dimension in self-similar sets. Ph.D. Thesis, University of WarwickGoogle Scholar
  2. [BK] Brin, M., Katok, A.: On local entropy. In: Geometric dynamics. Lecture notes in mathematics No.1007, Berlin, Heidelberg, New York: Springer, 1983, pp. 30–38Google Scholar
  3. [BU] Bedford, T., Urbański, M.: The box and Hausdorff dimension of self-affine sets. Ergodic Theory and Dynamical Systems.10, 627–644 (1990)Google Scholar
  4. [F1] Falconer, K.: The Hausdorff dimension of self-affine fractals. Math. Proc. Camb. Phil. Soc.103, 339–350 (1988)Google Scholar
  5. [F2] Falconer, K.: Fractal geometry-mathematical foundations and applications. New York: Wiley, 1990Google Scholar
  6. [G] Gu, X.: An upper bound for the Hausdorff dimension of a hyperbolic set. Nonlinearity4, 927–934 (1991)Google Scholar
  7. [GL] Gatzouras, D., Lalley, S.: Hausdorff and box dimensions of certain self-affine fractals. Indiana Univ. Math. J.41, 533–568 (1992)Google Scholar
  8. [H] Hutchinson, J.E.: Fractals and self similarity. Indiana Univ. Math. J.30, 713–747 (1981)Google Scholar
  9. [L] Ledrappier, F.: Some relations between dimension and Lyapunov exponents. Commun. Math. Phys.81, 229–238 (1981)Google Scholar
  10. [LY] Ledrappier, F., Young, L-S.: The metric entropy of diffeomorphisms. Ann. Math.122, 509–574 (1985)Google Scholar
  11. [Mc] McMullen, C.: The Hausdorff dimension of general Sierpinski carpets. Nagoya Math. J.96, 1–9 (1984)Google Scholar
  12. [MM] McCluskey, H., Manning, A.: Hausdorff dimension for horseshoe. Ergodic Theory and Dynamical Systems3, 251–260 (1983)Google Scholar
  13. [P] Petersen, K.: Ergodic theory. Cambridge: Cambridge University Press, 1983Google Scholar
  14. [R] Ruelle, D.: Bowen's formula for the Hausdorff dimension of self-similar sets. In: Scaling and self-similarity in physics, Progress in Physics7. Frohlich, J. (ed.). Boston: Birkhäuser, 1983Google Scholar
  15. [W] Walters, P.: An introduction to ergodic theory. Graduate Texts in Mathematics, vol.79. Berlin, Heidelberg, New York: Springer, 1981Google Scholar
  16. [Y] Young, L.-S.: Dimension, entropy and Lyapunov exponents. Ergodic Theory and Dynamical Systems2, 109–129 (1982)Google Scholar

Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • Huyi Hu
    • 1
  1. 1.Department of MathematicsUniversity of MarylandCollege ParkUSA

Personalised recommendations