Skip to main content
SpringerLink
Log in

On Hausdorff Dimension of Unimodal Attractors

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

There exists a universal constant σ<1 such that every attractor of every C 4 unimodal map with a non-degenerate critical point is an analytic manifold or its Hausdorff dimension is equal to or less than σ.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bruin, H., Keller, G., Nowicki, T., van Strien, S.: Wild Cantor attractors exist. Ann. of Math. (2) 143, no. 1, 97–130 (1996)

  2. Doyle, P.: On the bass note of a Schottky group. Acta. Math. 160, no. 3–4, 249–284 (1988)

  3. Feigenbaum, M.: Quantitative universality for a class of nonlinear transformations. J. Statist. Phys. 19, no. 1, 25–52 (1978)

  4. Graczyk, J., Sands, D., Światek, G.: Metric attractors for smooth unimodal maps. Annals of Math. 159, 725–740 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  5. Graczyk, J., Sands, D., Światek, G.: Decay of geometry for unimodal maps: negative Schwarzian case. Annals of Math.161, 613–677 (2005)

    Google Scholar 

  6. Graczyk, J., Światek, G.: Induced expansion for quadratic polynomials. Ann. Scient. Éc. Norm. Sup. 29, 399–482 (1996)

    MATH  MathSciNet  Google Scholar 

  7. Graczyk, J., Światek, G.: Critical circle maps near bifurcation. Commun. Math. Phys. 127, 227–260 (1996)

    Article  ADS  MathSciNet  Google Scholar 

  8. Graczyk, J., Światek, G.: The real Fatou conjecture. Princeton, NJ: Princeton University Press, 1998

  9. Grassberger, P.: On the Hausdorff dimension of fractal attractors. J. Statist. Phys. 26, no. 1, 173–179 (1981)

  10. Jakobson, M., Światek, G.: Metric properties of non-renormalizable S-unimodal maps. I. Induced expansion and invariant measures. Erg. Th. Dyn. Sys. 14, 721–755 (1994)

    MATH  MathSciNet  Google Scholar 

  11. Kozlovski, O.: Getting rid of the negative Schwarzian derivative condition. Ann. of Math. 152, 743–762 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  12. Lyubich, M.: Combinatorics, geometry and attractors of quasi-quadratic maps. Ann. of Math. 140, 347–404 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  13. Martens, M., de Melo, W., van Strien, S.: Julia-Fatou-Sullivan theory for real one-dimensional dynamics. Acta Math. 168, no. 3–4, 273–318 (1992)

    Google Scholar 

  14. de Melo, W., van Strien, S.: One-dimensional dynamics. Berlin-Heidelberg-New York: Springer-Verlag, 1993

  15. Milnor, J.: On the concept of attractor. Commun. Math. Phys. 99, no. 2, 177–195 (1985)

  16. Mihalache, N.: Hausdorff dimension of fractal attractors. Preprint, 2003

  17. Nowicki, T., Sands, D.: Non-uniform hyperbolicity and universal bounds for S-unimodal maps. Invent. Math. 132, no. 3, 633–680 (1998)

    Google Scholar 

  18. Patterson, S.J.: The limit set of a Fuchsian group. Acta Math. 136, no. 3–4, 241–273 (1976)

  19. Sullivan, D.: Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups. Acta Math. 153, no. 3–4, 259–277 (1984)

  20. Sullivan, D.: Bounds, quadratic differentials, and renormalization conjectures. American Mathematical Society centennial publications. Vol. II Providence, RI: Amer. Math. Soc., 1992, pp. 417–466

Download references

Author information

Authors and Affiliations

  1. Université de Paris-Sud, Mathématiques, 91405, Orsay, France

    J. Graczyk

  2. Mathematical Institute, Warwick University, England

    O.S. Kozlovski

Authors
  1. J. Graczyk
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. O.S. Kozlovski
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by G. Gallavotti

Rights and permissions

Reprints and permissions

About this article

Cite this article

Graczyk, J., Kozlovski, O. On Hausdorff Dimension of Unimodal Attractors. Commun. Math. Phys. 264, 565–581 (2006). https://doi.org/10.1007/s00220-006-1540-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-006-1540-9

Keywords

Search

Navigation