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On hausdorff dimension of some cantor attractors

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Abstract

We study what happens with the dimension of Feigenbaum-like attractors of smooth unimodal maps as the order of the critical point grows.

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References

  1. H. Bruin, G. Keller, T. Nowicki and S. Van Strien,Wild Cantor attractors exist, Annals of Mathematics143 (1996), 97–130.

    Article  MATH  MathSciNet  Google Scholar 

  2. P. Collet and J.-P. Eckmann,Iterated Maps on the Interval as Dynamical Systems, Progress in Physics, Birkhäuser, Boston, 1980.

    Google Scholar 

  3. M. Feigenbaum,Qualitative universality for a class of non-linear transformations, Journal of Statistical Physics19 (1978), 25–52.

    Article  MATH  MathSciNet  Google Scholar 

  4. M. Feigenbaum,The universal metric properties of non-linear transformations, Journal of Statistical Physics21 (1979), 669–706.

    Article  MATH  MathSciNet  Google Scholar 

  5. J. Graczyk and O. S. Kozlovski,Global universality in smooth unimodal maps, Warwick preprint 05/2002, February 2002.

  6. P. Grassberger,On the Hausdorff dimension of fractal attractors, Journal of Statistical Physics26 (1981), 173–179.

    Article  MathSciNet  Google Scholar 

  7. F. Ledrappier and M. Misiurewicz,Dimension of invariant measures for maps with exponent zero, Ergodic Theory and Dynamical Systems5 (1985), 595–610.

    Article  MATH  MathSciNet  Google Scholar 

  8. G. Levin and G. Świątek,Dynamics and universality of unimodal mappings with infinite criticality, matharxiv 0306033, 2003.

  9. W. De Melo and S. van Strien,One-dimensional Dynamics, Springer-Verlag, New York, 1993.

    MATH  Google Scholar 

  10. D. Mauldin and M. Urbanski,Dimensions and measures in infinite iterated function systems, Proceedings of the London Mathematical Society73 (1996), 105–154.

    Article  MATH  MathSciNet  Google Scholar 

  11. C. McMullen,Renormalization and 3-Manifolds which Fiber over the Circle, Annals of Mathematical Studies142, Princeton University Press, 1998.

  12. C. McMullen,Hausdorff dimension and conformal dynamics I: Kleinian groups and strong limits, Journal of Differential Geometry51 (1999), 471–515.

    MATH  MathSciNet  Google Scholar 

  13. C. McMullen,Hausdorff dimension and conformal dynamics II: Geometrically finite rational maps, Commentarii Mathematici Helvetici75 (2000), 535–593.

    Article  MATH  MathSciNet  Google Scholar 

  14. D. Sullivan,Bounds, quadratic differentials and renormalization conjectures, inMathematics into the Twenty-First Century, AMS Centennial Publications, 1991.

  15. S. Van Strien and T. Nowicki,Polynomial maps with a Julia set of positive Lebesgue measure: Fibonacci maps, manuscript, 1994.

  16. E. B. Vul, Ya. G. Sinai and K. M. Khanin,Feigenbaum universality and thermodynamical formalism, Russian Mathematical Surveys39 (1984), 1–40.

    Article  MATH  MathSciNet  Google Scholar 

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Levin, G., Przytycki, F. On hausdorff dimension of some cantor attractors. Isr. J. Math. 149, 185–197 (2005). https://doi.org/10.1007/BF02772540

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  • DOI: https://doi.org/10.1007/BF02772540

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