Download PDF - Quelques résultats sur la dimension de Hausdorff des ensembles de Julia des polynômes quadratiquesAll rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

Quelques résultats sur la dimension de Hausdorff des ensembles de Julia des polynômes quadratiques

Authors 1, 1

Affiliations

  1. MAPMO, URA CNRS 1803, Université d'Orléans, BP 6759, 45067 Orléans Cedex 2, France

Abstract

This paper deals with the Hausdorff dimension of the Julia set of quadratic polynomials. It is divided in two parts. The first aims to compute good numerical approximations of the dimension for hyperbolic points. For such points, Ruelle's thermodynamical formalism applies, hence computing the dimension amounts to computing the zero point of a pressure function. It is this pressure function that we approximate by a Monte-Carlo process combined with a shift method that considerably decreases the computational cost. The second part is a continuity result of the dimension on the real axis at the parabolic point 1/4 for Pc(z)=z2+c.

Bibliography

  1. J. Aaronson, M. Denker and M. Urbański, Ergodic theory for Markov fibered systems and parabolic rationnal maps, Trans. Amer. Math. Soc. 337 (1993), 495-548.
  2. N. Bouleau and D. Lépingle, Numerical Methods for Stochastic Processes, Wiley, 1994.
  3. R. Bowen, Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Math. 470, Springer, 1975.
  4. M. Denker and M. Urbański, Hausdorff and conformal measures on Julia sets with a rationally indifferent fixed point, J. London Math. Soc. (2) 43 (1991), 107-118.
  5. M. Denker and M. Urbański, Absolutely continuous invariant measures for expansive rational maps with rationally indifferent fixed points, Forum Math. 3 (1991), 561-579.
  6. A. Douady et J. Hubbard, Etude dynamique des polynômes complexes I, II, Publ. Math. d'Orsay 84-02, 85-04.
  7. L. Garnett, A computer algorithm for determining the Hausdorff dimension of certain fractals, Math. Comp. 51 (1988), 291-300.
  8. T. Ransford, Variation of Hausdorff dimension of Julia sets, Ergodic Theory Dynam. Systems 13 (1993), 167-174.
  9. M. Reed and B. Simon, Methods of Modern Mathematical Physics, tome IV, Academic Press, 1978.
  10. D. Ruelle, Repellers for real analytic maps, Ergodic Theory Dynam. Systems 2 (1982), 99-108.
  11. M. Shishikura, The boundary of the Mandelbrot set has Hausdorff dimension two, in: Complex Analytic Methods in Dynamical Systems, Astérisque 222 (1994), 389-405.
Fundamenta Mathematicae
1996/151/2
Export to PBN, Opens in new tab
Pages:
121-137
Main language of publication
French
Received
1995-06-26
Accepted
1996-04-04
Published
1996
Exact and natural sciences