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Lyapunov exponent and bounds for the Dausdorff dimension of Julia sets of polynomial maps

Показатель Ляпунова и границы для размерности Хаусдорфа для систем джулия полиномиальных отображений.

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Il Nuovo Cimento B (1971-1996)

Summary

We give some methods to get bounds for the Hausdorff dimension of the Julia setJ of polynomial hyperbolic maps. In this caseJ is an Axiom-A repeller and a quasi-self-similar fractal: using thermodynamic formalism it is possible to get some relations which involve the Hausdorff dimension, the escape rate and the Lyapunov exponent of the balanced measure onJ.

Riassunto

Si danno alcuni metodi per stimare la dimensione di Hausdorff dell’insieme di Julia di trasformazioni polinomiali iperboliche. In questo caso l’insieme di Julia è un repulsore Assioma-A con una struttura frattale autosimilare. Usando il formalismo termodinamico è possibile ottenere alcune relazioni fra la dimensione di Hausdorff, la velocità di fuga e l’esponente di Lyapunov della misura bilanciata sull’insieme di Julia.

Резюме

Мы приводим некоторые методы получения границ для размерности Хаусдорфа для системы ДжулияJ полиномиальных гиперболических отображений. Используя термодинамический формализм, можно получить некоторые соотношения, которые включают размерность Хаусдорфа, интенсивность утечки и показатель Ляпунова для сбалансированной меры наJ.

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References

  1. H. Brolin:Ark. Mat.,6, 103 (1965).

    Article  MathSciNet  Google Scholar 

  2. R. Mañé, P. Sád andD. Sullivan:Ann. Sci. Econ. Norm. Sup.,16, 193 (1983).

    MATH  Google Scholar 

  3. J. Guckenheimer: inProceedings of Symposium in Pure Mathematics, Am. Math. Soc.,14, 95 (1970).

  4. D. Ruelle:Thermodynamic Formalism (Addison-Wesley, Reading, Mass., 1978).

    Google Scholar 

  5. D. Ruelle:J. Ergodic Theory Dyn. Sys.,2, 99 (1982).

    Article  MathSciNet  MATH  Google Scholar 

  6. S. Pelikan:Am. Math. Soc.,292, 695 (1975).

    Article  MathSciNet  MATH  Google Scholar 

  7. L. Galgani, A. Giorgilli, D. Casati andL. Sironi:Phys. Lett. A,115, 5 (1986).

    MathSciNet  Google Scholar 

  8. S. McDonald, C. Grebogi, E. Ott andJ. A. Yorke:Physica (Utrecht) D,17, 125 (1985).

    Article  MathSciNet  ADS  MATH  Google Scholar 

  9. M’Ju. Ljubich:J. Ergodic Theory Dyn. Sys.,3, 351 (1983).

    MathSciNet  Google Scholar 

  10. P. Blanchard:Bull. Am. Math. Soc.,11 1, 85 (1984).

    Article  MathSciNet  Google Scholar 

  11. M. Tsuji:Potential Theory in Modern Function Theory (Maruzen, 1959).

  12. A. Manning:Ann. Math. (N. Y.),119, 425 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  13. P. Walters:Am. J. Math.,97, 937 (1975).

    Article  MathSciNet  Google Scholar 

  14. D. Sullivan: inProceedings of the International Conference on Dynamical Systems, Rio de Janeiro, Lectures Notes in Math.,1007 (1983).

  15. P. Walters:Am. Math. Soc.,236, 121 (1978).

    Article  MathSciNet  Google Scholar 

  16. P. Szépfalusy andT. Tél:Phys. Rev. A,34, 820 (1986).

    Article  Google Scholar 

  17. T. S. Pitcher andJ. R. Kinney:Ark. Mat.,8, 25 (1968).

    Article  MathSciNet  Google Scholar 

  18. L. Kadanoff andC. Tang:Proc. Natl. Acad. Sci. USA,81, 1276 (1984).

    Article  MathSciNet  ADS  Google Scholar 

  19. M. Widom, D. Bensimon, L. P. Kadanoff andS. J. Shenker:J. Stat. Phys.,32, 3 (1983).

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Dipartimento di Fisica dell’Università, Bologna, Italia

    S. Vaienti

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  1. S. Vaienti
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Vaienti, S. Lyapunov exponent and bounds for the Dausdorff dimension of Julia sets of polynomial maps. Nuov Cim B 99, 77–91 (1987). https://doi.org/10.1007/BF02827406

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  • DOI: https://doi.org/10.1007/BF02827406

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