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Hausdorff dimension of Julia sets of complex Hénon mappings

Published online by Cambridge University Press:  19 September 2008

A. Verjovsky  and
H. Wu
A. Verjovsky
Affiliation:
UFR de Mathématiques, Université des Sciences et Technologies de Lille 1, 59655 Villeneuve D'Ascq, Lille, France
H. Wu
Affiliation:
Einstein Chair, City University of New York, 33 West 42 Street, New York, NY 10036-8099, USA
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Abstract

The Hausdorff dimension of closed invariant sets under diffeomorphisms is an interesting concept as it is a measure of their complexity. The theory of holomorphic dynamical systems provides us with many examples of fractal sets and, in particular, a theorem of Ruelle [Ru1] shows that the Hausdorff dimension of the Julia set depends real analytically on f if f is a rational function of ℂ and the Julia set J of f is hyperbolic. In this paper we generalize Ruelle's result for complex dimension two and show the real analytic dependence of the Hausdorff dimension of the corresponding Julia sets of hyperbolic Hénon mappings.

Information

Type
Survey Article
Information
Ergodic Theory and Dynamical Systems , Volume 16 , Issue 4 , August 1996 , pp. 849 - 861
Copyright
Copyright © Cambridge University Press 1996

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References

REFERENCES

[BS]Bedford, E. and Smillie, J.. Polynomial diffeomorphisms of ℂ2: currents, equilibrium measure and hyperbolicity. Invent. Math. 103 (1991), 69–99.CrossRefGoogle Scholar
[Bo1]Bowen, R.. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, (Lecture Notes in Mathematics 470). Springer, 1973.Google Scholar
[Bo2]Bowen, R.. Hausdorff dimension of quasi circles. IHES Publ. Math. 50 (1979), 11–25.CrossRefGoogle Scholar
[Bo3]Bowen, R.. A horseshoe with positive measure. Invent. Math. 29 (1975), 203–204.CrossRefGoogle Scholar
[HP]Hirsch, M. W. and Pugh, C. C.. Stable manifolds and hyperbolic sets. Global Analysis, vol. XIV (Proc. Symp. in Pure Mathematics). AMS, Providence RI, 1970, 133–163.CrossRefGoogle Scholar
[Hu]Hubbard, J. H.. Hénon mappings in the complex domain. Chaotic Dynamics and Fractals. Barnsley, M. and Demko, S., eds. Academic, 1986.Google Scholar
[La]Lawson, H. B. JrMinimal Variables in Real and Complex Geometry. Seminaire de Mathématiques, Université de Montréal, 1974.Google Scholar
[Ma]Mañé, R.The Hausdorff dimension of horseshoes of diffeomorphisms of surfaces. Bull. Braz. Math. Soc. 20 (1990), 1–24.CrossRefGoogle Scholar
[MSS]Mañé, R.Sad, P., and Sullivan, D.. On the dynamics of rational maps. Ann. Sci. École Norm. Sup. 16 (1983), 193–217.CrossRefGoogle Scholar
[Mg]Manning, A.. A relation between Lyapunov exponents, Hausdorff dimension and entropy. Ergod. Th. & Dynam. Sys. 1 (1981), 451–459.CrossRefGoogle Scholar
[MM]Manning, A. and McClusky, H.. Hausdorff dimension for horseshoes. Ergod. Th. & Dynam. Sys. 3 (1983), 251–260.Google Scholar
[Mt]Marstrand, J. M.. The dimension of cartesian product sets. Proc. Cambridge Phil. Soc. 50 (1954), 198–202.CrossRefGoogle Scholar
[PV]Palis, J. and Viana, M.. On the continuity of Hausdorff dimension and limit capacity for horseshoes. Proc. Symp. on Dynamical Systems (Chile, 1986), (Lecture Notes in Mathematics 1331). Springer, 1988, pp. 150–160.Google Scholar
[Ru1]Ruelle, D.. Repellers for real analytic maps. Ergod. Th. & Dynam. Sys. 2 (1982), 99–107.CrossRefGoogle Scholar
[Ru2]Ruelle, D.. Generalized zeta-functions for Axiom A basic sets. Bull. Amer. Math. Soc. 82 (1976), 153–156.CrossRefGoogle Scholar
[Ru3]Ruelle, D.. Thermodynamic Formalism. Addison-Wesley, Reading MA, 1978.Google Scholar
[RS]Ruelle, D. and Sullivan, D.. Currents, flows and diffeomorphisms. Topology 14 (1975), 319–327.CrossRefGoogle Scholar
[Sh]Shub, M.. Global Stability of Dynamical Systems. Springer, 1987.CrossRefGoogle Scholar
[Sm]Smillie, J.. The entropy of polynomial diffeomorphisms of ℂ2. Ergod. Th. & Dynam. Sys. 10 (1990), 823–827.CrossRefGoogle Scholar
[Su]Sullivan, D.. Conformal dynamical systems. Proc. Conf. on Dynamical Systems (Rio de Janeiro, 1981), (Lecture Notes in Mathematics 1007). Springer, 1983, pp. 725–752.Google Scholar
[Wa]Walters, P.. An Introduction to Ergodic Theory. Springer, 1982.CrossRefGoogle Scholar
[Wu]Wu, H.. Complex stable manifolds of holomorphic diffeomorphisms. Indiana U. M. J. 4 (1993), 1349–1358.CrossRefGoogle Scholar