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Repellers for real analytic maps

Published online by Cambridge University Press:  13 August 2009

David Ruelle
David Ruelle
Affiliation:
Institut des Hautes Etudes Scientifiques, 35, Route de Chartres, 91440 Bures-sur-Yvette, France
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Abstract

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The purpose of this note is to prove a conjecture of D. Sullivan that when the Julia set J of a rational function f is hyperbolic, the Hausdorff dimension of J depends real analytically on f. We shall obtain this as corollary of a general result on repellers of real analytic maps (see corollary 5).

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 2 , Issue 1 , March 1982 , pp. 99 - 107
Copyright
Copyright © Cambridge University Press 1982

References

REFERENCES

[1]Bowen, R.. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. Lecture Notes in Math. no. 470. Springer: Berlin, 1975.CrossRefGoogle Scholar
[2]Bowen, R.. Hausdorff dimension of quasi-circles. Publ. Math. I.H.E.S. 50 (1979), 11–26.CrossRefGoogle Scholar
[3]Brolin, H.. Invariant sets under iteration of rational functions. Ark. Mat. 6 (1965), 103–144.CrossRefGoogle Scholar
[4]Chen, S.-S. & Manning, A.. The convergence of zeta functions for certain geodesic flows depends on their pressure. Math. Z. 176 (1981), 379–382.CrossRefGoogle Scholar
[5]Coven, E. M. & Reddy, W. L.. Positively expansive maps of compact manifolds. In Global Theory of Dynamical Systems. Lecture Notes in Math. no. 819, pp. 96–110. Springer: Berlin, 1980.CrossRefGoogle Scholar
[6]Grothendieck, A.. La théorie de Fredholm. Bull. Soc. Math. France. 84 (1956), 319–384.CrossRefGoogle Scholar
[7]Mandelbrot, B.. Fractal aspects of the iteration of z → λz(1 - z) for complex λ and z. Ann. N.Y. Acad. Sci. 357 (1980), 249–259.CrossRefGoogle Scholar
[8]Mayer, D. H.. The Ruelle-Araki Transfer Operator in Classical Statistical Mechanics. Lecture Notes in Physics no. 123. Springer: Berlin, 1980.Google Scholar
[9]Misiurewicz, M.. A short proof of the variational principle for a action on a compact space. Astérisque. 40 (1976), 147–157.Google Scholar
[10]Ruelle, D.. Generalized zeta-functions for Axiom A basic sets. Bull. Amer. Math. Soc. 82 (1976), 153–156.CrossRefGoogle Scholar
[11]Ruelle, D., Zeta-functions for expanding maps and Anosov flows. Invent. Math. 34 (1976), 231–242.CrossRefGoogle Scholar
[12]Ruelle, D.. Thermodynamic Formalism. Addison-Wesley: Reading, 1978.Google Scholar
[13]Sinai, la. G.. Construction of Markov partitions. Funkts. Analiz ego Pril. 2 (1968), 70–80. English translation. Funct. Anal. Appl. 2 (1968), 245–253.Google Scholar
[14]Sullivan, D.. Discrete conformal groups and measurable dynamics. In The Mathematical Heritage of Henri Poincaré. (To appear.)Google Scholar
[15]Sullivan, D.. Geometrically defined measures for conformal dynamical systems. (To appear.)Google Scholar
[16]Walters, P.. A variational principle for the pressure on continuous transformations. Amer. J. Math. 97 (1976), 937–971.CrossRefGoogle Scholar