no. 6
Dimension of the harmonic measure of non-homogeneous Cantor sets
[Dimension de la mesure harmonique d’ensembles de Cantor non-homogènes]
Batakis, Athanasios1
1 Université d’Orléans Département de Mathématiques MAPMO BP 6759 45067 Orléans cedex 2 (France)
Annales de l'Institut Fourier, Tome 56 (2006) no. 6, pp. 1617-1631.

Nous montrons que la dimension de la mesure harmonique du complémentaire d’ensembles de Cantor de type invariant par translation est une fonction continue des paramètres définissant ces ensembles. Ce résultat prolonge un précédent du même auteur et n’implique pas d’outils de la théorie ergotique, non-applicables dans notre configuration.

We prove that the dimension of the harmonic measure of the complementary of a translation-invariant type of Cantor sets is a continuous function of the parameters determining these sets. This results extends a previous one of the author and do not use ergotic theoretic tools, not applicables to our case.

DOI : 10.5802/aif.2222
Classification : 31A15, 28A80
Keywords: Harmonic measure, Cantor sets, fractals, Hausdorff dimension, entropy
Mot clés : Mesure Harmonique, Ensembles de Cantor, fractals, Dimension de Hausdorff, Entropie
Batakis, Athanasios 1

1 Université d’Orléans Département de Mathématiques MAPMO BP 6759 45067 Orléans cedex 2 (France) 
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Batakis, Athanasios. Dimension of the harmonic measure of non-homogeneous Cantor sets. Annales de l'Institut Fourier, Tome 56 (2006) no. 6, pp. 1617-1631. doi : 10.5802/aif.2222. http://www.numdam.org/articles/10.5802/aif.2222/

[1] Ancona, A. Principe de Harnack à la frontière et théorème de Fatou pour un opérateur elliptique dans un domaine Lipschitzien, Annales de l’ Institut Fourier, Grenoble, Volume 28 (1978) no. 4, pp. 169-213 | DOI | Numdam | MR | Zbl

[2] Balogh, Z.; Popovici, I.; Volberg, A. Conformally maximal polynomial-like dynamics and invariant harmonic measure, Ergodic Theory and Dynamical Systems, Volume 17 (1997) no. 1 , pp. 1-27 | DOI | MR | Zbl

[3] Batakis, A. Harmonic measure of some Cantor type sets, Ann. Acad. Sci. Fenn., Volume 21 (1996), pp. 255-270 | MR | Zbl

[4] Batakis, A. Théorie du potentiel : 1. Sur les domaines Poissoniens 2. Sur la mesure harmonique des ensembles de Cantor, Université de Paris-Sud (1997) (Ph. D. Thesis)

[5] Batakis, A. A continuity property of the dimension of harmonic measure of Cantor sets under perturbations, Ann. Inst. H. Poincaré Probab. Statist., Volume 36 (2000) no. 1, pp. 87-107 | DOI | Numdam | Zbl

[6] Batakis, A. On entropy and Hausdorff dimension of measures defined through a Markov process (2002) (Preprint)

[7] Batakis, A.; Heurteaux, Y. On relations between entropy and Hausdorff dimension of measures (2002) (To appear in the Asian Journal of Mathematics) | MR | Zbl

[8] Beardon, A. On the Hausdorff dimension of general Cantor sets, Proceedings of the Cambridge Philosophical Society, Volume 61 (1965), pp. 679-694 | DOI | MR | Zbl

[9] Billingsley, P. Ergodic Theory and Information, John Wiley & Sons, 1965 | MR | Zbl

[10] Binder, I.; Makarov, N.; Smirnov, S. Harmonic measure and polynomial Julia sets (2002) (to appear in Duke Mathematical Journal) | Zbl

[11] Brelot, M. Axiomatique des fonctions harmoniques, Les presses de l’Université de Montréal, 1969 | MR | Zbl

[12] Carleson, L. On the support of harmonic measure for sets of Cantor type, Ann. Acad. Sci. Fenn., Volume 10 (1985), pp. 113-123 | MR | Zbl

[13] Doob, J. L. Classical Potential Theory and Its Probabilistic Counterpart, Springer-Verlag New York, 1984 | MR | Zbl

[14] Fan, A.H. Sur la dimension des mesures, Studia Math., Volume 111 (1994), pp. 1-17 | Zbl

[15] Hall, P.; Heyde, C. C. Martingale theory and its applications, Probability and Mathematical Statistics, Academic Press (New York), 1980 | MR | Zbl

[16] Helms, L. L. Introduction to Potential Theory, John Wiley & Sons, 1969 | MR | Zbl

[17] Heurteaux, Y. Estimations de la dimension inférieure et de la dimension supérieure des mesures, Ann. Inst. H. Poincaré Probab. Statist., Volume 34 (1998), pp. 309-338 | DOI | Numdam | MR | Zbl

[18] Lyubich, M.; Volberg, A. A comparison of harmonic and balanced measures on Cantor repellors, Journal of Fourier Analysis and Applications (Special Issue J.-P. Kahane) (1995), pp. 379-399 | Zbl

[19] Makarov, N. Fine structure of harmonic measure, Saint Petersbourg Mathematical Journal, Volume 10 (1999), pp. 217-268 | MR | Zbl

[20] Makarov, N.; Volberg, A. On the harmonic measure of discontinuous fractals (1986) (Preprint LOMI E-6-86, Leningrad)

[21] Mattila, P. Geometric measure theory, Cambridge University Press, 1995

[22] Urbanski, M.; Zdunik, A. Hausdorff dimension of harmonic measure for self conformal maps (2000) (preprint IHES, Bures sur Yvette)

[23] Volberg, A. On harmonic measure of self-similar sets in the plane, Harmonic Analysis and Discrete Potential theory, Plenum Press, 1992 | MR

[24] Volberg, A. On the dimension of harmonic measure of Cantor-type repellers, Michigan Mathematical Journal, Volume 40 (1993), pp. 239-258 | DOI | MR | Zbl

[25] Young, L. Dimension, entropy and Lyapounov exponents, Ergod. Th. & Dynam. Sys., Volume 2 (1982), pp. 109-124 | DOI | MR | Zbl

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