Dimension of the harmonic measure of non-homogeneous Cantor sets
Annales de l'Institut Fourier, Volume 56 (2006) no. 6, p. 1617-1631
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.
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.
DOI : https://doi.org/10.5802/aif.2222
Classification:  31A15,  28A80
Keywords: Harmonic measure, Cantor sets, fractals, Hausdorff dimension, entropy
@article{AIF_2006__56_6_1617_0,
     author = {Batakis, Athanasios},
     title = {Dimension of the harmonic measure of non-homogeneous Cantor sets},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {56},
     number = {6},
     year = {2006},
     pages = {1617-1631},
     doi = {10.5802/aif.2222},
     zbl = {1113.31001},
     mrnumber = {2282670},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2006__56_6_1617_0}
}
Dimension of the harmonic measure of non-homogeneous Cantor sets. Annales de l'Institut Fourier, Volume 56 (2006) no. 6, pp. 1617-1631. doi : 10.5802/aif.2222. https://aif.centre-mersenne.org/item/AIF_2006__56_6_1617_0/

[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, Tome 28 (1978) no. 4, pp. 169-213 | Article | Numdam | MR 513885 | Zbl 0377.31001

[2] Balogh, Z.; Popovici, I.; Volberg, A. Conformally maximal polynomial-like dynamics and invariant harmonic measure, Ergodic Theory and Dynamical Systems, Tome 17 (1997) no. 1 , pp. 1-27 | Article | MR 1440765 | Zbl 0876.58036

[3] Batakis, A. Harmonic measure of some Cantor type sets, Ann. Acad. Sci. Fenn., Tome 21 (1996), pp. 255-270 | MR 1404086 | Zbl 0849.31005

[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., Tome 36 (2000) no. 1, pp. 87-107 | Article | Numdam | Zbl 0946.37018

[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 1946341 | Zbl 1046.28003

[8] Beardon, A. On the Hausdorff dimension of general Cantor sets, Proceedings of the Cambridge Philosophical Society, Tome 61 (1965), pp. 679-694 | Article | MR 177083 | Zbl 0145.05502

[9] Billingsley, P. Ergodic Theory and Information, John Wiley & Sons (1965) | MR 192027 | Zbl 0141.16702

[10] Binder, I.; Makarov, N.; Smirnov, S. Harmonic measure and polynomial Julia sets (2002) (to appear in Duke Mathematical Journal) | Zbl 1036.30017

[11] Brelot, M. Axiomatique des fonctions harmoniques, Les presses de l’Université de Montréal (1969) | MR 247124 | Zbl 0148.10401

[12] Carleson, L. On the support of harmonic measure for sets of Cantor type, Ann. Acad. Sci. Fenn., Tome 10 (1985), pp. 113-123 | MR 802473 | Zbl 0593.31004

[13] Doob, J. L. Classical Potential Theory and Its Probabilistic Counterpart, Springer-Verlag New York (1984) | MR 731258 | Zbl 0549.31001

[14] Fan, A.H. Sur la dimension des mesures, Studia Math., Tome 111 (1994), pp. 1-17 | Zbl 0805.28002

[15] Hall, P.; Heyde, C. C. Martingale theory and its applications, Academic Press (New York), Probability and Mathematical Statistics (1980) | MR 624435 | Zbl 0462.60045

[16] Helms, L. L. Introduction to Potential Theory, John Wiley & Sons (1969) | MR 261018 | Zbl 0188.17203

[17] Heurteaux, Y. Estimations de la dimension inférieure et de la dimension supérieure des mesures, Ann. Inst. H. Poincaré Probab. Statist., Tome 34 (1998), pp. 309-338 | Article | Numdam | MR 1625871 | Zbl 0903.28005

[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 0891.58024

[19] Makarov, N. Fine structure of harmonic measure, Saint Petersbourg Mathematical Journal, Tome 10 (1999), pp. 217-268 | MR 1629379 | Zbl 0909.30016

[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 1222465

[24] Volberg, A. On the dimension of harmonic measure of Cantor-type repellers, Michigan Mathematical Journal, Tome 40 (1993), pp. 239-258 | Article | MR 1226830 | Zbl 0797.30022

[25] Young, L. Dimension, entropy and Lyapounov exponents, Ergod. Th. & Dynam. Sys., Tome 2 (1982), pp. 109-124 | Article | MR 684248 | Zbl 0523.58024