Besicovitch subsets of self-similar sets

Ma, Ji-Hua; Wen, Zhi-Ying; Wu, Jun

Besicovitch subsets of self-similar sets
[Sous-ensembles de Besicovitch d'ensembles auto-similaires]

Ma, Ji-Hua ; Wen, Zhi-Ying ; Wu, Jun

Annales de l'Institut Fourier, Tome 52 (2002), p. 1061-1074 / Harvested from Numdam

Access to full text
Full (PDF)
Full (DJVU)

Résumé
Abstract

Soit $E$ un ensemble auto-similaire avec coefficients de similarité $r_{j} (0 \leq j \leq m - 1)$ et de dimension de Hausdorff $s$ , et soit $\vec{p} = (p_{0}, p_{1}) ... p_{m - 1}$ un vecteur de probabilité. Le sous-ensemble de type de Besicovitch de $E$ est défini par $E (\vec{p}) = \{x \in E : lim_{n \to \infty} \frac{1}{n} \sum_{k = 1}^{n} χ_{j} (x_{k}) = p_{j}, 0 \leq j \leq m - 1\},$ où $χ_{j}$ est la fonction indicatrice de l’ensemble ${j}$ . Soient $α = {dim}_{H} (E (\vec{p})) = {dim}_{P} (E (\vec{p})) = \frac{\sum_{j = 0}^{m - 1} p_{j} log p_{j}}{\sum_{j = 0}^{m - 1} p_{i} log r_{j}}$ et $g$ une fonction de jauge, on va démontrer dans cet article :(i) Si $\vec{p} = (r_{0}^{s}, r_{1}^{s}, \dots, r_{m - 1}^{s})$ , alors $ℋ^{s} (E (\vec{p})) = ℋ^{s} (E), 𝒫^{s} (E (\vec{p})) = 𝒫^{s} (E),$ de plus, $ℋ^{s} (E)$ et $𝒫^{s} (E)$ sont positifs et finis;(ii) Si $\vec{p}$ est un vecteur de probabilité différent de $(r_{0}^{s}, r_{1}^{s}, \dots, r_{m - 1}^{s})$ , alors on peut classer les fonctions de jauge comme suit : $ℋ^{g} (E (\vec{p})) = + \infty \Leftrightarrow \underset{t \to 0}{\lim^{¯}} \frac{log g (t)}{log t} \leq α; ℋ^{g} (E (\vec{p})) = 0 ⟺ \underset{t \to 0}{\lim^{¯}} \frac{log g (t)}{log t} > α,$ $𝒫^{g} (E (\vec{p})) = + \infty ⟺ \underset{t \to 0}{\lim_{̲}} \frac{log g (t)}{log t} \leq α; 𝒫^{g} (E (\vec{p})) = 0 ⟺ \underset{t \to 0}{\lim_{̲}} \frac{log g (t)}{log t} > α .$

Let $E$ be a self-similar set with similarities ratio $r_{j} (0 \leq j \leq m - 1)$ and Hausdorff dimension $s$ , let $\vec{p} (p_{0}, p_{1}) ... p_{m - 1}$ be a probability vector. The Besicovitch-type subset of $E$ is defined as $E (\vec{p}) = \{x \in E : lim_{n \to \infty} \frac{1}{n} \sum_{k = 1}^{n} χ_{j} (x_{k}) = p_{j}, 0 \leq j \leq m - 1\},$ where $χ_{j}$ is the indicator function of the set ${j}$ . Let $α = {dim}_{H} (E (\vec{p})) = {dim}_{P} (E (\vec{p})) = \frac{\sum_{j = 0}^{m - 1} p_{j} log p_{j}}{\sum_{j = 0}^{m - 1} p_{i} log r_{j}}$ and $g$ be a gauge function, then we prove in this paper:(i) If $\vec{p} = (r_{0}^{s}, r_{1}^{s}, \dots, r_{m - 1}^{s})$ , then $ℋ^{s} (E (\vec{p})) = ℋ^{s} (E), 𝒫^{s} (E (\vec{p})) = 𝒫^{s} (E),$ moreover both of $ℋ^{s} (E)$ and $𝒫^{s} (E)$ are finite positive;(ii) If $\vec{p}$ is a positive probability vector other than $(r_{0}^{s}, r_{1}^{s}, \dots, r_{m - 1}^{s})$ , then the gauge functions can be partitioned as follows $ℋ^{g} (E (\vec{p})) = + \infty \Leftrightarrow \underset{t \to 0}{\lim^{¯}} \frac{log g (t)}{log t} \leq α; ℋ^{g} (E (\vec{p})) = 0 ⟺ \underset{t \to 0}{\lim^{¯}} \frac{log g (t)}{log t} > α,$ $𝒫^{g} (E (\vec{p})) = + \infty ⟺ \underset{t \to 0}{\lim_{̲}} \frac{log g (t)}{log t} \leq α; 𝒫^{g} (E (\vec{p})) = 0 ⟺ \underset{t \to 0}{\lim_{̲}} \frac{log g (t)}{log t} > α .$

Publié le : 2002-01-01

MR 1926673 | Zbl 1024.28005

DOI : https://doi.org/10.5802/aif.1911
Classification: 28A80, 28A78, 26A30
Mots clés: mesures de perturbation, fonctions de jauge, ensemble de Besicovitch

@article{AIF_2002__52_4_1061_0,
     author = {Ma, Ji-Hua and Wen, Zhi-Ying and Wu, Jun},
     title = {Besicovitch subsets of self-similar sets},
     journal = {Annales de l'Institut Fourier},
     volume = {52},
     year = {2002},
     pages = {1061-1074},
     doi = {10.5802/aif.1911},
     mrnumber = {1926673},
     zbl = {1024.28005},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2002__52_4_1061_0}
}

Ma, Ji-Hua; Wen, Zhi-Ying; Wu, Jun. Besicovitch subsets of self-similar sets. Annales de l'Institut Fourier, Tome 52 (2002) pp. 1061-1074. doi : 10.5802/aif.1911. http://gdmltest.u-ga.fr/item/AIF_2002__52_4_1061_0/

Bibliographie

[1] A.S. Besicovitch On the sum of digits of real numbers represented in the dyadic system, Math. Ann, Tome 110 (1934), pp. 321-330 | Article | JFM 60.0949.01 | MR 1512941 | Zbl 0009.39503

[2] H.G. Eggleston The fractional dimension of a set defined by decimal properties, Quart. J. Math. Oxford Ser, Tome 20 (1949), pp. 31-36 | Article | MR 31026 | Zbl 0031.20801

[3] K.J. Falconer Techniques in Fractal Geometry, John Wiley and sons inc. (1997) | MR 1449135 | Zbl 0869.28003

[4] R. Kaufman A further example on scales of Hausdorff functions, J. London Math. Soc, Tome 8 (1974) no. 2, p. 585-586 | Article | MR 357721 | Zbl 0302.28015

[5] M. Moran; J. Rey Singularity of self-similar measures with respect to Hausdorff measures, Trans. of Amer. Math. Soc., Tome 350 (1998) no. 6, pp. 2297-2310 | Article | MR 1475691 | Zbl 0899.28002

[6] Y. Peres The self-affine carpets of McMullen and Bedford have infinite Hausdorff measure, Math. Proc. Camb. Phil. Soc, Tome 116 (1994), pp. 513-526 | Article | MR 1291757 | Zbl 0811.28005

[7] A.N. Shiryayev Probability, Springer-Verlag, New York (1984) | MR 737192 | Zbl 0536.60001

[8] J. Taylor The measure theory of random fractals, Math. Proc. Cambridge Philo. Soc, Tome 100 (1986), pp. 383-408 | Article | MR 857718 | Zbl 0622.60021