Besicovitch subsets of self-similar sets

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

doi:10.5802/aif.1911

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

Ma, Ji-Hua ¹ ; Wen, Zhi-Ying ² ; Wu, Jun ¹

¹ Wuhan University, Department of Mathematics, Wuhan 430072 (Rép. Pop. Chine)
² Tsinghua University, Department of mathematics, Beijing 10084 (Rép. Pop. Chine)

Annales de l'Institut Fourier, Tome 52 (2002) no. 4, pp. 1061-1074.

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}}

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)

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 .

MR Zbl

DOI : 10.5802/aif.1911

Classification : 28A80, 28A78, 26A30
Keywords: perturbation measures, gauge functions, Besicovitch set
Mot clés : mesures de perturbation, fonctions de jauge, ensemble de Besicovitch

Affiliations des auteurs :

Ma, Ji-Hua ¹ ; Wen, Zhi-Ying ² ; Wu, Jun ¹

¹ Wuhan University, Department of Mathematics, Wuhan 430072 (Rép. Pop. Chine)
² Tsinghua University, Department of mathematics, Beijing 10084 (Rép. Pop. Chine)

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     author = {Ma, Ji-Hua and Wen, Zhi-Ying and Wu, Jun},
     title = {Besicovitch subsets of self-similar sets},
     journal = {Annales de l'Institut Fourier},
     pages = {1061--1074},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {52},
     number = {4},
     year = {2002},
     doi = {10.5802/aif.1911},
     mrnumber = {1926673},
     zbl = {1024.28005},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.1911/}
}

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Ma, Ji-Hua; Wen, Zhi-Ying; Wu, Jun. Besicovitch subsets of self-similar sets. Annales de l'Institut Fourier, Tome 52 (2002) no. 4, pp. 1061-1074. doi : 10.5802/aif.1911. http://www.numdam.org/articles/10.5802/aif.1911/

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