Besicovitch subsets of self-similar sets

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

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

Soit $E$ un ensemble auto-similaire avec coefficients de similarité $r_j(0\le j\le m-1)$ et de dimension de Hausdorff $s$, et soit $\overrightarrow{p}=(p_0,p_1)...p_{m-1}$ un vecteur de probabilité. Le sous-ensemble de type de Besicovitch de $E$ est défini par

$$E\left(\overrightarrow{p}\right)=\left\{x\in E:\underset{n\to \infty }{lim}\frac{1}{n}\sum _{k=1}^n{\chi }_j\left(x_k\right)=p_j,0\le j\le m-1\right\},$$

où ${\chi }_j$ est la fonction indicatrice de l’ensemble $\left\{j\right\}$. Soient $\alpha ={dim}_H\left(E\left(\overrightarrow{p}\right)\right)={dim}_P\left(E\left(\overrightarrow{p}\right)\right)=\frac{{\sum }_{j=0}^{m-1}{p}_{j}log{p}_j}{{\sum }_{j=0}^{m-1}{p}_{i}log{r}_j}$ et $g$ une fonction de jauge, on va démontrer dans cet article :(i) Si $\overrightarrow{p}=(r_0^s,r_1^s,\cdots ,r_{m-1}^s)$, alors

$${\mathscr{H}}^s\left(E\left(\overrightarrow{p}\right)\right)={\mathscr{H}}^s\left(E\right),{𝒫}^s\left(E\left(\overrightarrow{p}\right)\right)={𝒫}^s\left(E\right),$$

de plus, ${\mathscr{H}}^s\left(E\right)$ et ${𝒫}^s\left(E\right)$ sont positifs et finis;(ii) Si $\overrightarrow{p}$ est un vecteur de probabilité différent de $(r_0^s,r_1^s,\cdots ,r_{m-1}^s)$, alors on peut classer les fonctions de jauge comme suit :

$${\mathscr{H}}^g\left(E\left(\overrightarrow{p}\right)\right)=+\infty \iff \underset{t\to 0}{\overline{\lim }}\frac{logg\left(t\right)}{logt}\le \alpha ;{\mathscr{H}}^g\left(E\left(\overrightarrow{p}\right)\right)=0⟺\underset{t\to 0}{\overline{\lim }}\frac{logg\left(t\right)}{logt}>\alpha ,$$

$${𝒫}^g\left(E\left(\overrightarrow{p}\right)\right)=+\infty ⟺\underset{t\to 0}{\underline{\lim }}\frac{logg\left(t\right)}{logt}\le \alpha ;{𝒫}^g\left(E\left(\overrightarrow{p}\right)\right)=0⟺\underset{t\to 0}{\underline{\lim }}\frac{logg\left(t\right)}{logt}>\alpha .$$

Let $E$ be a self-similar set with similarities ratio $r_j(0\le j\le m-1)$ and Hausdorff dimension $s$, let $\overrightarrow{p}(p_0,p_1)...p_{m-1}$ be a probability vector. The Besicovitch-type subset of $E$ is defined as

$$E\left(\overrightarrow{p}\right)=\left\{x\in E:\underset{n\to \infty }{lim}\frac{1}{n}\sum _{k=1}^n{\chi }_j\left(x_k\right)=p_j,0\le j\le m-1\right\},$$

where ${\chi }_j$ is the indicator function of the set $\left\{j\right\}$. Let $\alpha ={dim}_H\left(E\left(\overrightarrow{p}\right)\right)={dim}_P\left(E\left(\overrightarrow{p}\right)\right)=\frac{{\sum }_{j=0}^{m-1}{p}_{j}log{p}_j}{{\sum }_{j=0}^{m-1}{p}_{i}log{r}_j}$ and $g$ be a gauge function, then we prove in this paper:(i) If $\overrightarrow{p}=(r_0^s,r_1^s,\cdots ,r_{m-1}^s)$, then

$${\mathscr{H}}^s\left(E\left(\overrightarrow{p}\right)\right)={\mathscr{H}}^s\left(E\right),{𝒫}^s\left(E\left(\overrightarrow{p}\right)\right)={𝒫}^s\left(E\right),$$

moreover both of ${\mathscr{H}}^s\left(E\right)$ and ${𝒫}^s\left(E\right)$ are finite positive;(ii) If $\overrightarrow{p}$ is a positive probability vector other than $(r_0^s,r_1^s,\cdots ,r_{m-1}^s)$, then the gauge functions can be partitioned as follows

$${\mathscr{H}}^g\left(E\left(\overrightarrow{p}\right)\right)=+\infty \iff \underset{t\to 0}{\overline{\lim }}\frac{logg\left(t\right)}{logt}\le \alpha ;{\mathscr{H}}^g\left(E\left(\overrightarrow{p}\right)\right)=0⟺\underset{t\to 0}{\overline{\lim }}\frac{logg\left(t\right)}{logt}>\alpha ,$$

$${𝒫}^g\left(E\left(\overrightarrow{p}\right)\right)=+\infty ⟺\underset{t\to 0}{\underline{\lim }}\frac{logg\left(t\right)}{logt}\le \alpha ;{𝒫}^g\left(E\left(\overrightarrow{p}\right)\right)=0⟺\underset{t\to 0}{\underline{\lim }}\frac{logg\left(t\right)}{logt}>\alpha .$$

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

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