Besicovitch subsets of self-similar sets
Annales de l'Institut Fourier, Volume 52 (2002) no. 4, pp. 1061-1074.

Let $E$ be a self-similar set with similarities ratio ${r}_{j}\left(0\le j\le m-1\right)$ and Hausdorff dimension $s$, let $\stackrel{\to }{p}\left({p}_{0},{p}_{1}\right)...{p}_{m-1}$ be a probability vector. The Besicovitch-type subset of $E$ is defined as

 $E\left(\stackrel{\to }{p}\right)=\left\{x\in E:\underset{n\to \infty }{lim}\frac{1}{n}\sum _{k=1}^{n}{\chi }_{j}\left({x}_{k}\right)={p}_{j},\phantom{\rule{1em}{0ex}}0\le j\le m-1\right\},$
where ${\chi }_{j}$ is the indicator function of the set $\left\{j\right\}$. Let $\alpha ={dim}_{H}\left(E\left(\stackrel{\to }{p}\right)\right)={dim}_{P}\left(E\left(\stackrel{\to }{p}\right)\right)=\frac{{\sum }_{j=0}^{m-1}{p}_{j}log{p}_{j}}{{\sum }_{j=0}^{m-1}{p}_{i}log{r}_{j}}$ and $g$ be a gauge function, then we prove in this paper:(i) If $\stackrel{\to }{p}=\left({r}_{0}^{s},{r}_{1}^{s},\cdots ,{r}_{m-1}^{s}\right)$, then
 ${ℋ}^{s}\left(E\left(\stackrel{\to }{p}\right)\right)={ℋ}^{s}\left(E\right),\phantom{\rule{0.277778em}{0ex}}{𝒫}^{s}\left(E\left(\stackrel{\to }{p}\right)\right)={𝒫}^{s}\left(E\right),$
moreover both of ${ℋ}^{s}\left(E\right)$ and ${𝒫}^{s}\left(E\right)$ are finite positive;(ii) If $\stackrel{\to }{p}$ is a positive probability vector other than $\left({r}_{0}^{s},{r}_{1}^{s},\cdots ,{r}_{m-1}^{s}\right)$, then the gauge functions can be partitioned as follows
 ${ℋ}^{g}\left(E\left(\stackrel{\to }{p}\right)\right)=+\infty ⇔\underset{t\to 0}{\overline{\mathrm{lim}}}\frac{logg\left(t\right)}{logt}\le \alpha ;\phantom{\rule{4pt}{0ex}}{ℋ}^{g}\left(E\left(\stackrel{\to }{p}\right)\right)=0⟺\underset{t\to 0}{\overline{\mathrm{lim}}}\frac{logg\left(t\right)}{logt}>\alpha ,$
 ${𝒫}^{g}\left(E\left(\stackrel{\to }{p}\right)\right)=+\infty ⟺\underset{t\to 0}{\underline{\mathrm{lim}}}\frac{logg\left(t\right)}{logt}\le \alpha ;\phantom{\rule{4pt}{0ex}}{𝒫}^{g}\left(E\left(\stackrel{\to }{p}\right)\right)=0⟺\underset{t\to 0}{\underline{\mathrm{lim}}}\frac{logg\left(t\right)}{logt}>\alpha .$

Soit $E$ un ensemble auto-similaire avec coefficients de similarité ${r}_{j}\left(0\le j\le m-1\right)$ et de dimension de Hausdorff $s$, et soit $\stackrel{\to }{p}=\left({p}_{0},{p}_{1}\right)...{p}_{m-1}$ un vecteur de probabilité. Le sous-ensemble de type de Besicovitch de $E$ est défini par

 $E\left(\stackrel{\to }{p}\right)=\left\{x\in E:\underset{n\to \infty }{lim}\frac{1}{n}\sum _{k=1}^{n}{\chi }_{j}\left({x}_{k}\right)={p}_{j},\phantom{\rule{1em}{0ex}}0\le j\le m-1\right\},$
${\chi }_{j}$ est la fonction indicatrice de l’ensemble $\left\{j\right\}$. Soient $\alpha ={dim}_{H}\left(E\left(\stackrel{\to }{p}\right)\right)={dim}_{P}\left(E\left(\stackrel{\to }{p}\right)\right)=\frac{{\sum }_{j=0}^{m-1}{p}_{j}log{p}_{j}}{{\sum }_{j=0}^{m-1}{p}_{i}log{r}_{j}}$ et $g$ une fonction de jauge, on va démontrer dans cet article :(i) Si $\stackrel{\to }{p}=\left({r}_{0}^{s},{r}_{1}^{s},\cdots ,{r}_{m-1}^{s}\right)$, alors
 ${ℋ}^{s}\left(E\left(\stackrel{\to }{p}\right)\right)={ℋ}^{s}\left(E\right),\phantom{\rule{0.277778em}{0ex}}{𝒫}^{s}\left(E\left(\stackrel{\to }{p}\right)\right)={𝒫}^{s}\left(E\right),$
de plus, ${ℋ}^{s}\left(E\right)$ et ${𝒫}^{s}\left(E\right)$ sont positifs et finis;(ii) Si $\stackrel{\to }{p}$ est un vecteur de probabilité différent de $\left({r}_{0}^{s},{r}_{1}^{s},\cdots ,{r}_{m-1}^{s}\right)$, alors on peut classer les fonctions de jauge comme suit :
 ${ℋ}^{g}\left(E\left(\stackrel{\to }{p}\right)\right)=+\infty ⇔\underset{t\to 0}{\overline{\mathrm{lim}}}\frac{logg\left(t\right)}{logt}\le \alpha ;\phantom{\rule{4pt}{0ex}}{ℋ}^{g}\left(E\left(\stackrel{\to }{p}\right)\right)=0⟺\underset{t\to 0}{\overline{\mathrm{lim}}}\frac{logg\left(t\right)}{logt}>\alpha ,$
 ${𝒫}^{g}\left(E\left(\stackrel{\to }{p}\right)\right)=+\infty ⟺\underset{t\to 0}{\underline{\mathrm{lim}}}\frac{logg\left(t\right)}{logt}\le \alpha ;\phantom{\rule{4pt}{0ex}}{𝒫}^{g}\left(E\left(\stackrel{\to }{p}\right)\right)=0⟺\underset{t\to 0}{\underline{\mathrm{lim}}}\frac{logg\left(t\right)}{logt}>\alpha .$

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

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