Besicovitch subsets of self-similar sets
[Sous-ensembles de Besicovitch d'ensembles auto-similaires]
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 (0jm-1) et de dimension de Hausdorff s, et soit 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(p )=x E : lim n 1 n k=1 n χ j (x k ) = p j , 0 j m - 1,
χ j est la fonction indicatrice de l’ensemble {j}. Soient α=dim H (E(p ))=dim P (E(p ))= j=0 m-1 p j logp j j=0 m-1 p i logr j et g une fonction de jauge, on va démontrer dans cet article :(i) Si p =(r 0 s ,r 1 s ,,r m-1 s ), alors
s (E(p ))= s (E),𝒫 s (E(p ))=𝒫 s (E),
de plus, s (E) et 𝒫 s (E) sont positifs et finis;(ii) Si p est un vecteur de probabilité différent de (r 0 s ,r 1 s ,,r m-1 s ), alors on peut classer les fonctions de jauge comme suit :
g (E(p ))=+ lim ¯ t0 logg(t) logtα; g (E(p ))=0 lim ¯ t0 logg(t) logt>α,
𝒫 g (E(p ))=+ lim ̲ t0 logg(t) logtα;𝒫 g (E(p ))=0 lim ̲ t0 logg(t) logt>α.

Let E be a self-similar set with similarities ratio r j (0jm-1) and Hausdorff dimension s, let p (p 0 ,p 1 )...p m-1 be a probability vector. The Besicovitch-type subset of E is defined as

E(p )=x E : lim n 1 n k=1 n χ j (x k ) = p j , 0 j m - 1,
where χ j is the indicator function of the set {j}. Let α=dim H (E(p ))=dim P (E(p ))= j=0 m-1 p j logp j j=0 m-1 p i logr j and g be a gauge function, then we prove in this paper:(i) If p =(r 0 s ,r 1 s ,,r m-1 s ), then
s (E(p ))= s (E),𝒫 s (E(p ))=𝒫 s (E),
moreover both of s (E) and 𝒫 s (E) are finite positive;(ii) If p is a positive probability vector other than (r 0 s ,r 1 s ,,r m-1 s ), then the gauge functions can be partitioned as follows
g (E(p ))=+ lim ¯ t0 logg(t) logtα; g (E(p ))=0 lim ¯ t0 logg(t) logt>α,
𝒫 g (E(p ))=+ lim ̲ t0 logg(t) logtα;𝒫 g (E(p ))=0 lim ̲ t0 logg(t) logt>α.

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, Tome 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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