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)=xE:limn1nk=1nχj(xk)=pj,0jm-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),𝒫s(E(p))=𝒫s(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 ¯t0logg(t)logtα;g(E(p))=0 lim ¯t0logg(t)logt>α,
𝒫g(E(p))=+ lim ̲t0logg(t)logtα;𝒫g(E(p))=0 lim ̲t0logg(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)=xE:limn1nk=1nχj(xk)=pj,0jm-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),𝒫s(E(p))=𝒫s(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 ¯t0logg(t)logtα;g(E(p))=0 lim ¯t0logg(t)logt>α,
𝒫g(E(p))=+ lim ̲t0logg(t)logtα;𝒫g(E(p))=0 lim ̲t0logg(t)logt>α.

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},
     pages = {1061--1074},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {52},
     number = {4},
     year = {2002},
     doi = {10.5802/aif.1911},
     zbl = {1024.28005},
     mrnumber = {1926673},
     language = {en},
     url = {aif.centre-mersenne.org/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) no. 4, pp. 1061-1074. doi : 10.5802/aif.1911. https://aif.centre-mersenne.org/item/AIF_2002__52_4_1061_0/

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