. En, attendàattendà ce que p(n) croisse comme n d , c'est-` a-dire soit encadré par C 1 n d et C 2 n d , pour C 1 , C 2 deux constantes positives. La majoration est un résultat connu : c'est un théorème de Hansen et Robinson pour les pavages dits self-affine, qui s'´ enonce comme

. La-minoration-peutégalementpeutégalement-se-déduire-d-'un-résultat-de-solomyak, 4] : dans un pavage de substitution P sans période, il existe une constante C telle que si a et b sont deux amas de taille r de P qui sont un translatés l'un de l'autre (disons a = b + x avec x ? R d ), alors x ? Cr. Ainsi, les amas qui sont (( proches )) de a sont différents de a. Cela donne une borne inférieure sur la complexité. On donnera dans la suite de ce chapitre une preuve directe de l'encadrement de p pour un pavage de substitution. La preuve utilisera les propriétés d'invariance de la complexité par homéomorphisme bi-lipschitzien afin de simplifier leprobì eme, Cela dit, vol.66, issue.2

P. Si, alors on peut se restreindrè a des V i tels que pour tout i ? n, V i est un voisinage ouvert de x ientì erement inclus dans une arête de R n . De plus, si on note comme ci-dessus x i = (s i , s ? i , t), l'union des s n et des s ? n définit un mot fini de longueur n+1 centré en zéro

. Enfin, sont tous inclus dans des arêtes isométriquesisométriquesà [0, 1], chacun de ces intervalles définit I i de ]0, 1[. On appelle I leur intersection (finie) ; c'est un intervalle ouvert contenant t, Alors U (W, I) est une partie ouverte de ? inclue dans ? ?1 (V ), et contenant P

P. Si, la construction est similaire. On construit un sous-mot fini de m, lequel définit un voisinage W dans ? w

. Puis, dans [0, 1] (qui correspondent auxfì eches entrantes sur les graphes), et un certain nombre de voisinages de 0 dans [0, 1] (qui correspondent aux fì eches sortantes) On appelle I k tous les voisinages de 1 (avec i ? 1, n), et J k les voisinages de 0. On définit maintenant I comme l'intersection de tous les I k , et J comme celle de tous les J k . Finalement, on définit I ? = (J ? 1) ? I, qui est un voisinage de 0 dans R. Alors U (W, I ? ) est une partie ouverte de ? incluse dans ? ?1 (V ), et contenant P

L. Continuité-de-?-est-ainsi-montrée, Par compacité de ? w , il est automatique que ? est un homéomorphisme

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