Exact Hausdorff and packing measures of Cantor sets with overlaps

Abstract

Let K be the attractor of a linear iterated function system (IFS) S j (x) = ρ j x + b j , j = 1, . . . , m, on the real line R satisfying the generalized finite type condition (whose invariant open set O is an interval) with an irreducible weighted incidence matrix. This condition was recently introduced by Lau and Ngai [A generalized finite type condition for iterated function systems. Adv. Math. 208 (2007), 647-671] as a natural generalization of the open set condition, allowing us to include many important overlapping cases. They showed that the Hausdorff and packing dimensions of K coincide and can be calculated in terms of the spectral radius of the weighted incidence matrix. Let α be the dimension of K . In this paper, we state thatwhere H α denotes the α-dimensional Hausdorff measure and P α denotes the α-dimensional packing measure. This result extends a recent work of Olsen [Density theorems for Hausdorff and packing measures of self-similar sets. Aequationes Math. 75 (2008), 208-225] where the open set condition is required. We use these inequalities to obtain some precise density theorems for the Hausdorff and packing measures of K . Moreover, using these density theorems, we describe a scheme for computing H α (K ) exactly as the minimum of a finite set of elementary functions of the parameters of the IFS. We also obtain an exact algorithm for computing P α (K ) as the maximum of another finite set of elementary functions of the parameters of the IFS. These results extend previous ones by Ayer and Strichartz [Exact Hausdorff measure and intervals of maximum density for Cantor sets. Trans. Amer. Math. Soc. 351 (1999), 3725-3741] and by Feng [Exact packing measure of Cantor sets. Math. Natchr. 248-249 (2003), 102-109], respectively, and apply to some new classes allowing us to include Cantor sets in R with overlaps. Exact Hausdorff and packing measures of Cantor sets with overlaps 2633 Introduction and statement of resultsIn this paper we will analyze the behavior of the Hausdorff and packing measures of selfsimilar sets satisfying the generalized finite type condition, which is weaker than the open set condition. In particular, we will deal with the exact calculation of the Hausdorff and packing measures for a kind of Cantor sets in R with overlaps. The problem of calculating the dimension of the attractor of a self-similar iterated function system (IFS) is one of the most interesting objects in fractal geometry. During the past two decades there has been an enormous body of literature investigating this problem and wide-ranging generalizations thereof. See the books [2,4,14] and the references therein. Let {S j } m j=1 be an IFS of contractive similitudes on R d defined as S j (x) = ρ j R j x + b j j = 1, . . . , m, (1.1)

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