Density theorems for Hausdorff and packing measures of self-similar sets

Olsen, L.

doi:10.1007/s00010-007-2917-3

Cited by 10 publications

(20 citation statements)

References 11 publications

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“…A similar result in the SSC case was proved by Olsen in [14], and from which the explicit formula (1.5) was obtained.…”

Section: Introductionsupporting

confidence: 79%

“…The proof also needs an explicit formula for the packing measure. Actually, a similar formula for packing measure in the SSC case was also proved in [14], i.e., for each f ∈ M ∆ with ∆ > 0, the following formula holds.…”

Section: Introductionmentioning

confidence: 67%

“…This result has several applications on densities and can also be applied to compute the exact value of the packing measure P s (K) of K. Recall that in [14], Olsen also proved a density theorem for packing measure of self-similar sets which requires that the IFSs satisfy the SSC. In that setting, there exists r 0 > 0 such that the above formula holds for all x ∈ K and all r ∈ (0, r 0 ].…”

Section: Density Theorems For Packing Measure Of Self-similar Setsmentioning

confidence: 94%

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Continuity of packing measure functions of self-similar iterated function systems

Qiu¹

2011

Ergod. Th. Dynam. Sys.

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Link to this article: http://journals.cambridge.org/abstract_S0143385711000071How to cite this article: HUA QIU (2012). Continuity of packing measure functions of self-similar iterated function systems.Abstract. In this paper, we focus on the packing measures of self-similar sets. Let K be a self-similar set whose Hausdorff dimension and packing dimension equal s. We state that if K satisfies the strong open set condition with an open set O, thenwhere P s denotes the s-dimensional packing measure. We use this inequality to obtain some precise density theorems for the packing measures of self-similar sets. These theorems can be used to compute the exact value of the s-dimensional packing measure P s (K ) of K . Moreover, by using the above results, we show the continuity of the packing measure function of the attractors on the space of self-similar iterated function systems satisfying the strong separation condition. This result gives a complete answer to a question posed by Olsen in [15].

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“…A similar result in the SSC case was proved by Olsen in [14], and from which the explicit formula (1.5) was obtained.…”

Section: Introductionsupporting

confidence: 79%

Section: Introductionmentioning

confidence: 67%

Section: Density Theorems For Packing Measure Of Self-similar Setsmentioning

confidence: 94%

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Continuity of packing measure functions of self-similar iterated function systems

Qiu¹

2011

Ergod. Th. Dynam. Sys.

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“…The proof in [19] is quite different from that in [17]. Let K be a self-similar set in R d as described in (1.1) with the dimension α, under the OSC or the strong separation condition (SSC).…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Exact Hausdorff and packing measures of Cantor sets with overlaps

Qiu¹

2014

Ergod. Th. Dynam. Sys.

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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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“…In the last years, the exact measure has attracted the attention of the community. Very general results have been demonstrated by Olsen who has computed the exact Hausdorff and Packing measure of self-similar sets satisfying the open set condition [12]. Local densities are the main tool used by Olsen.…”

Section: Introductionmentioning

confidence: 97%