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

Qiu, Hua

doi:10.1017/s0143385711000071

Cited by 3 publications

(6 citation statements)

References 19 publications

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“…If r < 0:5, a continuous behavior of P s(r) is supported by the estimate values obtained in [28] (see Figure 1 in [28]). This fact is in concordance with Qiu´s result (see [34]) which establishes the continuity of the packing measure function of general self-similar sets satisfying the strong separation condition and, in particular, implies the continuity of P s(r) (S r ) for r < 0:5: However, the estimate of P s (S) obtained in this note (see (33)) is about half the approximate value corresponding to P s(r) (S r ) for r 0:5 (see [28]), which indicates that Qiu's conjecture might be false. The relation P s (S) 1 2 P s(r) (S r ) for r 0:5 might be caused by the di¤erences between the selected balls on each case.…”

Section: Lemmasupporting

confidence: 93%

“…Finally, a comparison of the current results (see (34) and ( 35)) with those obtained in [28] for the totally disconnected case leads to the open problem of the continuous dependence of the packing measure on the values of the contraction ratios. More precisely, let S r be the totally disconnected Sierpinski attractor with contraction ratio r < 0.5 and let P s(r) (S r ) be the corresponding packing measure of S r with s(r) := − log 3 log(r) .…”

Section: Numerical Resultsmentioning

confidence: 66%

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On the packing measure of the Sierpinski gasket

2018

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We show that the s-dimensional packing measure P s (S) of the Sierpinski gasket S; where s = log 3 log 2 is the similarity dimension of S; satis…es 1:6677 P s (S) 1:6713. The formula presented in Theorem 6 enables the achievement of the above measure bounds for this non-totally disconnected set as it shows that the symmetries of the Sierpinski gasket can be exploited to simplify the density characterization of P s obtained in [33] for self-similar sets satisfying the so-called Open Set Condition. Thanks to the reduction obtained in Theorem 6 we are able to handle the problem of computability of P s (S) with a suitable algorithm.

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Section: Lemmasupporting

confidence: 93%

Section: Numerical Resultsmentioning

confidence: 66%

On the packing measure of the Sierpinski gasket

2018

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“…These corollaries extend the results in [17,19,20]. Following the technique frame of [1,6], under suitable assumptions, we then give an algorithm for computing H α (K) and P α (K) exactly as the inverse of the maximal or minimal value of suitable finite sets of elementary functions of the parameters of the IFS respectively.…”

Section: Introduction and Statement Of Resultssupporting

confidence: 56%

“…A dual result for the packing measure was also proved which says that P α (K ∩ B(x, r)) ≥ (2r) α for each x ∈ K and small r > 0 if K satisfies the SSC. The latter result was generalized to the OSC case by the author recently (see [20]) in proving the continuity of the packing measure function of self-similar IFSs, which says that the above inequality actually holds for each B(x, r) contained in O with x ∈ K, where O is an open set associated with the OSC, satisfying O ∩ K = ∅. (There must exist such O since the OSC is equivalent to the strong OSC.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 79%

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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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Packing measure of super separated iterated function systems

Reid

2018

Geom Dedicata

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

Cited by 3 publications

References 19 publications

On the packing measure of the Sierpinski gasket

On the packing measure of the Sierpinski gasket

Exact Hausdorff and packing measures of Cantor sets with overlaps

Packing measure of super separated iterated function systems

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