Packing Measure, and its Evaluation for a Brownian Path

Taylor, Selwyn; Tricot, Claude

doi:10.2307/1999958

Cited by 80 publications

(124 citation statements)

References 5 publications

(10 reference statements)

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“…[15] or [3]. But P t give rise to a Borel measure, namely the t-dimensional packing measure P t (E) of E, as follows…”

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

confidence: 99%

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

Olsen

2008

Aequ. math.

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We analyze the local behaviour of the Hausdorff measure and the packing measure of self-similar sets. In particular, if K is a self-similar set whose Hausdorff dimension and packing dimension equal s, a special case of our main results says that if K satisfies the Open Set Condition, then there exists a number r 0 such thatfor all x ∈ K and all 0 < r < r 0 , where H s denotes the s-dimensional Hausdorff measure and P s denotes the s-dimensional packing measure. Inequality (1) and inequality (2) are used to obtain a number of very precise density theorems for Hausdorff and packing measures of self-similar sets. These density theorems can be applied to compute the exact value of the s-dimensional Hausdorff measure H s (K) and the exact value of the s-dimensional packing measure P s (K) of self-similar sets K.

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“…[15] or [3]. But P t give rise to a Borel measure, namely the t-dimensional packing measure P t (E) of E, as follows…”

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

confidence: 99%

“…The packing measure was introduced by Taylor & Tricot in [15] using centered δ-packings of open balls, and by Raymond & Tricot in [11] using centered δ-packings of closed balls. We refer the reader to [16] and [11] for more information on the packing measure and the packing dimension.…”

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

confidence: 99%

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

Olsen

2008

Aequ. math.

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“…The packing measure and packing dimension, introduced by Tricot [15], Taylor & Tricot [13,14] and Sullivan [12], play an important role in the study of fractal geometry in a manner dual to the Hausdorff measure and Hausdorff dimension (see [9] and [4] for further properties of the above measures and dimensions). However, because of the difficulty in the definition there are few results about the explicit computation of packing measures for fractal sets.…”

Section: Introductionmentioning

confidence: 99%

Exact packing measure of linear Cantor sets

Ding

2003

Mathematische Nachrichten

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Let K be the attractor of a linear iterated function system Sj x = ρjx + bj (j = 1, . . . , m) on the real line satisfying the open set condition (where the open set is an interval). It is well known that the packing dimension of K is equal to α, the unique positive solution y of the equation m j=1 ρ y j = 1; and the α-dimensional packing measure P α (K) is finite and positive. Denote by µ the unique self-similar measure for the IFS Sj m j=1 with the probability weight ρ α j m j=1 . In this paper, we prove that P α (K) is equal to the reciprocal of the so-called "minimal centered density" of µ, and this yields an explicit formula of P α (K) in terms of the parameters ρj, bj (j = 1, . . . , m). Our result implies that P α (K) depends continuously on the parameters whenever j ρj < 1.

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“…The packing measure and packing dimension were introduced by Tricot [13], Taylor and Tricot [11,12] and Sullivan [10]. As parameters to describe non-smooth sets, the packing measure and packing dimension play an important role in the study of fractal geometry in a manner dual to the Hausdorff measure and Hausdorff dimension (see [3,8] …”

Section: Introductionmentioning

confidence: 99%

Comparing Packing Measures to Hausdorff Measures on the Line

Ding

2002

Math. Nachr.

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For each 0 < s < 1, define where ${\cal P}^s$, ${\cal H}^s$ denote respectively the s‐dimensional packing measure and Hausdorff measure, and the infimum is taken over all the sets E ⊂ R with $0 < {\cal H}^s (E) < \infty$. In this paper we give a nontrivial estimation of c(s), namely, $2^s (1+v(s))^s \le c(s) \le 2^s \left (2 ^{1 \over s} - 1 \right )$ for each 0 < s < 1, where $v(s) = \min \left \{ 16 ^{- {{1} \over {1-s}}}, 8 ^{- {{1} \over {(1-s)^2}}} \right \}$. As an application, we obtain a lower density theorem for Hausdorff measures.

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Packing Measure, and its Evaluation for a Brownian Path

Cited by 80 publications

References 5 publications

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

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

Exact packing measure of linear Cantor sets

Comparing Packing Measures to Hausdorff Measures on the Line

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