Lacunarity of self-similar and stochastically self-similar sets

Gatzouras, D.

doi:10.1090/s0002-9947-99-02539-8

Cited by 61 publications

(65 citation statements)

References 24 publications

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“…The average Minkowski content (compare ) for instance is known to exist for any self‐similar set satisfying OSC, cf. , similarly the average S‐content (compare ) exists for any such set. For self‐conformal sets much more is known about the existence of average contents than about the nonaveraged counterparts, see, for example, .…”

Section: Introductionmentioning

confidence: 94%

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Localization results for Minkowski contents

Winter

2018

J. London Math. Soc.

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It was shown recently that the Minkowski content of a bounded set A in double-struckRd with volume zero can be characterized in terms of the asymptotic behaviour of the boundary surface area of its parallel sets Ar as the parallel radius r tends to 0. Here we discuss localizations of such results. The asymptotic behaviour of the local parallel volume of A relative to a suitable second set normalΩ can be understood in terms of the suitably defined local surface area relative to normalΩ. Also a measure version of this relation is shown: Viewing the Minkowski content as a locally determined measure, this local Minkowski content can be obtained as a weak limit of suitably rescaled surface measures of close parallel sets (local S‐content). Such measure relations had been observed before for self‐similar and some self‐conformal sets. They are now established for arbitrary closed sets, including even the case of unbounded sets. The results are based on a localization of Stachó's famous formula relating the boundary surface area of Ar to the derivative of the volume function at r.

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Section: Introductionmentioning

confidence: 94%

“…We expect that the assumed boundedness of the expression

ε^{D - d - γ} λ_{d} false(F_{ε} \cap G false)

as well as the assumption

{\bar{dim}}_{M} false(\partial O false) < D

in the statement above are always satisfied (implied by

\partial O \subset F

) and that this may be established with methods similar to the ones used in the self‐similar case in .…”

Section: Applicationsmentioning

confidence: 99%

Localization results for Minkowski contents

Winter

2018

J. London Math. Soc.

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“…Accordingly, other fractal parameters (box-counting dimension, lacunarity, etc.) have been introduced over time in order to solve these potential problems [ 2 , 21 ]. In particular, the different definitions of fractal dimension allow fractal modeling to be suitable for real-world applications.…”

Section: Preliminariesmentioning

confidence: 99%

Harmonic Sierpinski Gasket and Applications

Guariglia

2018

Entropy

180

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The aim of this paper is to investigate the generalization of the Sierpinski gasket through the harmonic metric. In particular, this work presents an antenna based on such a generalization. In fact, the harmonic Sierpinski gasket is used as a geometric configuration of small antennas. As with fractal antennas and Rényi entropy, their performance is characterized by the associated entropy that is studied and discussed here.

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“…This follows by a minor modification of the foregoing proof. The integral bound is of interest as it is automatically satisfied for a large family of self-similar fractal sets (see [19,Theorem 2…”

Section: If the Growth Condition Supmentioning

confidence: 99%

Markov uniqueness of degenerate elliptic operators

Robinson

Sikora²

2011

Annali Scuola Normale Superiore - Classe Di Scienze

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Let be an open subset of R d and H = − d i, j=1 ∂ i c i j ∂ j be a second-order partial differential operator on L 2 ( ) with domain C ∞ c ( ), where the coefficients c i j ∈ W 1,∞ ( ) are real symmetric and C = (c i j ) is a strictly positive-definite matrix over . In particular, H is locally strongly elliptic. We analyze the submarkovian extensions of H , i.e., the self-adjoint extensions that generate submarkovian semigroups. Our main result states that H is Markov unique, i.e., it has a unique submarkovian extension, if and only if cap (∂ ) = 0 where cap (∂ ) is the capacity of the boundary of measured with respect to H . The second main result shows that Markov uniqueness of H is equivalent to the semigroup generated by the Friedrichs extension of H being conservative.

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Lacunarity of self-similar and stochastically self-similar sets

Cited by 61 publications

References 24 publications

Localization results for Minkowski contents

Localization results for Minkowski contents

Harmonic Sierpinski Gasket and Applications

Markov uniqueness of degenerate elliptic operators

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