Pointwise tube formulas for fractal sprays and self-similar tilings with arbitrary generators

Lapidus, Michel L.; Pearse, Erin P. J.; Winter, Steffen

doi:10.1016/j.aim.2011.03.004

Cited by 41 publications

(88 citation statements)

References 29 publications

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“…Recall that Weyl's celebrated formula [41] .3), and that the corresponding term in the explicit formula dominates the remaining ones. This issue was discovered during collaboration with Steffen Winter on the forthcoming paper [24], and so it will be discussed in full detail in that paper. For example, a sufficient condition for Corollary 8.5 to hold is that D > d − 1 and…”

Section: Some Remarks On the Results In This Papermentioning

confidence: 99%

“…Not every generator G is monophase, so we introduce the pluriphase case in Definition 5.5. The most general case is discussed in [24]. In fact, even if G is polyhedral or convex, it still may not be monophase.…”

Section: The Generatorsmentioning

confidence: 99%

“…[4] and [37]. Note added in proof: Property (iii) of Definition 5.1 eventually proved to be unnecessary, and will be omitted in future work, e.g., [24].…”

Section: The Generatorsmentioning

confidence: 99%

“…In fact, even if G is polyhedral or convex, it still may not be monophase. Example 5.6 gives a set which is convex and pluriphase but not monophase and the Cantor Carpet discussed in [24,Ex. 4.1] gives a tiling with a nonconvex generator (in the shape of a Swiss cross) which is polyhedral but not even pluriphase.…”

Section: The Generatorsmentioning

confidence: 99%

“…In this paper, we treat only the special case of a generator G which has a Steiner-like representation whose coefficient functions κ k (G, ε) are constant functions of ε for 0 ≤ ε < g, in which case G is called monophase as in Definition 5.2. We treat more general generators in the forthcoming collaboration with Steffen Winter [24]. Definition 5.2.…”

Section: The Generatorsmentioning

confidence: 99%

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Tube Formulas and Complex Dimensions of Self-Similar Tilings

Lapidus

Pearse

2010

Acta Appl Math

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Abstract. We use the self-similar tilings constructed in [32] to define a generating function for the geometry of a self-similar set in Euclidean space. This tubular zeta function encodes scaling and curvature properties related to the complement of the fractal set, and the associated system of mappings. This allows one to obtain the complex dimensions of the self-similar tiling as the poles of the tubular zeta function and hence develop a tube formula for self-similar tilings in R d . The resulting power series in ε is a fractal extension of Steiner's classical tube formula for convex bodies K ⊆ R d . Our sum has coefficients related to the curvatures of the tiling, and contains terms for each integer i = 0, 1, . . . , d − 1, just as Steiner's does. However, our formula also contains a term for each complex dimension. This provides further justification for the term "complex dimension". It also extends several aspects of the theory of fractal strings to higher dimensions and sheds new light on the tube formula for fractals strings obtained in [30].

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Section: Some Remarks On the Results In This Papermentioning

confidence: 99%

Section: The Generatorsmentioning

confidence: 99%

“…[4] and [37]. Note added in proof: Property (iii) of Definition 5.1 eventually proved to be unnecessary, and will be omitted in future work, e.g., [24].…”

Section: The Generatorsmentioning

confidence: 99%

Section: The Generatorsmentioning

confidence: 99%

Section: The Generatorsmentioning

confidence: 99%

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Tube Formulas and Complex Dimensions of Self-Similar Tilings

Lapidus

Pearse

2010

Acta Appl Math

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Multifractal Tubes

Olsen

2013

Further Developments in Fractals and Related Fields

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Tube formulas refer to the study of volumes of r neighbourhoods of sets. For sets satisfying some (possible very weak) convexity conditions, this has a long history. However, within the past 20 years Lapidus has initiated and pioneered a systematic study of tube formulas for fractal sets. Following this, it is natural to ask to what extend it is possible to develop a theory of multifractal tube formulas for multifractal measures. In this paper we propose and develop a framework for such a theory. Firstly, we define multifractal tube formulas and, more generally, multifractal tube measures for general multifractal measures. Secondly, we introduce and develop two approaches for analysing these concepts for self-similar multifractal measures, namely:(1) Multifractal tubes of self-similar measures and renewal theory. Using techniques from renewal theory we give a complete description of the asymptotic behaviour of the multifractal tube formulas and tube measures of self-similar measures satisfying the Open Set Condition.(2) Multifractal tubes of self-similar measures and zeta-functions. Unfortunately, renewal theory techniques do not yield "explicit" expressions for the functions describing the asymptotic behaviour of the multifractal tube formulas and tube measures of self-similar measures. This is clearly undesirable. For this reason, we introduce and develop a second framework for studying multifractal tube formulas of self-similar measures. This approach is based on multifractal zeta-functions and allow us obtain "explicit" expressions for the multifractal tube formulas of self-similar measures, namely, using the Mellin transform and the residue theorem, we are able to express the multifractal tube formulas as sums involving the residues of the zeta-function.2000 Mathematics Subject Classification. 28A78, 37C30.where we have rewritten the Lebesgue measure. Motivated by the Renyi dimensions (i.e. (2.2) and (2.3)) and the above expression for V r (E), we define the multifractal Minkowski volume as follows. Namely, let r > 0 and E be a subset of R d . For real number q and a Borel measure µ on R d , we define the multifractal q Minkowski volume V q µ,r (E) of E with respect to the measure µ byNote, that if q = 0, then the q multifractal Minkowski volume V q µ,r (E) reduces to the usual Minkowski volume, i.e V 0 µ,r (E) = V r (E) . The importance of the Renyi dimensions in multifractal analysis together with the formal resemblance between the multifractal Minkowski volume V q µ,r (E) and the moments I q µ,r (E) used in the definition the Renyi dimensions may be seen as a justification for calling the quantity V q µ,r (E) for the multifractal Minkowski volume; a further justification for this terminology will be provided below.Using the multifractal Minkowski volume we can define multifractal Monkowski dimensions. For real number q and a Borel measure µ on R d , we define the lower and upper multifractal q Minkowski dimension of E, by(3.2)Again we note the close similarity between the multifractal Minkowski dimensions...

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Fractal Zeta Functions and Complex Dimensions: A General Higher-Dimensional Theory

Lapidus

Radunović²,

Žubrinić³

2015

Fractal Geometry and Stochastics V

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In 2009, the first author introduced a class of zeta functions, called 'distance zeta functions', which has enabled us to extend the existing theory of zeta functions of fractal strings and sprays (initiated by the first author and his collaborators in the early 1990s) to arbitrary bounded (fractal) sets in Euclidean spaces of any dimensions. A closely related tool is the class of 'tube zeta functions', defined using the tube function of a fractal set. These zeta functions exhibit deep connections with Minkowski contents and upper box (or Minkowski) dimensions, as well as, more generally, with the complex dimensions of fractal sets. In particular, the abscissa of (Lebesgue, i.e., absolute) convergence of the distance zeta function coincides with the upper box dimension of a set. We also introduce a class of transcendentally quasiperiodic sets, and describe their construction based on a sequence of carefully chosen generalized Cantor sets with two auxilliary parameters. As a result, we obtain a family of "maximally hyperfractal" compact sets and relative fractal drums (i.e., such that the associated fractal zeta functions have a singularity at every point of the critical line of convergence). Finally, we discuss the general fractal tube formulas and the Minkowski measurability criterion obtained by the authors in the context of relative fractal drums (and, in particular, of bounded subsets of R N ).Keywords. Fractal set, fractal zeta functions, distance zeta function, tube zeta function, geometric zeta function of a fractal string, Minkowski content, Minkowski measurability, upper box (or Minkowski) dimension, complex dimensions of a fractal set, holomorphic and meromorphic functions, abscissa of convergence, quasiperiodic function, quasiperiodic set, relative fractal drum, fractal tube formulas.

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Pointwise tube formulas for fractal sprays and self-similar tilings with arbitrary generators

Cited by 41 publications

References 29 publications

Tube Formulas and Complex Dimensions of Self-Similar Tilings

Tube Formulas and Complex Dimensions of Self-Similar Tilings

Multifractal Tubes

Fractal Zeta Functions and Complex Dimensions: A General Higher-Dimensional Theory

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