NDLI: Tube Formulas and Complex Dimensions of Self-Similar Tilings

Content Provider	Springer Nature Link
Author	Pearse, Erin P. J. Lapidus, Michel L.
Copyright Year	2010
Abstract	We use the self-similar tilings constructed in (Pearse in Indiana Univ. Math J. 56(6):3151–3169, 2007) to define a generating function for the geometry of a self-similar set in Euclidean space. This tubularzeta 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 tubularzeta function and hence develop a tube formula for self-similar tilings in ℝd. The resulting power series in εis a fractal extension of Steiner’s classical tube formula for convex bodies K⊆ℝd. 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 (Lapidus and van Frankenhuijsen in Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings, 2006).
Ending Page	136
Page Count	46
Starting Page	91
File Format	PDF
ISSN	01678019
e-ISSN	15729036
Journal	Acta Applicandae Mathematica
Issue Number	1
Volume Number	112
Language	English
Publisher	Springer Netherlands
Publisher Date	2010-06-25
Publisher Place	Dordrecht
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Curvature matrix Generating function for the geometry Other Dirichlet series and zeta functions Theoretical, Mathematical and Computational Physics Distributional explicit formula Fractals Entropy and other invariants Mathematics Statistical Physics, Dynamical Systems and Complexity Fractal string Fourier coefficients, Fourier series of functions with special properties, special Fourier series Computer Science Self-similar tiling Mixed volumes and related topics Tube formula Complex dimensions Lattices and convex bodies in $n$ dimensions Inradius Zeta functions Classical almost periodic functions, mean periodic functions Length, area, volume Convex sets in $n$ dimensions (including convex hypersurfaces) Length, area, volume, other geometric measure theory Mechanics Fractal Steiner formula Contents, measures, outer measures, capacities Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. Explicit formulas Hausdorff and packing measures
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics

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