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).
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The work of MLL was partially supported by the US National Science Foundation under the research grants DMS-0070497 and DMS-0707524. The work of EPJP was partially supported by the University of Iowa Department of Mathematics NSF VIGRE grant DMS-0602242.
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Lapidus, M.L., Pearse, E.P.J. Tube Formulas and Complex Dimensions of Self-Similar Tilings. Acta Appl Math 112, 91–136 (2010). https://doi.org/10.1007/s10440-010-9562-x
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DOI: https://doi.org/10.1007/s10440-010-9562-x
Keywords
- Complex dimensions
- Zeta functions
- Tube formula
- Fractal Steiner formula
- Inradius
- Self-similar tiling
- Curvature matrix
- Generating function for the geometry
- Distributional explicit formula
- Fractal string