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Open AccessArticle

Minkowski Dimension and Explicit Tube Formulas for p-Adic Fractal Strings

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Department of Mathematics, University of California, Riverside, CA 92521-0135, USA
2
Department of Mathematics, Hawai‘i Pacific University, Honolulu, HI 96813-2785, USA
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Department of Mathematics, Utah Valley University, Orem, UT 84058-5999, USA
*
Author to whom correspondence should be addressed.
Fractal Fract 2018, 2(4), 26; https://doi.org/10.3390/fractalfract2040026
Received: 23 August 2018 / Accepted: 5 September 2018 / Published: 4 October 2018
The theory of complex dimensions describes the oscillations in the geometry (spectra and dynamics) of fractal strings. Such geometric oscillations can be seen most clearly in the explicit volume formula for the tubular neighborhoods of a p-adic fractal string L p , expressed in terms of the underlying complex dimensions. The general fractal tube formula obtained in this paper is illustrated by several examples, including the nonarchimedean Cantor and Euler strings. Moreover, we show that the Minkowski dimension of a p-adic fractal string coincides with the abscissa of convergence of the geometric zeta function associated with the string, as well as with the asymptotic growth rate of the corresponding geometric counting function. The proof of this new result can be applied to both real and p-adic fractal strings and hence, yields a unifying explanation of a key result in the theory of complex dimensions for fractal strings, even in the archimedean (or real) case. View Full-Text
Keywords: fractal geometry; p-adic analysis; p-adic fractal strings; zeta functions; complex dimensions; Minkowski dimension; fractal tubes formulas; p-adic self-similar strings; Cantor; Euler and Fibonacci strings fractal geometry; p-adic analysis; p-adic fractal strings; zeta functions; complex dimensions; Minkowski dimension; fractal tubes formulas; p-adic self-similar strings; Cantor; Euler and Fibonacci strings
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Lapidus, M.L.; Lũ’, H.; Van Frankenhuijsen, M. Minkowski Dimension and Explicit Tube Formulas for p-Adic Fractal Strings. Fractal Fract 2018, 2, 26.

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