Minkowski Dimension and Explicit Tube Formulas for p-Adic Fractal Strings
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 u...
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2018-10-01
|
| Series: | Fractal and Fractional |
| Subjects: | |
| Online Access: | http://www.mdpi.com/2504-3110/2/4/26 |
| _version_ | 1823933329068523520 |
|---|---|
| author | Michel L. Lapidus Hùng Lũ’ Machiel van Frankenhuijsen |
| author_facet | Michel L. Lapidus Hùng Lũ’ Machiel van Frankenhuijsen |
| author_sort | Michel L. Lapidus |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-12-16T22:04:26Z |
| format | Article |
| id | doaj.art-96605d8f1412426ea8d9d117ca4266aa |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-12-16T22:04:26Z |
| publishDate | 2018-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj.art-96605d8f1412426ea8d9d117ca4266aa2022-12-21T22:14:31ZengMDPI AGFractal and Fractional2504-31102018-10-01242610.3390/fractalfract2040026fractalfract2040026Minkowski Dimension and Explicit Tube Formulas for p-Adic Fractal StringsMichel L. Lapidus0Hùng Lũ’1Machiel van Frankenhuijsen2Department of Mathematics, University of California, Riverside, CA 92521-0135, USADepartment of Mathematics, Hawai‘i Pacific University, Honolulu, HI 96813-2785, USADepartment of Mathematics, Utah Valley University, Orem, UT 84058-5999, USAThe 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.http://www.mdpi.com/2504-3110/2/4/26fractal geometryp-adic analysisp-adic fractal stringszeta functionscomplex dimensionsMinkowski dimensionfractal tubes formulasp-adic self-similar stringsCantorEuler and Fibonacci strings |
| spellingShingle | Michel L. Lapidus Hùng Lũ’ Machiel van Frankenhuijsen Minkowski Dimension and Explicit Tube Formulas for p-Adic Fractal Strings Fractal and Fractional 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 |
| title | Minkowski Dimension and Explicit Tube Formulas for p-Adic Fractal Strings |
| title_full | Minkowski Dimension and Explicit Tube Formulas for p-Adic Fractal Strings |
| title_fullStr | Minkowski Dimension and Explicit Tube Formulas for p-Adic Fractal Strings |
| title_full_unstemmed | Minkowski Dimension and Explicit Tube Formulas for p-Adic Fractal Strings |
| title_short | Minkowski Dimension and Explicit Tube Formulas for p-Adic Fractal Strings |
| title_sort | minkowski dimension and explicit tube formulas for p adic fractal strings |
| topic | 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 |
| url | http://www.mdpi.com/2504-3110/2/4/26 |
| work_keys_str_mv | AT michelllapidus minkowskidimensionandexplicittubeformulasforpadicfractalstrings AT hunglu minkowskidimensionandexplicittubeformulasforpadicfractalstrings AT machielvanfrankenhuijsen minkowskidimensionandexplicittubeformulasforpadicfractalstrings |