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Minkowski measurability and exact fractal tube formulas for \(p\)-adic self-similar strings. (English) Zbl 1321.37081

Carfì, David (ed.) et al., Fractal geometry and dynamical systems in pure and applied mathematics I: Fractals in pure mathematics. Selected papers based on three conferences following the passing of Benoît Mandelbrot in October 2010. 1st PISRS 2011 international conference on analysis, fractal geometry, dynamical systems and economics, Messina, Italy, November 8–12, 2011, AMS special session on fractal geometry in pure and applied mathematics, in memory of Benoît Mandelbrot, Boston, MA, USA, January 2012, AMS special session on geometry and analysis on fractal spaces, Honolulu, HI, USA, March 2012. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-9147-6/pbk; 978-1-4704-1082-7/ebook). Contemporary Mathematics 600, 161-184 (2013).
Summary: The theory of \(p\)-adic fractal strings and their complex dimensions was developed by the first two authors [J. Fixed Point Theory Appl. 3, No. 1, 181–190 (2008; Zbl 1234.28011); \(p\)-Adic Numbers Ultrametric Anal. Appl. 1, No. 2, 167–180 (2009; Zbl 1187.28014); Contemp. Math. 551, 163–206 (2011; Zbl 1276.37053)], particularly in the self-similar case, in parallel with its archimedean (or real) counterpart developed by the first and third author in [Fractal geometry and number theory. Complex dimensions of fractal strings and zeros of zeta functions. Boston, MA: Birkhäuser (2000; Zbl 0981.28005); Fractal geometry, complex dimensions and zeta functions. Geometry and spectra of fractal strings. New York, NY: Springer (2006; Zbl 1119.28005)]]. Using the fractal tube formula obtained by the authors for \(p\)-adic fractal strings in [“Minkowski dimension and explicit tube formulas for \(p\)-adic fractal strings, preprint, (2012)], we present here an exact volume formula for the tubular neighborhood of a \(p\)-adic self-similar fractal string \(\mathcal L_p\), expressed in terms of the underlying complex dimensions. The periodic structure of the complex dimensions allows one to obtain a very concrete form for the resulting fractal tube formula. Moreover, we derive and use a truncated version of this fractal tube formula in order to show that \(\mathcal L_p\) is not Minkowski measurable and obtain an explicit expression for its average Minkowski content. The general theory is illustrated by two simple examples, the 3-adic Cantor string and the 2-adic Fibonacci strings, which are nonarchimedean analogs (introduced in [Zbl 1234.28011; Zbl 1187.28014]) of the real Cantor and Fibonacci strings studied in [Zbl 0981.28005].
For the entire collection see [Zbl 1276.00022].

MSC:

37P20 Dynamical systems over non-Archimedean local ground fields
11M41 Other Dirichlet series and zeta functions
26E30 Non-Archimedean analysis
28A12 Contents, measures, outer measures, capacities
28A80 Fractals
11M06 \(\zeta (s)\) and \(L(s, \chi)\)
11K41 Continuous, \(p\)-adic and abstract analogues
30G06 Non-Archimedean function theory
32P05 Non-Archimedean analysis
46S10 Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis
47S10 Operator theory over fields other than \(\mathbb{R}\), \(\mathbb{C}\) or the quaternions; non-Archimedean operator theory
81Q65 Alternative quantum mechanics (including hidden variables, etc.)
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