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Minkowski dimension and explicit tube formulas for -adic fractal strings
The local 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 -adic fractal string , 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
-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 -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.Comment: 34 pages, 1 figure. arXiv admin note: substantial text overlap with
arXiv:1105.2966 This is the final version of an original research article on
the Minkowski dimension and explicit tube formulas for -adic fractal
strings. It is appeared in the open access journal Fractal Fractiona
Fractal Strings and Multifractal Zeta Functions
For a Borel measure on the unit interval and a sequence of scales that tend
to zero, we define a one-parameter family of zeta functions called multifractal
zeta functions. These functions are a first attempt to associate a zeta
function to certain multifractal measures. However, we primarily show that they
associate a new zeta function, the topological zeta function, to a fractal
string in order to take into account the topology of its fractal boundary. This
expands upon the geometric information garnered by the traditional geometric
zeta function of a fractal string in the theory of complex dimensions. In
particular, one can distinguish between a fractal string whose boundary is the
classical Cantor set, and one whose boundary has a single limit point but has
the same sequence of lengths as the complement of the Cantor set. Later work
will address related, but somewhat different, approaches to multifractals
themselves, via zeta functions, partly motivated by the present paper.Comment: 32 pages, 9 figures. This revised version contains new sections and
figures illustrating the main results of this paper and recent results from
others. Sections 0, 2, and 6 have been significantly rewritte
Random fractal strings: their zeta functions, complex dimensions and spectral asymptotics
In this paper a string is a sequence of positive non-increasing real numbers which sums to one. For our purposes a fractal string is a string formed from the lengths of removed sub-intervals created by a recursive decomposition of the unit interval. By using the so called complex dimensions of the string, the poles of an associated zeta function, it is possible to obtain detailed information about the behaviour of the asymptotic properties of the string. We consider random versions of fractal strings. We show that using a random recursive self-similar construction it is possible to obtain similar results to those for deterministic self-similar strings. In the case of strings generated by the excursions of stable subordinators, we show that the complex dimensions can only lie on the real line. The results allow us to discuss the geometric and spectral asymptotics of one-dimensional domains with random fractal boundary
Fractal Complex Dimensions, Riemann Hypothesis and Invertibility of the Spectral Operator
A spectral reformulation of the Riemann hypothesis was obtained in [LaMa2] by
the second author and H. Maier in terms of an inverse spectral problem for
fractal strings. This problem is related to the question "Can one hear the
shape of a fractal drum?" and was shown in [LaMa2] to have a positive answer
for fractal strings whose dimension is c\in(0,1)-\{1/2} if and only if the
Riemann hypothesis is true. Later on, the spectral operator was introduced
heuristically by M. L. Lapidus and M. van Frankenhuijsen in their theory of
complex fractal dimensions [La-vF2, La-vF3] as a map that sends the geometry of
a fractal string onto its spectrum. We focus here on presenting the rigorous
results obtained by the authors in [HerLa1] about the invertibility of the
spectral operator. We show that given any , the spectral operator
, now precisely defined as an unbounded normal
operator acting in a Hilbert space , is `quasi-invertible'
(i.e., its truncations are invertible) if and only if the Riemann zeta function
does not have any zeroes on the line . It follows
that the associated inverse spectral problem has a positive answer for all
possible dimensions , other than the mid-fractal case when ,
if and only if the Riemann hypothesis is true.Comment: To appear in: "Fractal Geometry and Dynamical Systems in Pure and
Applied Mathematics", Part 1 (D. Carfi, M. L. Lapidus, E. P. J. Pearse and M.
van Frankenhuijsen, eds.), Contemporary Mathematics, Amer. Math. Soc.,
Providence, RI, 2013. arXiv admin note: substantial text overlap with
arXiv:1203.482
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