Complex dimensions of fractals and meromorphic extensions of fractal zeta functions. Lapidus, M. L., Radunovic, G., & Zubrinic, D. JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 453(1):458–484, September, 2017. Place: 525 B ST, STE 1900, SAN DIEGO, CA 92101-4495 USA Publisher: ACADEMIC PRESS INC ELSEVIER SCIENCE Type: Articledoi abstract bibtex We study meromorphic extensions of distance and tube zeta functions, as well as of geometric zeta functions of fractal strings. The distance zeta function zeta(A) (s) := integral(A delta) d(x, A)(s-N) dx, where delta \textgreater 0 is fixed and d(x, A) denotes the Euclidean distance from x to A, has been introduced by the first author in 2009, extending the definition of the zeta function zeta L associated with bounded fractal strings L = (l(j)) j \textgreater= 1 to arbitrary bounded subsets A of the N-dimensional Euclidean space. The abscissa of Lebesgue (i.e., absolute) convergence D(zeta(A)) coincides with D :=(dim) over bar (B) A, the upper box (or Minkowski) dimension of A. The (visible) complex dimensions of A are the poles of the meromorphic continuation of the fractal zeta function (i.e., the distance or tube zeta function) of A to a suitable connected neighborhood of the “critical line” \Re s = D\. We establish several meromorphic extension results, assuming some suitable information about the second term of the asymptotic expansion of the tube function vertical bar A(t)vertical bar as t -\textgreater 0(+), where At is the Euclidean t-neighborhood of A. We pay particular attention to a class of Minkowski measurable sets, such that vertical bar A(t)vertical bar = t(N-D) (M + O(t(r))) as t -\textgreater 0(+), with gamma \textgreater 0, and to a class of Minkowski nonmeasurable sets, such that vertical bar A(t)vertical bar = t(N-D) (G(log t(-1)) + O(t(gamma))) as t -\textgreater 0(+), where G is a nonconstant periodic function and gamma \textgreater 0. In both cases, we show that zeta(A) can be meromorphically extended (at least) to the open right half-plane \Re s \textgreater D-gamma\ and determine the corresponding visible complex dimensions. Furthermore, up to a multiplicative constant, the residue of zeta(A) evaluated at s = D is shown to be equal to M (the Minkowski content of A) and to the mean value of G (the average Minkowski content of A), respectively. Moreover, we construct a class of fractal strings with principal complex dimensions of any prescribed order, as well as with an infinite number of essential singularities on the critical line \Re s = D\. Finally, using an appropriate quasiperiodic version of the above construction, with infinitely many suitably chosen quasiperiods associated with a two-parameter family of generalized Cantor sets, we construct “maximally-hyperfractal” compact subsets of R-N, for N \textgreater= 1 arbitrary. These are compact subsets of R-N such that the corresponding fractal zeta functions have nonremovable singularities at every point of the critical line \Re s = D\. (C) 2017 Elsevier Inc. All rights reserved.
@article{lapidus_complex_2017,
title = {Complex dimensions of fractals and meromorphic extensions of fractal zeta functions},
volume = {453},
issn = {0022-247X},
doi = {10.1016/j.jmaa.2017.03.059},
abstract = {We study meromorphic extensions of distance and tube zeta functions, as well as of geometric zeta functions of fractal strings. The distance zeta function zeta(A) (s) := integral(A delta) d(x, A)(s-N) dx, where delta {\textgreater} 0 is fixed and d(x, A) denotes the Euclidean distance from x to A, has been introduced by the first author in 2009, extending the definition of the zeta function zeta L associated with bounded fractal strings L = (l(j)) j {\textgreater}= 1 to arbitrary bounded subsets A of the N-dimensional Euclidean space. The abscissa of Lebesgue (i.e., absolute) convergence D(zeta(A)) coincides with D :=(dim) over bar (B) A, the upper box (or Minkowski) dimension of A. The (visible) complex dimensions of A are the poles of the meromorphic continuation of the fractal zeta function (i.e., the distance or tube zeta function) of A to a suitable connected neighborhood of the “critical line” \{Re s = D\}. We establish several meromorphic extension results, assuming some suitable information about the second term of the asymptotic expansion of the tube function vertical bar A(t)vertical bar as t -{\textgreater} 0(+), where At is the Euclidean t-neighborhood of A. We pay particular attention to a class of Minkowski measurable sets, such that vertical bar A(t)vertical bar = t(N-D) (M + O(t(r))) as t -{\textgreater} 0(+), with gamma {\textgreater} 0, and to a class of Minkowski nonmeasurable sets, such that vertical bar A(t)vertical bar = t(N-D) (G(log t(-1)) + O(t(gamma))) as t -{\textgreater} 0(+), where G is a nonconstant periodic function and gamma {\textgreater} 0. In both cases, we show that zeta(A) can be meromorphically extended (at least) to the open right half-plane \{Re s {\textgreater} D-gamma\} and determine the corresponding visible complex dimensions. Furthermore, up to a multiplicative constant, the residue of zeta(A) evaluated at s = D is shown to be equal to M (the Minkowski content of A) and to the mean value of G (the average Minkowski content of A), respectively. Moreover, we construct a class of fractal strings with principal complex dimensions of any prescribed order, as well as with an infinite number of essential singularities on the critical line \{Re s = D\}. Finally, using an appropriate quasiperiodic version of the above construction, with infinitely many suitably chosen quasiperiods associated with a two-parameter family of generalized Cantor sets, we construct “maximally-hyperfractal” compact subsets of R-N, for N {\textgreater}= 1 arbitrary. These are compact subsets of R-N such that the corresponding fractal zeta functions have nonremovable singularities at every point of the critical line \{Re s = D\}. (C) 2017 Elsevier Inc. All rights reserved.},
language = {English},
number = {1},
journal = {JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS},
author = {Lapidus, Michel L. and Radunovic, Goran and Zubrinic, Darko},
month = sep,
year = {2017},
note = {Place: 525 B ST, STE 1900, SAN DIEGO, CA 92101-4495 USA
Publisher: ACADEMIC PRESS INC ELSEVIER SCIENCE
Type: Article},
keywords = {Dirichlet integral, Distance zeta function, Fractal set, Meromorphic extension, Minkowski content, Principal complex dimensions},
pages = {458--484},
}
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The abscissa of Lebesgue (i.e., absolute) convergence D(zeta(A)) coincides with D :=(dim) over bar (B) A, the upper box (or Minkowski) dimension of A. The (visible) complex dimensions of A are the poles of the meromorphic continuation of the fractal zeta function (i.e., the distance or tube zeta function) of A to a suitable connected neighborhood of the “critical line” \\Re s = D\\. We establish several meromorphic extension results, assuming some suitable information about the second term of the asymptotic expansion of the tube function vertical bar A(t)vertical bar as t -\\textgreater 0(+), where At is the Euclidean t-neighborhood of A. We pay particular attention to a class of Minkowski measurable sets, such that vertical bar A(t)vertical bar = t(N-D) (M + O(t(r))) as t -\\textgreater 0(+), with gamma \\textgreater 0, and to a class of Minkowski nonmeasurable sets, such that vertical bar A(t)vertical bar = t(N-D) (G(log t(-1)) + O(t(gamma))) as t -\\textgreater 0(+), where G is a nonconstant periodic function and gamma \\textgreater 0. In both cases, we show that zeta(A) can be meromorphically extended (at least) to the open right half-plane \\Re s \\textgreater D-gamma\\ and determine the corresponding visible complex dimensions. Furthermore, up to a multiplicative constant, the residue of zeta(A) evaluated at s = D is shown to be equal to M (the Minkowski content of A) and to the mean value of G (the average Minkowski content of A), respectively. Moreover, we construct a class of fractal strings with principal complex dimensions of any prescribed order, as well as with an infinite number of essential singularities on the critical line \\Re s = D\\. Finally, using an appropriate quasiperiodic version of the above construction, with infinitely many suitably chosen quasiperiods associated with a two-parameter family of generalized Cantor sets, we construct “maximally-hyperfractal” compact subsets of R-N, for N \\textgreater= 1 arbitrary. These are compact subsets of R-N such that the corresponding fractal zeta functions have nonremovable singularities at every point of the critical line \\Re s = D\\. (C) 2017 Elsevier Inc. 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The distance zeta function zeta(A) (s) := integral(A delta) d(x, A)(s-N) dx, where delta {\\textgreater} 0 is fixed and d(x, A) denotes the Euclidean distance from x to A, has been introduced by the first author in 2009, extending the definition of the zeta function zeta L associated with bounded fractal strings L = (l(j)) j {\\textgreater}= 1 to arbitrary bounded subsets A of the N-dimensional Euclidean space. The abscissa of Lebesgue (i.e., absolute) convergence D(zeta(A)) coincides with D :=(dim) over bar (B) A, the upper box (or Minkowski) dimension of A. The (visible) complex dimensions of A are the poles of the meromorphic continuation of the fractal zeta function (i.e., the distance or tube zeta function) of A to a suitable connected neighborhood of the “critical line” \\{Re s = D\\}. We establish several meromorphic extension results, assuming some suitable information about the second term of the asymptotic expansion of the tube function vertical bar A(t)vertical bar as t -{\\textgreater} 0(+), where At is the Euclidean t-neighborhood of A. We pay particular attention to a class of Minkowski measurable sets, such that vertical bar A(t)vertical bar = t(N-D) (M + O(t(r))) as t -{\\textgreater} 0(+), with gamma {\\textgreater} 0, and to a class of Minkowski nonmeasurable sets, such that vertical bar A(t)vertical bar = t(N-D) (G(log t(-1)) + O(t(gamma))) as t -{\\textgreater} 0(+), where G is a nonconstant periodic function and gamma {\\textgreater} 0. In both cases, we show that zeta(A) can be meromorphically extended (at least) to the open right half-plane \\{Re s {\\textgreater} D-gamma\\} and determine the corresponding visible complex dimensions. Furthermore, up to a multiplicative constant, the residue of zeta(A) evaluated at s = D is shown to be equal to M (the Minkowski content of A) and to the mean value of G (the average Minkowski content of A), respectively. Moreover, we construct a class of fractal strings with principal complex dimensions of any prescribed order, as well as with an infinite number of essential singularities on the critical line \\{Re s = D\\}. Finally, using an appropriate quasiperiodic version of the above construction, with infinitely many suitably chosen quasiperiods associated with a two-parameter family of generalized Cantor sets, we construct “maximally-hyperfractal” compact subsets of R-N, for N {\\textgreater}= 1 arbitrary. These are compact subsets of R-N such that the corresponding fractal zeta functions have nonremovable singularities at every point of the critical line \\{Re s = D\\}. (C) 2017 Elsevier Inc. 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