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Fractal zeta functions and complex dimensions of relative fractal drums

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Abstract

The theory of “zeta functions of fractal strings” has been initiated by the first author in the early 1990s and developed jointly with his collaborators during almost two decades of intensive research in numerous articles and several monographs. In 2009, the same author introduced a new class of zeta functions, called “distance zeta functions,” which since then has enabled us to extend the existing theory of zeta functions of fractal strings and sprays to arbitrary bounded (fractal) sets in Euclidean spaces of any dimension. A natural and closely related tool for the study of distance zeta functions is the class of “tube zeta functions,” defined using the tube function of a fractal set. These three classes of zeta functions, under the name of “fractal zeta functions,” exhibit deep connections with Minkowski contents and upper box dimensions, as well as, more generally, with the complex dimensions of fractal sets. Further extensions include zeta functions of relative fractal drums, the box dimension of which can assume negative values, including minus infinity. We also survey some results concerning the existence of the meromorphic extensions of the spectral zeta functions of fractal drums, based in an essential way on earlier results of the first author on the spectral (or eigenvalue) asymptotics of fractal drums. It follows from these results that the associated spectral zeta function has a (nontrivial) meromorphic extension, and we use some of our results about fractal zeta functions to show the new fact according to which the upper bound obtained for the corresponding abscissa of meromorphic convergence is optimal.

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  1. Department of Mathematics, University of California, 900 University Avenue, 231 Surge Building, Riverside, CA, 92521-0135, USA

    Michel L. Lapidus

  2. Department of Applied Mathematics, University of Zagreb, Faculty of Electrical Engineering and Computing, Unska 3, 10000, Zagreb, Croatia

    Goran Radunović & Darko Žubrinić

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  1. Michel L. Lapidus
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Correspondence to Michel L. Lapidus.

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To Professor Haïm Brezis, with profound admiration, on the occasion of his 70th birthday.

Avec mes sincères remerciements, pour votre amitié, vos enseignements, et votre exemple édifiant. Votre étudiant, Michel.

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Lapidus, M.L., Radunović, G. & Žubrinić, D. Fractal zeta functions and complex dimensions of relative fractal drums. J. Fixed Point Theory Appl. 15, 321–378 (2014). https://doi.org/10.1007/s11784-014-0207-y

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