Naslov
Fractal zeta functions and complex dimensions: A general higher-dimensional theory

Autori
Lapidus, Michel L. ; Radunović, Goran ; Žubrinić, Darko

Vrsta, podvrsta i kategorija rada
Poglavlja u knjigama, pregledni

Knjiga
Fractal Geometry and Stochastics V

Urednik/ci
C. Bandt, K. Falconer and M. Zähle

Izdavač
Birkhäuser ; Springer

Grad
Basel : Boston : Berlin

Godina
2015

Raspon stranica
229-257

ISBN
978-3-319-18659-7

Ključne riječi
Fractal set, fractal zeta functions, distance zeta function, tube zeta function, geometric zeta function of a fractal string, Minkowski content, Minkowski measurability, upper box (or Minkowski) dimension, complex dimensions of a fractal set, holomorphic and meromorphic functions, abscissa of convergence, quasiperiodic function, quasiperiodic set, relative fractal drum, fractal tube formulas

Sažetak
In 2009, the first author introduced a class of zeta functions, called `distance zeta functions', which has enabled us to extend the existing theory of zeta functions of fractal strings and sprays (initiated by the first author and his collaborators in the early 1990s) to arbitrary bounded (fractal) sets in Euclidean spaces of any dimensions. A closely related tool is the class of `tube zeta functions', defined using the tube function of a fractal set. These zeta functions exhibit deep connections with Minkowski contents and upper box (or Minkowski) dimensions, as well as, more generally, with the complex dimensions of fractal sets. In particular, the abscissa of (Lebesgue, i.e., absolute) convergence of the distance zeta function coincides with the upper box dimension of a set. We also introduce a class of transcendentally quasiperiodic sets, and describe their construction based on a sequence of carefully chosen generalized Cantor sets with two auxilliary parameters. As a result, we obtain a family of ``maximally hyperfractal'' compact sets and relative fractal drums (i.e., such that the associated fractal zeta functions have a singularity at every point of the critical line of convergence). Finally, we discuss the general fractal tube formulas and the Minkowski measurability criterion obtained by the authors in the context of relative fractal drums (and, in particular, of bounded subsets of $\mathbb{; ; ; R}; ; ; ^N$).

Izvorni jezik
Engleski

Znanstvena područja
Matematika

Napomena
Based on a plenary lecture given by Professor Michel L. Lapidus at the conference Fractal Geometry and Stochastics V, Tabarz, Germany.

POVEZANOST RADA