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Minkowski measurability criteria for compact sets and relative fractal drums in Euclidean spaces


Lapidus L., Michel; Radunović Goran; Žubrinić Darko
Minkowski measurability criteria for compact sets and relative fractal drums in Euclidean spaces // Proceedings of the 6th Cornell Conference on Analysis, Probability, and Mathematical Physics on Fractals (2019) (znanstveni, prihvaćen)


Naslov
Minkowski measurability criteria for compact sets and relative fractal drums in Euclidean spaces

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

Vrsta, podvrsta
Radovi u časopisima, znanstveni

Izvornik
Proceedings of the 6th Cornell Conference on Analysis, Probability, and Mathematical Physics on Fractals (2019)

Status rada
Prihvaćen

Ključne riječi
Mellin transform ; complex dimensions of a relative fractal drum ; relative fractal drum ; fractal set ; box dimension ; fractal zeta func tions ; distance zeta function ; tube zeta function ; fractal string ; Minkowski content ; Minkowski measurability criterion ; Minkowski measurable set ; residue ; meromorphic extension ; gauge-Minkowski measurability ; singularities of fractal zeta functions

Sažetak
We establish a Minkowski measurability criterion for a large class of relative fractal drums (or, in short, RFDs), in Euclidean spaces of arbitrary dimension in terms of their complex dimensions, which are defined as the poles of their associated fractal zeta functions. Relative fractal drums represent a far-reaching generalization of bounded subsets of Euclidean spaces as well as of fractal strings studied extensively by the first author and his collaborators. In fact, the Minkowski measur ability criterion established here is a generalization of the corresponding one obtained f or fractal strings by the first author and M. van Frankenhuijsen. Similarly as in the case of fractal strings, the criterion established here is formulated in terms of the locations of the principal complex dimensions associated with the relative drum under consideration. These complex dimensions are defined as poles or, more generally, singularities of the corresponding distance (or tube) zeta function. We also reflect on the notion of gauge-Minkowski measurability of RFDs and establish several results connecting it to the nature and location of the complex dimensions. (This is especially useful when the underlying scaling does not follow a classic power law.) We illustrate our results and their applications by means of a number of interesting examples.

Izvorni jezik
Engleski

Znanstvena područja
Matematika



POVEZANOST RADA


Projekt / tema
HRZZ-IP-2014-09-2285 - Geometrijska, ergodička i topološk a analiza nisko-dimenzionalnih dinamičkih sustava (Siniša Slijepčević, )

Ustanove
Fakultet elektrotehnike i računarstva, Zagreb,
Prirodoslovno-matematički fakultet, Zagreb