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Relative fractal drums, complex dimensions and geometric oscillations

We give an overview of the higher-dimensional theory of complex dimensions for relative fractal drums. Relative fractal drums or, in short, RFDs are a far reaching and convenient generalization of compact sets in Euclidean spaces. For such objects we associate a fractal zeta function which we call the distance (or Lapidus) zeta function. The corresponding complex dimensions of the RFD are then defined as the poles (or more general singularities) of the associated distance zeta function. These complex dimensions generalize the classical Minkowski dimension and are connected to the intrinsic geometric oscillations of the RFD. Possible application of the theory could be found in fractal analysis of bifurcations of dynamical systems. This is a joint work with Michel L. Lapidus and Darko Zubrinic.

Mellin transform; complex dimensions of a relative fractal drum; relative fractal drum; fractal set; box dimension; fractal zeta function; distance zeta function; tube zeta function; fractal string; Minkowski content; Minkowski measurable set; fractal tube formula; residue; meromorphic extension

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