engleski

An overview of the theory of complex dimensions and fractal zeta functions

We will give an overview of the main results of the new higher-dimensional theory of complex dimensions valid for arbitrary subsets of Euclidean spaces of any dimension. This theory has been developed in a series of papers and in a research monograph ”Fractal Zeta Functions and Fractal Drums: Higher-Dimensional Theory of Complex Dimensions” coauthored by M. L. Lapidus, G.Radunović and D. Žubrinić. The theory gives a far-reaching generalization of the one-dimensional theory (for fractal strings) developed by M. L. Lapidus, M.van Frankenhuijsen and their numerous collaborators. The complex dimensions of a given set are defined as the poles (or more general singularities) of the (distance or tube) fractal zeta function associated with the given set and they generalize the well-known notion of the Minkowski dimension. We will define and give the main properties and results for the distance fractal zeta function. Although the complex dimensions are defined analytically, we will show that they have a deep geometric meaning connected to the fractal nature of the given set and the intrinsic oscillations in its geometry. Namely, this can be seen from the so-called fractal tube formulas which, under appropriate assumptions, give an asymptotic expansion of the Lebesgue measure of the delta-neighborhood of the given set (when delta is close to zero) in terms of its complex dimensions. We will also reflect on some of the possible applications of the theory in studying dynamical systems and their bifurcations.

fractal zeta function ; complex dimensions ; Minkowski content ; box dimension

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