Fractal analysis of unbounded sets in Euclidean spaces and Lapidus zeta functions

In this thesis , we consider relative fractal drums and their corresponding Lapidus fractal zeta functions, as well as a generalization of this notions to the case of unbounded sets at infinity. Relative fractal drums themselves are a generalization of the notion of a bounded subset in an Euclidean space. Here, we continue the ongoing research into their properties and the higher-dimensional theory of their fractal zeta functions and complex dimensions which started as a collaboration between M. L. Lapidus and D. Žubrinic in 2009 with the later addition of the author of this thesis. The theory of complex dimensions is already well developed for fractal strings; that is, for fractal subsets of the real line. The complex dimensions of a relative fractal drum are defined as poles of a meromorphic continuation of its corresponding distance or tube zeta function. Complex dimensions of a relative fractal drum generalize, in a way, the notion its box (or Minkowski) dimension. More precisely, under some mild conditions, the value of the box dimension of a relative fractal drum is a pole of its corresponding fractal zeta function with maximal real part. Moreover, the residue computed at this pole is closely related to its Minkowski content. Here we derive important results which further justify the notion of ‘complex dimensions’ and connect it to fractal properties of a given relative fractal drum. More precisely, we establish fractal tube formulas for a class of relative fractal drums which express their relative tube function; that is, the Lebesgue measure of their relative δ-neighborhood for small values of δ, as a sum over the residues of their fractal zeta function. These formulas are given with or without an error term and hold pointwise or distributionally depending on the growth properties of the corresponding fractal zeta function. The importance of these formulas is that they show how the complex dimensions are related to the asymptotic development of the relative tube function of a given relative fractal drum. As an application we derive a Minkowski measurability criterion for a large class of relative fractal drums. Furthermore, we also show that the complex dimensions of a relative fractal drum are, as expected, invariant to the dimension of the ambient space. We introduce a further generalization of the theory of complex dimensions to the context of unbounded sets at infinity which can be used as a new approach of applying fractal analysis to unbounded subsets in Euclidean spaces. This is done for unbounded sets of finite Lebesgue measure by introducing the notions of Minkowski content at infinity and Minkowski (or box) dimension at infinity which describe their fractal properties. Furthermore, we proceed by introducing an appropriate Lapidus (or distance) zeta function at infinity and show that it is well connected with the fractal properties of unbounded sets. We proceed by constructing interesting examples of quasiperiodic sets at infinity with arbitrary number (even infinite) of quasiperiods that exhibit complex fractal behavior. We also address the natural question which arises when dealing with unbounded sets and their fractal properties; that is, establish results about the fractal properties of their images under the one-point compactification and under the geometric inversion. Furthermore, we also investigate fractal properties of unbounded sets of infinite Lebesgue measure by introducing notions of the parametric φ-shell Minkowski content at infinity and the corresponding parametric φ-shell Minkowski (or box) dimension at infinity and we establish results connecting these notions with the distance zeta function at infinity. Finally we demonstrate how fractal analysis of unbounded sets via the geometric inversion may be applied to investigate bifurcations of dynamical systems occurring at infinity.