Fractal zeta functions at infinity and the $ϕ$-shell Minkowski content

Abstract

We study fractal properties of unbounded domains with infinite Lebesgue measure via their complex fractal dimensions. These complex dimensions are defined as poles of a suitable defined Lapidus fractal zeta function at infinity and are a generalization of the Minkowski dimension for a special kind of a degenerated relative fractal drum at infinity. It is a natural generalization of a similar approach applied to unbounded domains of finite Lebesgue measures investigated previously by the author. In this case we adapt the definition of Minkowski content and dimension at infinity by introducing the so-called $ϕ$-shells where $ϕ$ is a real parameter. We show that the new notion of the upper $ϕ$-shell Minkowski dimension is independent on the paramter $ϕ$ and well adapted to the fractal zeta function at infinity. We also study how the new definition connects with the one-point compactification and the classical fractal properties of the corresponding compactified domain. We also give a construction of maximally hyperfractal and of quasperiodic domains at infinity of infinite Lebesgue measure.