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

TL;DR

This paper extends fractal analysis to unbounded domains with infinite measure by defining complex dimensions via zeta functions at infinity, introducing the $$-shell Minkowski content, and exploring their properties and applications.

Contribution

It introduces a novel approach to fractal properties at infinity using complex dimensions and $$-shell Minkowski content, generalizing previous finite measure methods.

Findings

The upper $$-shell Minkowski dimension is independent of $$.

The new fractal dimension connects with classical properties through domain compactification.

Constructs of maximally hyperfractal and quasiperiodic domains at infinity are provided.

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 $\phi$-shells where $\phi$ is a real parameter. We show that the new notion of the upper $\phi$-shell Minkowski dimension is independent on the paramter $\phi$ 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.