Tube Formulae for Generalized von Koch Fractals through Scaling Functional Equations

TL;DR

This paper develops a general framework using scaling functional equations to analyze the geometric properties of generalized von Koch fractals, including their tube functions and complex dimensions.

Contribution

It introduces a novel method to solve scaling functional equations for fractals, extending analysis to a broader class of self-similar structures with varied polygons and ratios.

Findings

Derived explicit formulas for tubular neighborhood volumes.

Identified the complex dimensions of generalized von Koch fractals.

Extended classical fractal analysis to more flexible self-similar systems.

Abstract

In this work, we provide a treatment of scaling functional equations in a general setting involving fractals arising from sufficiently nice self-similar systems in order to analyze the tube functions, tube zeta functions, and complex dimensions of relative fractal drums. Namely, we express the volume of a tubular neighborhood in terms of scaled copies of itself and a remainder term and then solve this expression by means of the tube zeta functions. We then apply our methods to analyze these generalized von Koch fractals, which are a class of fractals that allow for different regular polygons and scaling ratios to be used in the construction of the standard von Koch curve and snowflake. In particular, we describe the volume of an inner tubular neighborhoods and the possible complex dimensions of such fractal snowflakes.