Intersections of moving fractal sets

TL;DR

This paper investigates how the intersection properties of moving fractal sets depend on their relative position, revealing a self-affine behavior characterized by a Hurst exponent linked to their scaling properties.

Contribution

It provides a generic analytical expression for the Hurst exponent governing intersections of moving fractals, supported by proofs and Monte-Carlo simulations.

Findings

Mass of intersection is a self-affine function of relative position.

Hurst exponent depends on the scaling exponents of the sets.

Analytical results are validated through simulations.

Abstract

Intersection of a random fractal or self-affine set with a linear manifold or another fractal set is studied, assuming that one of the sets is in a translational motion with respect to the other. It is shown that the mass of such an intersection is a self-affine function of the relative position of the two sets. The corresponding Hurst exponent h is a function of the scaling exponents of the intersecting sets. A generic expression for h is provided, and its proof is offered for two cases --- intersection of a self-affine curve with a line, and of two fractal sets. The analytical results are tested using Monte-Carlo simulations.