On the intersection of fractal cubes

TL;DR

This paper studies the intersections of fractal k-cubes, providing a representation theorem, dimension formulas, and conditions for finiteness, cardinality, and topological properties like being a dendrite.

Contribution

It introduces a graph-directed system approach to analyze fractal cube intersections, deriving new formulas and conditions for their measure, cardinality, and topological structure.

Findings

Representation of intersections as attractors of graph-directed systems

Dimension formulas for intersections

Conditions for finite measure and specific cardinalities

Abstract

We consider the intersections of fractal k-cubes of order n and intersections of their respective opposite l-faces. The main result of the paper is the theorem on representation of such intersection as the attractor of a graph-directed system of similarities in terms of the sets of units corresponding to these cubes and intersections of pairs of l-faces. As a corollary, we prove dimension formula for the intersection and the condition of finiteness of its measure. Another corollary gives the conditions under which the intersections have the given cardinality. Applying these techniques, we obtain the conditions under which a fractal k-cube has the finite intersection property and the conditions under which the fractal cube is a dendrite.