Graph-Directed Sprays and Their Tube Volumes via Functional Equations

TL;DR

This paper introduces a generalized concept of sprays for graph-directed fractals, deriving a tube formula via functional equations to better analyze their geometric properties.

Contribution

It extends the classical spray notion to include multiple generators, enabling application to graph-directed fractals and deriving related volume formulas.

Findings

Established a functional equation for the volume of inner neighborhoods.

Derived a tube formula for generalized sprays.

Enhanced analytical tools for fractal geometry.

Abstract

The notion of sprays introduced by Lapidus and his co-workers has proved useful in the context of fractal tube formulas. In the present note, we propose a more general concept of sprays, where we allow several generators to make them more convenient for applications to graph-directed fractals. Using a simple functional equation satisfied by the volume of the inner $\varepsilon$-neighborhood of such a generalized spray, we establish a tube formula for them.