Graph-directed sprays and their tube volumes via functional equations

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 define a notion of a graph-directed spray, associated with a weighted directed graph. Using a simple functional equation satisfied by the volume of the inner $\epsilon$-neighborhood of such a graph-directed spray, we establish a tube formula for them. We give also an example to illustrate the application of this notion to the computation of the tube volume of graph-directed fractals.