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, where we allow the generators of the spray to be pluriphase. We give also an example to illustrate the application of this notion to the computation of the tube volume of graph-directed fractals.