Tube Formulas for Graph-Directed Fractals

Abstract

Lapidus and Pearse proved recently an interesting formula about the volume of tubular neighborhoods of fractal sprays, including the self-similar fractals. We consider the graph-directed fractals in the sense of graph self-similarity of Mauldin-Williams within this framework of Lapidus-Pearse. Extending the notion of complex dimensions to the graph-directed fractals we compute the volumes of tubular neighborhoods of their associated tilings and give a simplified and pointwise proof of a version of Lapidus-Pearse formula, which can be applied to both self-similar and graph-directed fractals.

Minkowski content and fractal Euler characteristic for conformal graph directed systems

“…If we allow the OSC to be satisfied with a different feasible open set, then often the system can still be represented by a cGDS. [3] π [4] π [5] π [6] L 1,1 L 2,1 Figure 2. Primary gaps of the limit set of the cGDS from Ex.…”

Graph-directed sprays and their tube volumes via functional equations

“…Beyond the self-similar fractals, the next family of interest is the class of graphdirected fractals. For tube volumes of graph-directed fractals there are formulas (see [2]), which are analogous to the tube formulas of Lapidus and Pearse ([8]). So far as we know, the notion of spray has not been extended to the graph-directed setting.…”

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