Cookie-Cutter-Like Sets with Graph-Directed Construction

In this chapter, we extend the cookie-cutter-like construction introduced by Ma, Rao, and Wen to the case having the graph-directed construction which is introduced by Mauldin and Williams and obtain a new class of fractals, which can be used to study the dimensions of the spectrum of discrete Schrodinger operators. Under suitable assumptions we prove that this class of fractals possesses the properties of bounded variation, bounded distortion, bounded covariation, and the existence of Gibbs-like measures. With these properties we give expressions for the Hausdorff dimensions, box dimensions, and packing dimensions of the fractals. We also discuss the continuous dependence of the dimensions on the defining data.