Nonlinear Iterated Function Systems And Fractal Properties Of The Spectrum Of Schr(?)dinger Operator

In this paper, we study two problems from fractal geometry and operator theory:the fractal properties of the spectrum of a class of one-dimensional Schro¨dinger oper-ator with Sturm potentials, nolinear iterated function systems generated by a sequenceof digraphs.The structure of the spectrum of the Schro¨dinger operator is very important inmathematical physics. The studies are focused on a special potential–Sturmian poten-tial. The results involve the Lebesgue measure, the estimation of the fractal dimensionunder the condition that the frequency is ration number or its continued fraction expan-sion is periodic. We use the technique in dynamical systems and Cookie-Cutter-Likesets to get that the spectral generating bands possess properties of bounded variationand bounded covariation, then we show that there exists Gibbs-like measure on thespectrum. As an application, we show the connection of the pre-dimension of the spec-trum with the Hausdor? dimension and upper boxing dimension of the spectrum.The study of fractals is very widely. From the view of generative mechanism, itinclude self similar sets, self a?ne sets, graph-directed sets, cookie-cutter sets, moransets, random fractals, and the attractors and repellors of dynamical systems. We com-bine the methods of graph-directed system and cookie-cutter-like system to generate anew class of fractals, we prove that this class of fractals have the properties of boundedvariation, bound distortion, and bounded covariation. We also prove that there existGibbs-like measures on that sets. And we discuss Hausdor? Hausdor? dimensions,box dimensions, Packing dimensions, and the continuous dependence of the dimen-sions on the defining data.