Some Remarks on the Hausdorff and Spectral Dimension of V-Variable Nested Fractals
New families of random fractals, referred to as V-variable fractals (developed by Barnsley, Hutchinson, and Stenflo), are presented. In order to define and investigate a Laplacian and a Brownian motion–or, equivalently– a Dirichlet form on them, we introduce the class of V-variable nested fractals. These are nested fractals over families of iterated function systems with nested attractors and a uniform set of essential fixed points. So, the underlying graphs in the construction steps form sequences of comparable resistance networks. Dirichlet forms are defined ω-wise in a canonical way. In a survey style, we explain how to get Hausdorff and spectral dimension of such fractals by applying results of Furstenberg and Kesten on limits of products of random matrices.
KeywordsBrownian Motion Dirichlet Form Iterate Function System Variable Fractal Sierpinski Gasket
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