Dimensions of Points in Self-similar Fractals

Self-similar fractals arise as the unique attractors of iterated function systems (IFSs) consisting of finitely many contracting similarities satisfying an open set condition. Each point x in such a fractal F arising from an IFS S is naturally regarded as the “outcome” of an infinite coding sequence T (which need not be unique) over the alphabet Σk = {0, . . . , k − 1}, where k is the number of contracting similarities in S. A classical theorem of Moran (1946) and Falconer (1989) states that the Hausdorff and packing dimensions of a self-similar fractal coincide with its similarity dimension, which depends only on the contraction ratios of the similarities. The theory of computing has recently been used to provide a meaningful notion of the dimensions of individual points in Euclidean space. In this paper, we use (and extend) this theory to analyze the dimensions of individual points in fractals that are computably self-similar, meaning that they are unique attractors of IFSs that are computable and satisfy the open set condition. Our main theorem states that, if F ⊆ Rn is any computably self-similar fractal and S is any IFS testifying to this fact, then the dimension identities dim(x) = sdim(F ) dimS (T ) ∗Department of Computer Science, Iowa State University, Ames, IA 50011 USA. [email protected]. Research supported in part by National Science Foundation Grants 0344187, 0652569, and 0728806 and by Spanish Government MEC Project TIN 200508832-C03-02. Part of this author’s research was performed during a sabbatical at the University of Wisconsin and two visits at the University of Zaragoza. †Departamento de Informática e Ingenieŕıa de Sistemas, Maŕıa de Luna 1, Universidad de Zaragoza, 50018 Zaragoza, SPAIN. [email protected]. Research supported in part by Spanish Government MEC Project TIN 2005-08832-C03-02.