Dynamics of Self-Similar Tilings

This paper investigates dynamical systems arising from the action by translations on the orbit closures of self-similar and self-affine tilings of R d : The main focus is on spectral properties of such systems which are shown to be uniquely ergodic. We establish criteria for weak mixing and pure discrete spectrum for wide classes of such systems. They are applied to a number of examples which include tilings with polygonal and fractal tile boundaries; systems with pure discrete, continuous and mixed spectrum. . 1991 Mathematics Subject Classification. Primary 52C20, 52C22, 28D15, 58F19; Secondary 82D25. Key words and phrases: tiling, measure-preserving group action, discrete spectrum. y Department of Mathematics, Box 354350, University of Washington, Seattle, WA 98195; E-mail [solomyak@math.washington.edu]. Supported in part by NSF Grants 9201369 and 9500744, and the University of Washington Graduate School Fund. Introduction In this paper we study the properties of dynamica..