Ergodic Theory and Dynamical Systems

Dynamics of self-similar tilings

a1 Department of Mathematics, Box 354350, University of Washington, Seattle, WA 98195, USA (e-mail:


This paper investigates dynamical systems arising from the action by translations on the orbit closures of self-similar and self-affine tilings of ${\Bbb R}^d$. 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.

(Received March 13 1995)
(Revised July 14 1995)


Supported in part by NSF Grants 9201369 and 9500744.