DYNAMICS OF SELF-SIMILAR TILINGS

This paper investigates dynamical systems arising from the action by t ranslations on the orbit closures of self-similar and self-affine tili ngs of R-d. Th, main focus is on spectral properties of such systems w hich are shown to be uniquely ergodic. We establish criteria for weak mixing and pure discrete spectrum for wide classes of such systems. Th ey are applied to a number of examples which include tilings with poly gonal and fractal tile boundaries; systems with pure discrete, continu ous and mixed spectrum.