Self-affine tiles in ℝn

Abstract

A self-affine tile in ℝⁿ is a set T of positive measure with A(T) = ∪ <inf>d ∈ script D</inf> (T + d), where A is an expanding n × n real matrix with |det(A)| = m an integer, and script D = {d, d<inf>2</inf>, ..., d<inf>m</inf>} ⊆ ℝⁿ is a set of m digits. It is known that self-affine tiles always give tilings of ℝn by translation. This paper extends known characterizations of digit sets script D yielding self-affine tiles. It proves several results about the structure of tilings of ℝⁿ possible using such tiles, and gives examples showing the possible relations between self-replicating tilings and general tilings, which clarify results of Kenyon on self-replicating tilings. © 1996 Academic Press, Inc.