Generating Method of Boundaries of Digit Tiles

which satisfies A (T ) = ∑Ni=1 (T + d i ) , called a digit tile (see Figure 1). The digit tiles are found in tiling theory, elementary number theory, wavelets, ergodic theory etc. Several basic properties of the compact set T are investigated by many articles of Bandt [3], Gilbert [8], Katai and Szabo [10], Kenyon [11, 12], Knuth [13], Lagarias and Wang [4, 15], Vince [6, 18] and Sadahiro et al [16] etc. The boundary of the compact set T is usually fractal. The purpose of this paper is to give a generating method of the fractal boundary of the set T . For 2-dimensional digit tiles, Dekking propose the recurrent set method which uses the endomorphism of free groups of rank 2 (Dekking [4, 5], Ito and Ohtsuki [9]). In this paper, we extend Dekking’s method by using the higher dimensional endomorphism, which is an endomorphism of a Z-module Gd−1 introduced by [1], [2], [7], [17], instead of using the endomorphism of the free group Fd . In fact, from an integer expanding matrix A and a digit set DA = {d1, . . . , dN }, we define a tiling mapΘ : Gd → Gd by