Boundaries of Disk-Like Self-affine Tiles

Abstract

Let T := T(A,D) be a disk-like self-affine tile generated by an integral expanding matrix A and a consecutive collinear digit set D, and let f(x) = x2+px+q be the characteristic polynomial of A. In the paper, we identify the boundary ∂T with a sofic system by constructing a neighbor graph and derive equivalent conditions for the pair (A,D) to be a number system. Moreover, by using the graph-directed construction and a device of pseudo-norm ω, we find the generalized Hausdorff dimension dimωH(∂T) = 2 log ρ(M)/log{pipe}q{pipe} where ρ is the spectral radius of certain contact matrix M. Especially, when A is a similarity, we obtain the standard Hausdorff dimension dimH(∂T) = 2 log ρ/log{pipe}q{pipe} where ρ is the largest positive zero of the cubic polynomial x3-({pipe}p{pipe}-1)x2-({pipe}q{pipe}-{pipe}p{pipe})x-{pipe}q{pipe}, which is simpler than the known result. © 2013 Springer Science+Business Media New York.