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 dimHω(∂T)=2logρ(M)/log∣q∣ where
ρ(M) is the spectral radius of certain contact matrix M. Especially,
when A is a similarity, we obtain the standard Hausdorff dimension dimH(∂T)=2logρ/log∣q∣ where ρ is the largest positive zero of
the cubic polynomial x3−(∣p∣−1)x2−(∣q∣−∣p∣)x−∣q∣, which is simpler than
the known result.Comment: 26 pages, 11 figure