Boundaries of Disk-Like Self-affine Tiles

Let be a disk-like self-affine tile generated by an integral expanding matrix and a consecutive collinear digit set , and let be the characteristic polynomial of . In the paper, we identify the boundary with a sofic system by constructing a neighbor graph and derive equivalent conditions for the pair to be a number system. Moreover, by using the graph-directed construction and a device of pseudo-norm , we find the generalized Hausdorff dimension where is the spectral radius of certain contact matrix . Especially, when is a similarity, we obtain the standard Hausdorff dimension where is the largest positive zero of the cubic polynomial , which is simpler than the known result.