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 dim ωH (∂T) = 2 log ρ (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 dim H(∂T) = 2 log ρ / 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. Copyright © 2013 Springer Science+Business Media New York.
|Journal||Discrete & Computational Geometry|
|Publication status||Published - May 2013|
CitationLeung, K.-S., & Luo, J. J. (2013). Boundaries of disk-like self-affine tiles. Discrete & Computational Geometry, 50(1), 194-218.
- Self-affine tile
- Sofic system
- Number system
- Neighbor graph
- Contact matrix
- Graph-directed set
- Hausdorff dimension