Boundaries of Disk-like Self-affine Tiles

TL;DR

This paper characterizes the boundary of disk-like self-affine tiles using sofic systems, derives conditions for number systems, and computes their Hausdorff dimensions with explicit formulas, especially for similarity matrices.

Contribution

It introduces a new method to identify tile boundaries with sofic systems and provides explicit formulas for their Hausdorff dimensions, improving upon previous results.

Findings

Boundary of tiles is identified with a sofic system.

Derived explicit conditions for the pair (A, D) to form a number system.

Computed the Hausdorff dimension of the boundary explicitly.

Abstract

Let $T:= T(A, {\mathcal D})$ be a disk-like self-affine tile generated by an integral expanding matrix $A$ and a consecutive collinear digit set ${\mathcal D}$, and let $f(x)=x^{2}+px+q$ be the characteristic polynomial of $A$. In the paper, we identify the boundary $\partial T$ with a sofic system by constructing a neighbor graph and derive equivalent conditions for the pair $(A,{\mathcal D})$ to be a number system. Moreover, by using the graph-directed construction and a device of pseudo-norm $\omega$, we find the generalized Hausdorff dimension $\dim_H^{\omega} (\partial T)=2\log \rho(M)/\log |q|$ where $\rho(M)$ is the spectral radius of certain contact matrix $M$. Especially, when $A$ is a similarity, we obtain the standard Hausdorff dimension $\dim_H (\partial T)=2\log \rho/\log |q|$ where $\rho$ is the largest positive zero of the cubic polynomial $x^{3}-(|p|-1)x^{2}-(|q|-|p|)x-|q|$, which is simpler than the known result.