Boundaries of Disk-like Self-affine Tiles

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.

26 pages, 11 figures

References in corpus (1)

• Connectedness of planar self-affine sets associated with non-consecutive collinear digit sets

Cited by in corpus (3)

• On the connectedness of planar self-affine sets
• Connectedness of self-affine sets with product digit sets
• Elementary fractal geometry. Networks and carpets involving irrational rotations