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Let A be an expanding matrix with characteristic polynomial $$f(x)=x^2+px+3$$ and $$\mathcal {D}=\{0,v,\ell v+kAv\}$$ be a digit set where $$\ell ,k\in {\mathbb {Z}},\ v\in {\mathbb {R}}^2$$ so that $$\{v,Av\}$$ is linearly independent. It is well known that there exists a unique self-affine fractal T satisfying $$AT=T+\mathcal {D}$$ . In this paper, we give a complete characterization for the connectedness of T by using radix expansion.

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., Comment: 26 pages, 11 figures

We consider the disklikeness of the planar self-affine tile T T generated by an integral expanding matrix A A and a consecutive collinear digit set D = { 0 , v , 2 v , ⋯ , ( | q | − 1 ) v } ⊂ Z 2 {\mathcal {D}}= \{0, v, 2v, \cdots , (|q|-1)v \}\subset {\Bbb {Z}}^2 . Let f ( x ) = x 2 + p x + q f(x)=x^{2}+ p x+ q be the characteristic polynomial of A A . We show that the tile T T is disklike if and only if 2 | p | ≤ | q + 2 | 2|p|\leq |q+2| . Moreover, T T is a hexagonal tile for all the cases except when p = 0 p=0 , in which case T T is a square tile. The proof depends on certain special devices to count the numbers of nodal points and neighbors of T T and a criterion of Bandt and Wang (2001) on disklikeness.