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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 study the connectedness of the planar self-affine sets $T(A,{\mathcal{D}})$ generated by an integer expanding matrix $A$ with $|\det(A)|=3$ and a non-collinear digit set ${\mathcal D}=\{0, v, kAv\}$ where $k\in {\mathbb Z}\setminus\{0\}$ and $v\in {\mathbb Z}^2$ such that $\{v, Av\}$ is linearly independent. By checking the characteristic polynomials of $A$ case by case, we obtain a criterion concerning only $k$ to determine the connectedness of $T(A,{\mathcal{D}})$., 13 pages, 10 figures

In the paper, we focus on the connectedness of planar self-affine sets $T(A,{\mathcal{D}})$ generated by an integer expanding matrix $A$ with $|\det (A)|=3$ and a collinear digit set ${\mathcal{D}}=\{0,1,b\}v$, where $b>1$ and $v\in {\mathbb{R}}^2$ such that $\{v, Av\}$ is linearly independent. We discuss the domain of the digit $b$ to determine the connectedness of $T(A,{\mathcal{D}})$. Especially, a complete characterization is obtained when we restrict $b$ to be an integer. Some results on the general case of $|\det (A)|> 3$ are obtained as well., 15 pages, 10 figures