We consider a class of planar self-affine tiles T that are generated by the lower triangular expanding matrices and the product-form digit sets. We give necessary and sufficient conditions for T to be connected and disk-like. Also for the disconnect case, we give a condition that enumerates the number of connected components of T .

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For a conformal iterated function system satisfying the bounded distortion property and the weak separation condition, we prove a formula for the Hausdorff dimension of the attractor, and establish the equality between the Hausdorff dimension and the growth dimension. Furthermore, a relation between the open set condition and the weak separation condition is given.

Let {\mathscr H}_K denote the family of homogenous IFSs that satisfy the open set condition (OSC) and generate the same self-similar set K , we call the IFSs in {\mathscr H}_K isotopic , and give the isotopic class {\mathscr H}_K a multiplication operation defined by composition. The finitely generated property of {\mathscr H}_K was first studied by Feng and Wang on \mathbb{R} [FW], and by the authors on \mathbb{R}^d under the strong separation condition [DL]. In this paper, we continue the investigation of the isotopic class on \mathbb{R}^d . By using a new technique with the OSC, we prove that {\mathscr H}_K is finitely generated if either (i) K is totally disconnected, or (ii) the convex hull {\rm Co}(K) is a polytope, and there exists a line L passing through a vertex of Co (K) such that L\cap K is a totally disconnected infinite set. The conditions are easy to check and are satisfied by many standard self-similar sets.