of the IFS has non-empty interior. More precisely,

-   If $0.832 < {\lambda}< \mu < 1$ then $A_{{\lambda}, \mu}$, the attractor for the (positive) real case, has non-empty interior [@HS Corollary 1.3].

-   If $2^{-1/2} \approx 0.707 < {\lambda}< \mu < 1$ then $A_{-{\lambda}, \mu}$, the attractor for the (mixed) real case, has non-empty interior.

-   If $0.832 < \nu  < 1$ then $A_{\nu}$, the attractor for the Jordan block case, has non-empty interior.

-   If $2^{-1/4} \approx 0.841 < |\kappa|  < 1$ with $\kappa = a+ b i \not\in {\mathbb{R}}$ then $A_{{\kappa}}$, the attractor for the complex case, has non-empty interior.

The remaining three cases are shown in Section \[sec:interior\]. The last case relies upon an argument of V. Kleptsyn [@MO].

Some non-explicit results are known in the complex case. Let $\kappa = a + b i$ and consider the attractor $A'_{\kappa}$ satisfying $A'_{\kappa}= {\kappa}A'_{\kappa}\cup ({\kappa}A'_{\kappa}+ 1)$ which is clearly similar to $A_{\kappa}$.

Let ${\kappa}\not\in {\mathbb{R}}$ be sufficiently close to $1$ in absolute value. Then $A'_{{\kappa}}\supset \{z : |z| \le 1\}$.

Since $A'_{\kappa}$ tends to a segment in $\mathbb R$ as the imaginary part of ${\kappa}$ tends to 0 in the Hausdorff metric (with any fixed real part), it 