uniqueness'
---

[^1] [^2]

Introduction {#sec:intro}
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Consider two self-affine linear contraction maps $T_m, T_p: {\mathbb{R}}^2 \to {\mathbb{R}}^2$: $$\label{eq:main}
T_m(v) =  M v - u \ \ \mathrm{and}\ \ T_p(v) = M v + u,$$ where $M$ is a $2\times2$ real matrix with both eigenvalues less than 1 in modulus and $u\neq0$. Here “$m$” is for “minus” and “$p$” is for “plus”. We are interested in the iterated function system (IFS) generated by $T_m$ and $T_p$. Then, as is well known, there exists a unique non-empty compact set $A$ such that $A = T_m(A) \cup T_p(A)$.

The properties that we are interested in (non-empty interior of $A$ and the set of uniqueness) do not change if we consider a conjugate of $T_m$ and $T_p$. That is, if we consider $g \circ T_i \circ g^{-1}$ instead of the $T_i$ where $g$ is any invertible linear map from ${\mathbb{R}}^2 \to {\mathbb{R}}^2$. As such, we can assume that $M$ is a $2 \times 2$ matrix in one of three forms: $$\begin{pmatrix}         {\lambda}& 0  \\ 0 & \mu \end{pmatrix},
\begin{pmatrix}         \nu & 1  \\ 0 & \nu \end{pmatrix},
\begin{pmatrix}         a & b  \\ -b & a \end{pmatrix}.$$

We will call the first of these 