[*the real case*]{}, the second [*the Jordan block case*]{}, and the last [*the complex case*]{}.

We will say that the IFS is [*degenerate*]{} if it is restricted to a one-dimensional subspace of ${\mathbb{R}}^2$. This will occur if any of the eigenvalues are $0$. It will also occur if ${\lambda}= \mu$ in the real case, or (equivalently) if $b = 0$ in the complex case. If our IFS is non-degenerate, then $u$ can be chosen to be a cyclic vector for $M$, i.e., such that the span of $\{M^n u \mid n\ge0\}$ is all of $\mathbb R^2$ (which we will assume henceforth).

In [@HS] the authors studied the real case with ${\lambda}, \mu > 0$. Properties of the complex case have been studied extensively since the seminal paper [@BH] - see, e.g., [@Cal] and references therein. Note that most authors concentrate on the connectedness locus, i.e., pairs $(a,b)$ such that the attractor $A$ is connected.

In the present paper we study all three of the above cases, allowing us to make general claims. Our main result is

\[thm:interior\] If all eigenvalues of $M$ are between $2^{-1/4} \approx 0.8409$ and $1$ in absolute value, and the IFS is non-degenerate, then the attractor 