with “digits” 0 and 1. Note that if ${\kappa}\in {\mathbb{R}}$ then the resulting IFS is real (and degenerate).

Throughout we will refer to $[i_1\dots i_k]$ as the [*cylinder*]{} of all $(a_i)_0^\infty\in\{m,p\}^{\mathbb N}$ such that $a_j = i_j$ for $j = 1, \dots, k$. We note that this is a compact subset of $\{m,p\}^{\mathbb{N}}$ under the usual product topology.

Attractors with interior {#sec:interior}
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The first question that we are interested in is, when does $A$ have interior. For the real and Jordan block case we look at a related, albeit somewhat easier, question: when is $(0,0)$ contained in the interior of $A$? We will say that $(-{\lambda}, \mu)$ for the mixed real case is in ${{\mathcal{Z}}}_{\mathbb{R}}$ if $(0,0) \in \mathrm{int}(A_{-{\lambda}, \mu})$. An equivalent definition is given for ${{\mathcal{Z}}}_J$ for the Jordan block case.

In fact, the real case (both mixed and positive) and the Jordan block case are both special cases of a more general result – see Theorem \[thm:tool2\] below.

Consider a contraction matrix $M$ with all real eigenvalues such that any duplicate eigenvalue is within the same Jordan block. That is, let $J_{{\lambda}, k}$ be the $k \times k$ Jordan block $$J_{{\lambda}, k} = \begin{pmatrix}
                         {\lambda}& 1   &        