A self-affine tile is a compact set $T$ in $\mathbb R^n$ of positive Lebesgue measure for which there is an $n\times n$ expanding matrix $A$ (i.e. all its eigenvalues have modulus greater than 1) such that the affinely inflated copy $A(T)$ of $T$ can be perfectly tiled with essentially disjoint translates of $T$.

Thus we have $$ A(T) = \cup_{i=1}^m (T+d_i); \mathcal D= d_1,d_2,\dots,d_m $$

where $|det(A)| =|\mathcal D|= m$

Results of Kenyon (Projecting the one-dimensional Sierpinski gasket Projecting the one-dimensional Sierpinski gasket. Israel J. Math. 97 (1997), 221--238.) and Lagarias and Wang (Self-affine tiles in $R^n$. Adv. Math. 121 (1996), no. 1, 21--49) tells that such sets always can be used to give a translational tiling of $R^n$ and has boundary of measure zero and has nonempty interiors.

Thus in one dimension we can think of them as a union of intervals (possibly infinitely many ).

My question is :-

Is there a characterization of self-affine tiles in $\mathbb R$ which are union of finitely many intervals ?

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