Two-dimensional self-affine sets with interior points, and the set of uniqueness

Abstract

Let $M$ be a $2\times 2$ real matrix with both eigenvalues less than~1 in modulus. Consider two self-affine contraction maps from ${\mathbb{R}}^2\to {\mathbb{R}}^2$, \begin{equation*} T_m(v) = M v - u \ \ \mathrm{and}\ \ T_p(v) = M v + u, \end{equation*} where $u\ne 0$. We are interested in the properties of the attractor of the iterated function system (IFS) generated by $T_m$ and $T_p$, i.e., the unique non-empty compact set $A$ such that $A=T_m(A)\cup T_p(A)$. Our two main results are as follows: 1. If both eigenvalues of $M$ are between $2^{-1/4}\approx 0.8409$ and $1$ in absolute value, and the IFS is non-degenerate, then $A$ has non-empty interior. 2. For almost all non-degenerate IFS, the set of points which have a unique address is of positive Hausdorff dimension -- with the exceptional cases fully described as well. This paper continues our work begun in [11].

29 pages, 7 figures

References in corpus (1)

• On a family of self-affine sets: topology, uniqueness, simultaneous expansions

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