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arXiv · 1410.4101

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

Kevin G. Hare·Nikita Sidorov
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Abstract

Let $β_1,β_2>1$ and $T_i(x,y) = \bigl(\frac{x+i}{β_1}, \frac{y+i}{β_2}\bigr),\ i\in\{\pm1\}$. Let $A := A_{β_1, β_2}$ be the unique compact set satisfying $A = T_{1}(A) \cup T_{-1}(A)$. In this paper we give a detailed analysis of $A$, and the parameters $(β_1, β_2)$ where$A$ satisfies various topological properties. In particular, we show that if $β_1<β_2<1.202$,then $A$ has a non-empty interior, thus significantly improving the bound from [1]. In the opposite direction,we prove that the connectedness locus for this family studied in [16] is not simply connected.We prove that the set of points of $A$ which have a unique address has positive Hausdorff dimension for all $(β_1,β_2)$.Finally, we investigate simultaneous $(β_1,β_2)$-expansions of reals, which were the initial motivation for studying this family in [5].

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Kevin G. Hare, Nikita Sidorov. 2015-05-11. On a family of self-affine sets: topology, uniqueness, simultaneous expansions. https://doi.org/10.1017/etds.2015.41

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