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

Abstract

Let ${\beta }_1,{\beta }_2>1$ and $T_i(x,y)=(\frac{x+i}{{\beta }_1},\frac{y+i}{{\beta }_2}),i\in \{\pm 1\}$. Let $A:=A_{{\beta }_1,{\beta }_2}$ 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 $({\beta }_1,{\beta }_2)$ where$A$ satisfies various topological properties. In particular, we show that if ${\beta }_1<{\beta }_2<1.202$,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 $({\beta }_1,{\beta }_2)$.Finally, we investigate simultaneous $({\beta }_1,{\beta }_2)$-expansions of reals, which were the initial motivation for studying this family in [5].

44 pages, 18 figures

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