关于一族自仿射集：拓扑，唯一性，同时展开
On a family of self-affine sets: topology, uniqueness, simultaneous expansions
摘要：
设$β_1,β_2>1$和$T_i(x,y) = \bigl(\frac{x+i}{β_1}, \frac{y+i}{β_2}\bigr),\ i\in\{\pm1\}$。设$A := A_{β_1, β_2}$是满足$A = T_{1}(A) \cup T_{-1}(A)$的唯一紧集.本文对$A$进行了详细的分析，给出了参数$(β_1, β_2)$，其中$A$满足各种拓扑性质。特别地，我们证明了如果$β_1<β_2<1.202$，则$A$具有非空内部，从而显著地改进了[1]的界。在相反的方向上，我们证明了[16]中所研究的这个族的连通轨迹不是单连通的，我们证明了$A$中具有唯一地址的点集对所有$(β_1,β_2)$都具有正的豪斯多夫维数，最后我们研究了实数的同时$(β_1,β_2)$ -展开，这是[5]中研究这个族的最初动机.
英文摘要：
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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