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

Let $\unicode[STIX]{x1D6FD}_{1},\unicode[STIX]{x1D6FD}_{2}>1$ and $T_{i}(x,y)=((x+i)/\unicode[STIX]{x1D6FD}_{1},(y+i)/\unicode[STIX]{x1D6FD}_{2}),i\in \{\pm 1\}$ . Let $A:=A_{\unicode[STIX]{x1D6FD}_{1},\unicode[STIX]{x1D6FD}_{2}}$ 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 $(\unicode[STIX]{x1D6FD}_{1},\unicode[STIX]{x1D6FD}_{2})$ where $A$ satisfies various topological properties. In particular, we show that if $\unicode[STIX]{x1D6FD}_{1}