Topological isomorphism for rank-1 systems

We define the Polish space $\mathcal{R}$ of non-degenerate rank-1 systems. Each non-degenerate rank-1 system can be viewed as a measure-preserving transformation of an atomless, $\sigma$-finite measure space and as a homeomorphism of a Cantor space. We completely characterize when two non-degenerate rank-1 systems are topologically isomorphic. We also analyze the complexity of the topological isomorphism relation on $\mathcal{R}$, showing that it is $F_{\sigma}$ as a subset of $\mathcal{R} \times \mathcal{R}$ and bi-reducible to $E_0$. We also explicitly describe when a non-degenerate rank-1 system is topologically isomorphic to its inverse.