Orbit equivalence of Cantor minimal systems

In this paper we study the descriptive complexity of the topological orbit equvalence relation for some Borel classes of Cantor minimal systems. Specifically, we study the Borel class of all Cantor minimal systems with only finitely many ergodic measures, and show that the orbit equivalence for this class is Borel bireducible with the equivalence relation $=^+$. We prove the same for the subclass of regular $\{0, 1\}$-Toeplitz subshifts or that of the uniquely ergodic minimal subshifts. We also study the orbit equivalence for the Borel class of minimal subshifts of finite topological rank. Denote by $R_n$ the orbit equivalence for minimal subshifts of topological rank $n\geq 2$. We prove that for any $n\geq 2$, $R_n$ is virtually countable, i.e., Borel reducible to a countable Borel equivalence relation. Moreover, $R_2$ is virtually amenable. On the other hand, $R_n$ is not smooth when $n\geq 2$, is not virtually hyperfinite when $n\geq 4$, and is not virtually treeable when $n\geq 5$. For any $n\geq 2$, our contructions yield uniquely ergodic minimal subshifts of topological rank exactly $n$.