Classification of the crossed product $C(M)\times_θ\Z_p$ for certain pairs $(M,θ)$

Abstract

Let $M$ be a separable compact Hausdorff space with $\dim M\le 2$ and $θ\colon M\to M$ be a homeomorphism with prime period $p$ ($p\ge 2$). Set $M_θ=\{x\in M| θ(x)=x\}\not=\varnothing$ and $M_0=M\backslash M_θ$. Suppose that $M_0$ is dense in $M$ and $\mathrm H^2(M_0/θ,\Z)\cong 0$, $\mathrm H^2(χ(M_0/θ),\Z)\cong 0$. Let $M'$ be another separable compact Hausdorff space with $\dim M'\le 2$ and $θ'$ be the self--homeomorphism of $M'$ with prime period $p$. Suppose that $M_0'=M'\backslash M_{θ'}'$ is dense in $M'$. Then $C(M)\times_θ\Z_p\cong C(M')\times_{θ'}\Z_p$ iff there is a homeomorphism $F$ from $M/θ$ onto $M'/θ'$ such that $F(M_θ)=M'_{θ'}$. Thus, if $(M,θ)$ and $(M',θ')$ are orbit equivalent, then $C(M)\times_θ\Z_p\cong C(M')\times_{θ'}\Z_p$.