Let M be a separable compact Hausdorff space with $\dim M\leq2$ and $\Theta:M\to M$ be a homeomorphism with prime period $p$ ($p\geq 2$). Set $M_{\theta}=\{x\in M\mid\theta(x)=x\}\neq\varnothing$ and $M_0=M\setminus M_{\theta}$. Suppose that $M_0$ is dense in M and $H^2(M_0/\theta,Z)\cong 0$, $H^2(\chi(M_O/\theta),Z)\cong0$. Let $M'$ be another separable compact Hausdorff space with $\dim M'\leq2$ and $\theta'$ be the self-homeomorphism of $M'$ with prime period $p$. Suppose that $M_0'=M'\setminus M'_{\theta'}$ is dense in $M'$. Then $C(M)\times_{\theta}Z_p\cong C(M')\times_{\theta'}Z_p$ if there is a homeomorphism $F$ from $M/{\theta}$ onto $M'/0'$ such that $F(M_{\theta}) = M'_{\theta'}$. Thus, if $(M,\theta)$ and $(M', \theta')$ are orbit equivalent, then $C(M)\times_{\theta}Z_p\cong C(M')_{\theta'}Z_p$.