Let $\beta:\,{{S}^{2n+1}}\,\to \,{{S}^{2n+1}}$ be a minimal homeomorphism $\left( n\,\ge \,1 \right)$. We show that the crossed product $C\left( {{S}^{2n+1}} \right)\,{{\rtimes }_{\beta}}\mathbb{Z}$ has rational tracial rank at most one. Let $\Omega $ be a connected, compact, metric space with finite covering dimension and with ${{H}^{1}}\left( \Omega ,\,\mathbb{Z} \right)\,=\,\left\{ 0 \right\}$. Suppose that ${{K}_{i}}\left( C\left( \Omega \right) \right)\,=\,\mathbb{Z}\,\oplus \,{{G}_{i}}$, where ${{G}_{i}}$ is a finite abelian group, $i\,=\,0,\,1$. Let $\beta :\,\Omega \,\to \,\Omega $ be a minimal homeomorphism. We also show that $A\,=\,C\left( \Omega \right)\,{{\rtimes }_{\beta}}\,\mathbb{Z}$ has rational tracial rank at most one and is classifiable. In particular, this applies to the minimal dynamical systems on odd dimensional real projective spaces. This is done by studying minimal homeomorphisms on $X\,\times \,\Omega $, where $X$ is the Cantor set.

We introduce the notion of tracial equivalence for $C^*$-algebras. Let $A$ and $B$ be two unital separable $C^*$-algebras. If they are tracially equivalent, then there are two sequences of asymptotically multiplicative contractive completely positive linear maps $\phi_n:A\to B$ and $\psi_n:B\to A$ with a tracial condition such that $\{\phi_n\circ\psi_n\}$ and $\{\psi_n\circ\phi_n\}$ are tracially approximately inner. Let $A$ and $B$ be two unital separable simple $C^*$-algebras with tracial topological rank zero. It is proved that $A$ and $B$ are tracially equivalent if and only if $A$ and $B$ have order isomorphic ranges of tracial states. For the Cantor minimal systems $(X_1,\sigma_1)$ and $(X_2,\sigma_2)$, using a result of Giordano, Putnam and Skau, we show that two such dynamical systems are (topological) orbit equivalent if and only if the associated crossed products $C(X_1)\times_{\sigma_1}\mathbb{Z}$ and $C(X_2)\times_{\sigma_2}\mathbb{Z}$ are tracially equivalent.