On diagonal actions of free group on the Cantor set

We study diagonal actions $\varphi:\mathbb{F}_2\curvearrowright\partial\mathbb{F}_2\times K$ on the Cantor set which are given by $\varphi_a=\partial_a\times\alpha,\varphi_b=\partial_b\times\beta$. Under some restrictions on $\alpha,\beta$ we compute $K_*(C(\partial\mathbb{F}_2\times K)\rtimes_r\mathbb{F}_2)$. As an application in the case of $\alpha$ is Denjoy homeomorphism of the Cantor set and $\beta=id$ we will show that $C(\partial\mathbb{F}_2\times K)\rtimes_r\mathbb{F}_2$ is Kirchberg algebra with $K_*(C(\partial\mathbb{F}_2\times K)\rtimes_r\mathbb{F}_2)=(\mathbb{Z}^\infty,\mathbb{Z}^\infty)$. Also we will check that $C^*$-crossed product by Denjoy homeomorphism on the Cantor set is $C^*$-algebra generated by weighted shift, namely $C(K)\rtimes\mathbb{Z}\cong C^*(T_x)$ where $x\in \{1,2\}^\mathbb{Z}$ is two-sided Fibonacci sequence.