On universal continuous actions on the Cantor set

Using the notion of proper Cantor colorings we prove the following theorem. For any countably infinite group $\Gamma$, there exists a free continuous action $\zeta: \Gamma \curvearrowright C$ on the Cantor set, which is universal in the following sense: for any free Borel action $\alpha: \Gamma \curvearrowright X$ on the standard Borel space, there exists an injective Borel map $\Theta_\alpha: X\to C$ such that $\Theta_\alpha\circ \alpha=\zeta \circ \Theta_\alpha$. We extend our theorem for (nonfree) Borel $(\Gamma,Z)$-actions, where $Z$ is a uniformly recurrent subgroup.