$C^1$-Smooth $\Omega$-Stable Skew Products and Completely Geometrically Integrable Self-Maps of 3D-Tori, I: $\Omega$-Stability

We prove here the criterion of $C^1$- $\Omega$-stability of self-maps of a 3D-torus, which are skew products of circle maps. The $C^1$- $\Omega$-stability property is studied with respect to homeomorphisms of skew products type. We give here an example of the $\Omega$-stable map on a 3D-torus and investigate approximating properties of maps under consideration.