\(C^{1}\)-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, whichare 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.