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

Abstract

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.