\(C^1\)-smooth \(\Omega\)-stable skew products and completely geometrically integrable self-maps of 3D-tori. I: \(\Omega\)-stability (Q6580674)

This paper investigates nonhyperbolic endomorphisms of the three-dimensional torus. Specifically, this means a study of \(C^1\)-smooth stable skew products of circle maps. The author's results generalize other work on \(C^1\)-stability of skew products of interval maps on a compact rectangle in the plane described, for example, in [the author, Russian Math. Surveys 72, No 1, 101--178 (2017; Zbl 1381.37047)] as well as in several previous papers of the author.\N\NThe goal here is to begin to increase the dimension of the phase space of the skew products under consideration. The author proves \(C^1\)-\(\Omega\)-stability of skew products on a three-dimensional torus with respect to homeomorphisms of skew-product type. The results are formulated using properties of stability in different functional spaces of the \(C^1\)-smooth quotient map and \(C^1\)-smooth fiber maps.\N\NThe paper concludes with an example of an \(\Omega\)-stable map on the three-dimensional torus, and some considerations for approximating properties of \(C^1\) \(\Omega\)-stable skew products of circle maps.