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

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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.

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Notes

  1. Following [1], everywhere when studying the \(\Omega\)-stability property of maps on manifolds with an edge, we assume that the edge is invariant with respect to maps under consideration.

  2. In what follows, the subscript in the entry \(T^{j},j=1\mbox{ or }2,\) will mean the coordinates with which \(T^{j}\) is provided.

  3. A map \(\psi\in C^{1}(S^{1})\) is said to be expanding if for every \(x\in S^{1}\) the inequality \(|\psi^{\prime}(x)|>1\) holds [24, Ch. 1, §1.3].

  4. A quasiminimal set is the closure of a recurrent (but not periodic) trajectory [31, Ch. 5, §5].

  5. This property means that for a finite cover of \(K(\psi)\) by intervals of the length \(\leqslant\theta\) there is a periodic orbit that intersects every interval of this cover, at least, at one point.

  6. If \(\Psi\in SP^{0}(T^{3})\), then the equality \(\text{Per}(\widehat{\Psi})=pr_{\widehat{x}_{2}}\big{(}\text{Per}(\Psi)\big{)}\) may not hold. Among the points of the set \(\text{Per}(\widehat{\Psi})\) there may be points that belong to the projection on \(T^{2}_{\widehat{x}_{2}}\) of the set of uniformly recurrent points, but not the set of periodic points of \(\Psi\) (cf. [25]). For such points \(\widehat{x}_{2}\) it is possible that \(\big{(}\text{Per}(\Psi)\big{)}(\widehat{x}_{2})=\emptyset.\) As for continuous skew products on 3D-cells, the following equalities are valid: \(\text{Per}(\widehat{\Psi})=pr_{\widehat{x}_{2}}\big{(}\text{Per}(\Psi)\big{)}\) and \(\text{Per}\big{(}\psi_{3,\widehat{x}_{2},m(\widehat{x}_{2})}\big{)}=\big{(}\text{Per}(\Psi)\big{)}(\widehat{x}_{2})\).

  7. By Definition 6, Jacobson’s theorem and Proposition 4, it means that all fiber maps belong to \(\bf L_{2}\) and have the same degree \((\)for the degree of fiber maps, see also [25]\()\).

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Funding

This research is supported by the Russian Science Foundation (RSF), grant No. 24-21-00242, https://rscf.ru/en/project/24-21-00242/.

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  1. Nizhny Novgorod State University, pr. Gagarina 23, 603022, Nizhny Novgorod, Russia

    Lyudmila S. Efremova

  2. Moscow Institute of Physics and Technology, Institutsky per. 9, 141701, Dolgoprudny, Russia

    Lyudmila S. Efremova

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MSC2010

37C05, 37C20, 37D30, 37Exx

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Efremova, L.S. \(C^{1}\)-Smooth \(\Omega\)-Stable Skew Products and Completely Geometrically Integrable Self-Maps of 3D-Tori, I: \(\Omega\)-Stability. Regul. Chaot. Dyn. 29, 491–514 (2024). https://doi.org/10.1134/S1560354724520010

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