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Simplest Skew Products on \(\boldsymbol{n}\)-Dimensional (\(\boldsymbol{n\geq 2}\)) Cells, Cylinders and Tori

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Abstract

We describe the structure of the nonwandering set of continuous skew products on \(n\)-dimensional (\(n\geq 2\)) cells, cylinders and tori in the following two cases: 1) the set of periodic points is not empty (for self-maps of cylinders and tori), and it is closed for all maps under consideration; 2) the set of periodic points is empty for self-maps of cylinders and tori, and the nonwandering set is minimal.

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Notes

  1. We also use the usual term ‘‘wandering set’’ for the set \(M\setminus\Omega(F)\) (see §3).

  2. In the current time there are no examples of skew products with \(n\)-dimensional phase spaces for \(n\geq 4\) and \(n_{*}\)-dimensional \(\omega\)-limit sets for arbitrary \(2\leq n_{*}\leq n-1\), although the paper [17] contains the algorithm (based on divergent series), which makes it possible to construct examples of these maps.

  3. If the case \((ii_{2.1})\) is realized then \(\delta^{*}=\varepsilon,\) and we consider positive subtrajectories with respect to \(F^{2^{\nu}q}\) of points \((x_{1}^{m},x_{2}^{m})\) and \((x_{1}^{0},x_{2}^{m})\).

  4. In the case \((ii_{2.2})\) the set \(W^{u}(x_{1}^{*},f_{1}^{2^{\gamma}q})\setminus\{x_{1}^{*}\}\) contains countable number of elements of the sequence \(\{x_{1}^{p}\}_{p\geq 1}\)

  5. In the case of existence of a positive number \(\delta,\) \(\delta>\varepsilon\) (here \(\delta^{*}=\varepsilon\)), it is necessary to consider sequential images of the point \((x_{1}^{p(\beta^{\prime})},x_{2}^{p(\beta^{\prime})})\) under corresponding maps.

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Authors and Affiliations

  1. Lobachevsky Nizhny Novgorod State University, 603950, Nizhny Novgorod, Russia

    L. S. Efremova

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

    L. S. Efremova

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Efremova, L.S. Simplest Skew Products on \(\boldsymbol{n}\)-Dimensional (\(\boldsymbol{n\geq 2}\)) Cells, Cylinders and Tori. Lobachevskii J Math 43, 1598–1618 (2022). https://doi.org/10.1134/S1995080222100080

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