On limit sets of simplest skew products defined on multidimensional cells

L. S. Efremova ; M. A. Shalagin

Izvestiya VUZ. Applied Nonlinear Dynamics, Tome 32 (2024) no. 6, pp. 796-815.

Résumé

The purpose of this work is to describe two important types of limit sets of the most simple skew products of interval maps, the phase space of each of which is a compact $n$-dimensional cell ($n\geqslant 2$): firstly, a non-wandering set and, secondly, $\omega$-limit sets of trajectories. Methods. A method for investigating of a nonwandering set (new even for the two-dimensional case) is proposed, based on the use of the concept of $C_0-\Omega$-blow up in continuous closed interval maps, and the concept of $C_0-\Omega$-blow up introduced in the work in the family of continuous fibers maps. To describe the $\omega$-limit sets, the technique of special series constructed for the trajectory and containing an information about its asymptotic behavior is used. Results. A complete description is given of the nonwandering set of the continuous simplest skew product of the interval maps, that is, a continuous skew product on a compact $n$-dimensional cell, the set of (least) periods of periodic points of which is bounded. The results obtained in the description of a nonwandering set are used in the study of $\omega$-limit sets. The paper describes a topological structure of $\omega$-limit sets of the maps under consideration. Sufficient conditions have been found under which the $\omega$-limit set of the trajectory is a periodic orbit, as well as the necessary conditions for the existence of one-dimensional $\omega$-limit sets. Conclusion. Further development of the $C_0-\Omega$-blow up technique in the family of maps in fibers will allow us to describe the structure of a nonwandering set of skew products of one-dimensional maps, in particular, with a closed set of periodic points defined on the simplest manifolds of arbitrary finite dimension. Further development of the theory of special divergent series constructed in the work will allow us to proceed to the description of $\omega$-limit sets of arbitrary dimension $d$, where $2 \leqslant d \leqslant n - 1$, $n\geqslant 3$, in the simplest skew products.

Détail

Keywords: skew product, nonwandering set, $C_0-\Omega$-blow up, $\omega$-limit set, fixed point, periodic point

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