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Remarks on the Nonwandering Set of Skew Products with a Closed Set of Periodic Points of the Quotient Map

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Nonlinear Maps and their Applications

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 57))

Abstract

Exact formulas are given for the description of the nonwandering set of a continuous skew product of interval maps with a closed set of periodic points in the base. Special multifunctions, such as the Ω-function and suitable functions for the Ω-function, are used. A corollary is given for the nonwandering set of the C 1-smooth skew product of interval maps with a closed set of periodic points.

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Notes

  1. 1.

    By introducing the multifunctions \(\eta ^{\prime}_{l_{n}}\) and \(\eta ^{\prime}_{l_{n},\,1}\), one avoids complications about the possible breakdown of the equality \(\varOmega (\tilde{g}_{x}^{{2}^{i} }) =\varOmega (\tilde{g}_{x}^{{2}^{i-1} })\) (see [8]).

  2. 2.

    This theorem in the above form is announced44 in [15]. As the equality (6.5) shows, results concerning the structure of the nonwandering set from [10, 12] are correct for the restriction of the map F on the set Per(f) × I2.

  3. 3.

    If \(y -\alpha ^{\prime}_{2} = 0\) (\(\beta ^{\prime}_{2} - y = 0\)). Then the neighborhood U 2(y) is the right-hand (left-hand) neighborhood of the point y.

  4. 4.

    An analogous result can be found in [7], in which is stated the existence of a C ∞-smooth skew product of maps of an interval of type \(\prec {2}^{\infty }\) with one-dimensional attracting set. In [7], the skew product is realized as the shift map along the trajectories (defined for every t) of the corresponding nonautonomous system of differential equations with C ∞-smooth right-hand sides. This means that considerations are relative to R 3, and oscillations of the trajectory in a neighborhood of its limit set are “distributed” along the unbounded axis t. In the consideration of a skew product in a rectangle of the plane xOy, there is no opportunity to “distribute” oscillations of a trajectory possessing a one-dimensional attracting set. As a result, this leads to oscillations of the partial derivative \(\frac{\partial } {\partial x}g_{x}(y)\) and its unboundedness in a neighborhood of the attracting set, although the map g x (y) can be a C ∞-map with respect to y (but not with respect to the union of the variables x and y) [13].

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Acknowledgements

The Author is partially supported by the Federal Target Program “Scientific and Scientific-Pedagogical Personnel of Innovative Russia” (2009–2013) of the Education Ministry of Russia, grant No 14.B37.21.0361.

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Authors and Affiliations

  1. Department of Differential Equations and Mathematical Analysis, Faculty of Mathematics and Mechanics, University of Nizhny Novgorod, Gagarin Ave., 23a, build. 6., 603950, Nizhny Novgorod, Russia

    Lyudmila S. Efremova

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  1. Lyudmila S. Efremova
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Correspondence to Lyudmila S. Efremova .

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Editors and Affiliations

  1. Colégio Luís António Verney, University of Évora CIMA-UE, Évora, Portugal

    Clara Grácio

  2. LAAS-INSA, Toulouse, France

    Daniele Fournier-Prunaret

  3. Tokushima University, Tokushima, Japan

    Tetsushi Ueta

  4. E. E. Eng., Tokushima University, Tokushima, Japan

    Yoshifumi Nishio

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Efremova, L.S. (2014). Remarks on the Nonwandering Set of Skew Products with a Closed Set of Periodic Points of the Quotient Map. In: Grácio, C., Fournier-Prunaret, D., Ueta, T., Nishio, Y. (eds) Nonlinear Maps and their Applications. Springer Proceedings in Mathematics & Statistics, vol 57. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-9161-3_6

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