Small perturbations of smooth skew products and Sharkovsky's theorem

Abstract

In this paper we prove that the famous Sharkovsky's Theorem holds in the class of maps obtained from $C^1$-smooth skew products of interval maps with Ω-stable quotients by means of small $C^1$-smooth perturbations.

Keywords:

• Sharkovsky's order
• periodic point
• periodic orbit
• skew product
• Ω-stable quotient
• perturbation

2010 Mathematics Subject Classifications:

• 37Exx
• 37Cxx

Acknowledgments

The author thanks Professor Vladimir N. Belykh for fruitful discussions. The author is obliged to the Reviewers for their careful reading of the paper and constructive criticism which contributed to beneficial amendments in the text.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 In this paper we give the proof of results announced in [14], where the Author's report at the Conference ‘Mathematical Physics, Dynamical Systems and Infinite-Dimensional Analysis’ (17–21 June 2019, Dolgoprudny, Russia) is published.

2 Definitions of nonwandering points and sets one can find in [17, Part 1, Chap. 3, § 3.3].

3 The definition of a lamination one can find in [3].

4 The term ‘invariant lamination’ means that the map under consideration transforms a fibre to a fibre.

5 A quasiminimal set is the closure of a recurrent (but not periodic) trajectory [22, Ch. 5, § 5].

6 In Lemma 3.2 the more strong claim is proved: fibres that start from the points $(x,a_2)$, where $x$ is $f$-periodic point, are $C^1$-smooth.

7 The definition of a local lamination one can find in [3].

Additional information

Funding

This work is supported in the part by Ministry of Science and Education of RF, project No 1.3287.2017.