Ramified continua as global attractors of C¹-smooth self-maps of a cylinder close to skew products

Abstract

In the paper we consider $C^1$-smooth self-maps of a cylinder close to $C^1$-smooth skew products (and satisfying some additional conditions). We study such geometric property of the maps, as existence of $C^1$-smooth invariant local lamination, and apply this geometric property to the proof of the geometric integrability of maps under consideration. Using obtained results we construct the example of the family of $C^1$-smooth maps close to skew products so that each map from this family admits the global attractor, which is a one-dimensional ramified continuum with a complicated topological structure. The global attractor of every map from the family under consideration consists of arcs of two types. On the unique circle (which is the arc of first type) the map is mixing; on arcs of second type of different lengths homeomorphic to a closed interval (the family of such arcs has continuum cardinality) the map is not mixing. The topological structure of the global attractor and dynamical properties of trajectories on the attractor lead to the property of dense intermittency (in the complement to the attractor) of attraction sets of different ω-limit sets, the union of which coincides with the global attractor.

Keywords:

• Skew product
• small perturbation
• local lamination
• invariant foliation
• maximal attractor
• ramified continuum

AMS Subject Classifications:

• 37Exx
• 37Cxx

Acknowledgements

The author thanks the reviewers for useful remarks and comments.

Correction Statement

This article has been corrected with minor changes. These changes do not impact the academic content of the article.

Disclosure statement

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

Notes

1 A map $\phi \in C^1\left(S^1\right)$ is said to be expanding if for every $x\in S^1$ the inequality holds $|{\phi }^{\mathrm{\prime }}\left(x\right)|>1$ [48, Ch. 1, Section 1.3].

2 A quasiminimal set is the closure of a recurrent (but not periodic) trajectory [47, Ch. 5, Section 5].

3 The maximal smoothness of a curve can be higher than r.

4 The inclusion (21(21) $F\left({\gamma }_{x^{\mathrm{\prime }}}\right)\subseteq {\gamma }_{\psi \left(x^{\mathrm{\prime }}\right)}$(21) ) means the invariance of a local lamination.