A quasiperiodically forced skew-product on the cylinder without fixed-curves

TL;DR

This paper constructs a counterexample in quasiperiodically forced skew-product systems on the cylinder, demonstrating that Sharkovski Theorem does not hold when restricted to curves, and showing the existence of systems without invariant curves.

Contribution

It provides the first known example of a quasiperiodic skew product on the cylinder with a periodic orbit of curves but no invariant curves, challenging previous extensions of Sharkovski Theorem.

Findings

Counterexample with a period-2 orbit of curves

Existence of quasiperiodic skew products without invariant curves

Disproof of Sharkovski Theorem extension to curves

Abstract

In [FJJK] the Sharkovski\u{i} Theorem was extended to periodic orbits of strips of quasiperiodic skew products in the cylinder. In this paper we deal with the following natural question that arises in this setting: Does Sharkovski\u{i} Theorem holds when restricted to curves instead of general strips?mWe answer this question in the negative by constructing a counterexample: We construct a map having a periodic orbit of period 2 of curves (which is, in fact, the upper and lower circles of the cylinder) and without any invariant curve. In particular this shows that there exist quasiperiodic skew products in the cylinder without invariant curves.