Small perturbations of smooth skew products and Sharkovsky's theorem

Small C 1-smooth perturbations of skew products and the partial integrability property

“…1. This paper is the direct continuation of the work [18], where stability of the Sharkovsky's order respectively small C 1 -smooth perturbations of skew products of interval maps is proved. Results of [18] are announced in [19], where the part of the Author's report at the Conference "Mathematical Physics, Dynamical Systems and Infinite-Dimensional Analysis" (17-21 June 2019, Dolgoprudny, Russia) devoted to periodic orbits of C 1 -smooth maps defined below is presented.…”

“…Following [18], [19] we suppose in this paper that the map (1) is C 1 -smooth on I, and the map f : I 1 → I 1 is so that the conditions hold: (i f ) f (∂ I 1 ) ⊂ ∂ I 1 , where ∂ (•) is the boundary of a set; (ii f ) f is the Ω-stable in the space of C 1 -smooth self-maps of the interval I 1 with the invariant boundary.…”

“…In [18] it is shown also that the autonomous discrete dynamical system (1) is connected with the nonautonomous discrete dynamical system generated by skew products of interval maps. So, one can represent the value of n-th (n ≥ 1) iteration of the map F in every initial point (x 0 , y 0 ) ∈ I as the composition of values of various skew products of interval maps in the corresponding points:…”

“…In different problems of dynamical systems theory only existence of an invariant lamination (but not an invariant foliation!) can be proved (see, e.g., [4], [18]). Therefore, it is naturally to introduce the following concept of the partial integrability for discrete dynamical systems.…”