## 1 Introduction

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.

*F*of a closed rectangle

*I*=

*I*

_{1}×

*I*

_{2}into itself, where

*I*

_{1},

*I*

_{2}are closed intervals of the straight line

**R**,

^{1}*I*= [

_{k}*a*] for

_{k}, b_{k}*k*= 1, 2, and

*F*satisfies the equality

*g*(

_{x}*y*) for

*g*(

*x, y*), where (

*x, y*) is an arbitrary point of the rectangle

*I*.

In the paper [31] the integrable system of differential equations is constructed so that this system approximates the Lorenz system of differential equations [3]. The paper [31] generated the wave of interest to dynamical systems (1) (see, e.g., [5], [29] – [30]) for the case of the discontinuous Lorenz map *f* of the closed interval *I*_{1} into itself [8], [26].

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.

We suppose also that the *C*^{1}-smooth function *μ* (of variables *x* and *y*) satisfies the boundary conditions:

- (
*i*) the equalities_{μ}*μ*(*x, a*_{2}) =*μ*(*x, b*_{2}) = 0 are valid for every*x*∈*I*_{1}; and the equalities*μ*(*a*_{1},*y*) =*μ*(*b*_{1},*y*) = 0 are valid for every*y*∈*I*_{2}.

By the properties (*i _{f}*) and (

*i*) the set ({

_{μ}*a*

_{1}} ×

*I*

_{2}) ∪ ({

*b*

_{1}} ×

*I*

_{2}) is

*F*-invariant. If, in addition, the inclusion

*g*(

_{x}*∂ I*

_{2}) ⊂

*∂ I*

_{2}holds for all

*x*∈

*I*

_{1}then the union of the horizontal intervals

*I*

_{1}

*×*{

*a*

_{2}} and

*I*

_{1}

*×*{

*b*

_{2}} is

*F*-invariant too.

2. Give the list of the functional spaces connected with the map (1).

Let
*C*^{1}-smooth maps of the interval *I*_{1} into itself satisfying conditions (*i _{f}*) – (

*ii*). The standard

_{f}*C*

^{1}-norm

*|| · ||*

_{1,1}of the linear normalized space of

*C*

^{1}-smooth maps of the interval

*I*

_{1}into the straight line

**R**generates the

^{1}*C*

^{1}-topology in

*C*

^{1}-topology in

*ɛ*> 0 and

By the *C*^{1}- Ω-stability of the map *f* (see the condition (*ii _{f}*)) for any

*δ*> 0 there exists an

*ɛ*-neighborhood

*f*in the space

*f*by means of a homeomorphism which is

*δ*-close in the

*C*

^{0}-topology of the uniform convergence to the identity map of the nonwandering set

^{a}of the map

*f*.

Let
*C*^{1}-smooth maps of the rectangle *I* into the interval *I*_{1} endowed with the standard *C*^{1}-norm *|| · ||*_{1,(1,1)} of the linear normalized space of *C*^{1}-smooth maps of the rectangle *I* into the straight line **R ^{1}** that contains the interval

*I*

_{1}. This norm induces the

*C*

^{1}-topology in the space

*ɛ*-balls

*ɛ*> 0.

We suppose that the function *μ* = *μ*(*x, y*) satisfies the following “condition of smallness”:

- (
*ii*)_{μ}*||μ||*_{1,(1,1)}<*ɛ,*where*ɛ*is found for*δ*> 0 by the property of the*C*^{1}- Ω-stability of*f*.

*|| · ||*

_{1,1}for every

*y*∈

*I*

_{2}and

*|| · ||*

_{1,(1,1)}:

*f*of one variable can be considered as the function of two variables of the type

*f*○

*pr*

_{1}, where

*pr*

_{1}:

*I*→

*I*

_{1}is the natural projection of

*I*on

*I*

_{1}. Hence, by the condition (

*ii*) and the inequality (2) we have

_{μ}*y*∈

*I*

_{2}.

*C*

^{1}-smooth maps (1) such that the function

*f*satisfies conditions (

*i*) – (

_{f}*ii*), and

_{f}*μ*satisfies conditions (

*i*) – (

_{μ}*ii*). Endow

_{μ}*C*

^{1}-norm

*|| · ||*

_{1}of the linear normalized space of

*C*

^{1}-smooth maps of the rectangle

*I*into the plane

**R**. The base of the

^{2}*C*

^{1}-topology generated by this norm, is given by the system of

*ɛ*-balls

*F*for every

*ɛ*> 0 and

*C*

^{1}-smooth perturbations (satisfying conditions (

*i*) – (

_{μ}*ii*)) of the

_{μ}*C*

^{1}-smooth skew product of interval maps of the type

*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:

_{yi}:

*I*→

*I*for every

*i*(0 ≤

*i*≤

*n*− 1) is presented in the form

Denote by
*C*^{1}-smooth skew products of interval maps with quotients satisfying conditions (*i _{f}*) – (

*ii*). Endow this space with the

_{f}*C*

^{1}-topology generated by the standard

*C*

^{1}-norm

*|| · ||*

_{1}. The structure of the functional space

*∂ I*of the rectangle

*I*is invariant. Then the inclusions are valid:

*C*

^{1}-topology in

*ɛ*-balls

*ɛ*> 0 and

*i*) every map

_{μ}4. There is a vast literature devoted to different integrability aspects of dynamical systems both with continuous time (see, e.g., [7], [23] – [24]), and with discrete time (see, e.g., [1] – [2], [35] – [36]). Originally, the concept of integrability of dynamical systems with discrete time was introduced for systems obtained by digitization of known differential equations [1] – [2], [35] – [36]). But there are discrete dynamical systems that do not belong to this class. We consider here precisely this case.

Remind the following Birkhoff's thought: “If we try to formulate the exact definition of integrability then we see that many definitions are possible, and every of them is of the specific theoretical interest” [6].

Our definition of integrability of dynamical systems with discrete time given in [4] (see also [16]) follows the paper [20] and generalizes the definition from [20] given for polynomials and rational maps, on the case of arbitrary maps. (The last set contains maps that can not be obtained by the procedure of digitization of differential equations.)

*G*of some (open or closed) domain Π ⊂

**R**to itself is

^{2}*integrable*if there exists a self-map

*ψ*of an interval

*J*of the real line

**R**such that

^{1}*G*is semiconjugate with

*ψ*by means of a continuous surjection

In the framework of the suggested approach in the paper [17] the definition of integrability is introduced for some multifunctions.

As it follows from Definition 1 skew products of interval maps are integrable maps. Here
^{b}.

*[4]. Let* Π *be a convex connected compact subset of***R ^{2}**

*such that the section of*Π

*by an arbitrary line y*=

*const (if it is non-empty) is a non-degenerate interval, and let G*: Π → Π

*be a continuous map. Then G is integrable in the sense of Definition 1 by means of a continuous surjection*

*J that is one-to-one with respect to x (here J is a closed interval of*

**R**

^{1}*) if and only if some homeomorphism reduces G to a skew product of interval maps defined in a compact planar rectangle.*

Definition 1 distinguishes such feature of integrable dynamical systems satisfying conditions of Theorem 1, as the existence of an invariant foliation. This property is the key point of the proof of the integrability property of a dynamical system.

Point out that the existence of a continuous invariant foliation for Lorenz type maps is proven in [3], and existence of a *C*^{1}-smooth invariant foliation (with *C*^{2}-smooth fibers) for these maps is proven in [34].

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.

*G*defined on some (open or closed) domain Π of the plane

**R**with values in Π is

^{2}*partially integrable*if there exist a closed invariant set

*A*⊂ Π (

*A*≠ Π), a self-map

*ψ*of an interval

*J*of the real line

**R**and a closed invariant set

^{1}*B*⊂

*J*(

*B*≠

*J*) such that

*G*is semiconjugate with

_{|A}*ψ*by means of a continuous surjection

_{|B}5. In this paper we investigate stability of the integrability property of skew products of interval maps under small *C*^{1}-smooth perturbations satisfying conditions (*i _{μ}*) – (

*ii*). We obtain here (sufficient) conditions of partial integrability for maps from the space

_{μ}## 2 Preliminaries

This section contains the relevant definitions and results on dynamics both of continuous maps and *C*^{1}-smooth Ω-stable maps of a closed interval.

1. We begin from the famous Sharkovsky's Theorem [32].

*[32] If a continuous map f*:

*I*

_{1}→

*I*

_{1}

*contains a periodic orbit of a (least) period m*> 1

*then it contains also periodic orbits of every (least) period n, where n precedes m (n*≺

*m) in the Sharkovsky's order:*

In accordance with the Sharkovsky's Theorem, the space of continuous self-maps of a closed interval can be presented as the union of three subspaces (see, e.g., [33]): the first of which consists of the maps of type ≺ 2^{∞}, that is, the maps that have periodic points with (least) periods {1,2,2^{2},...,2^{ν}}, where 0 ≤ *ν* < +∞; the second subspace consists of the maps of type 2^{∞}, that is, maps whose periodic points have (least) periods {1,2,2^{2},...,2^{i},2^{i+1},...}; the third subspace consists of the maps of type ≻ 2^{∞}, that is, maps with periodic points that possess (least) periods outside the set {2^{i}}_{i ≥0}.

In this paper we consider maps from the space
*f* satisfies the following additional condition: (*iii _{f}*)

*f*has type ≻ 2

^{∞}.

The above condition (*iii _{f}*) means that

*f*demonstrates a chaotic behavior (see, e.g., [33]).

2. Formulate the properties of *C*^{1}-smooth Ω-stable maps of a closed interval, give the definition of Σ-stability and remind the properties of the Σ-stable maps of a closed interval.

*[21], [27] Let**and satisfy the condition* (*iii _{f}*).

*Then*

- (3.1)
*the nonwandering set*Ω(*f*)*is the union of a finite number of hyperbolic periodic points (that form the rarefied set*Ω_{r}(*f*)*) and a finite number of locally maximal (i.e., maximal quasiminimal*^{c}*sets in some their neighborhood) hyperbolic perfect nowhere dense sets (that form the perfect set*Ω_{p}(*f*)*));* - (3.2)
*periodic points are everywhere dense in the set*Ω_{p}(*f*)*; moreover, for every natural number m*≥ 2*periodic points with multiple m (least) periods are everywhere dense in*Ω_{p}(*f*)*;* - (3.3)
*there are numbers α*=*α*(*f*) > 0*and c*=*c*(*f*) > 1*so that for every x*∈ Ω_{p}(*f*)*and n*≥ 1*the inequality |*(*f*(^{n}*x*))′*|*>*αc*Ω^{n}holds (that is,_{p}(*f*)*is the repelling hyperbolic set);* - (3.4)
*the subspace*${\tilde{C}}_{\omega}^{1}\left({I}_{1}\right)$ *of maps satisfying condition*(*iii*)_{f}*is open and everywhere dense in the containing it space of C*^{1}*-smooth self-maps of the closed interval I*_{1}*of type*≻ 2^{∞}*with the invariant boundary.*

*[21], [27] Let*

*and satisfy the condition*(

*iii*).

_{f}*Then the equality holds:*

*Here*Ω

^{s}(

*f*)

*is the nonempty finite invariant set of all f-sinks. The set*Ω

^{u}(

*f*)

*is invariant and equals the union of*Ω

_{p}(

*f*)

*with a finite (possibly empty) set that consists of isolated sources in the set of f-periodic points*.

In the set *I*_{1} \ Ω(*f* ) (just as in the set *I*_{1} \ Ω_{p}(*f* )) the points of attraction domains of *f*-sinks are everywhere dense.

Point out that for the map
*f*^{−1}(Ω(*f* )) ⊂ Ω(*f* ) can be false (although the equality *f* (Ω(*f* )) = Ω(*f* ) holds). Here *f*^{−1} (·) means the first complete preimage of a set.

*f*, and with respect to

*f*

^{−1}, and contains the set Ω

^{u}(

*f*) as its subset. For this goal we need the attraction domain of all

*f*-sinks:

*Orb*(

*x, f*) is the periodic orbit of the sink

*x*∈ Ω

^{s}(

*f*),

*D*(

*Orb*(

*x, f*)) is the immediate attraction domain of the periodic orbit

*Orb*(

*x, f*),

*D*(

^{−i}*Orb*(

*x, f*)) is

*i*-th complete primage of the immediate attraction domain of the periodic orbit

*Orb*(

*x, f*).

*D*(

*Orb*(

*x, f*)) of the periodic orbit

*Orb*(

*x, f*) consists of

*m*(

*m*is the (least) period of

*x*) pairwise disjoint open (in the topology of the segment

*I*

_{1})

*f*-invariant intervals

^{m}*D*

_{f j(x)}(0 ≤

*j*≤

*m*− 1) such that every of these intervals contains the point

*f*(

^{j}*x*) :

Complete *f*-invariance of the immediate attraction domain^{d}*D*(*Orb*(*x, f* )) implies correctness of the definition of preimages *D ^{−i}*(

*Orb*(

*x, f*)) (

*D*(

^{−i}*Orb*(

*x, f*)) ≠ ∅) for every

*i*≥ 0.

By the condition (*iii _{f}*) the set Δ(

*f*) is a countable union (see the claim (3.1) of Lemma 3) of pairwise disjoint intervals (open in the topology of the closed interval

*I*

_{1}); Δ(

*f*) is invariant both with respect to

*f*, and with respect to

*f*

^{−1}.

We suppose further that the set *Cr*(*f* ) of *f*-critical points satisfies the condition

- (
*iv*)_{f}*Cr*(*f*) ⊂ Δ° (*f*), where Δ° (*f*) is the interior of the set Δ (*f*).

*[33] Let**satisfy conditions* (*iii _{f}*) – (

*iv*).

_{f}*Then one of the following three cases is realized for the boundary ∂*(

*D*

_{f j(x)})

*of every interval D*

_{f j(x)}(0 ≤

*j*≤

*m −*1

*):*

- (5.1)
*∂*(*D*_{f j(x)})*consists of two f*^{m}-fixed points; - (5.2)
*points of ∂*(*D*_{f j(x)})*form a periodic orbit of (least) period*2*with respect to f*^{m}; - (5.3)
*one of the points of ∂*(*D*_{f j(x)})*is f*.^{m}-fixed point source, and the other is its preimage with respect to f^{m}

*f*and

*f*

^{−1}:

^{u}(

*f*) ⊂ Σ(

*f*), and Ω

^{s}(

*f*) ∩ Σ (

*f*) = ∅.

*stable (in the C*

^{1}

*-topology)*if for every

*δ*> 0 there exists an

*ɛ*-neighborhood

*f*, that is, the equality

*h*: Σ(

*f*) → Σ(

*ϕ*). Here

*h*is

*δ*-close to the identity map on Σ(

*f*) in the

*C*

^{0}-topology.

The following claim is the direct corollary of Definition 3.

*Let**satisfy conditions* (*iii _{f}*) – (

*iv*).

_{f}*Then f is*Σ

*-stable in the C*

^{1}

*-topology*.

By [21] the set of maps from
*iii _{f}*) – (

*iv*) contains the open everywhere dense subset of

_{f}*C*

^{2}-smooth maps with nondegenerate critical points.

## 3 Sufficient conditions of partial integrability of the map (1)

In this section we prove the main result of the paper.

*Let the quotient f of the skew product of interval maps**satisfy conditions* (*iii _{f}*) – (

*iv*).

_{f}*Then for any δ*> 0

*there exists an ɛ-neighborhood*

*of the map*Φ

_{0}

*in the space*

*such that every map*

*obtained from*Φ

_{0}

*by means of the C*

^{1}

*-perturbation μ*=

*μ*(

*x, y*),

*where μ satisfies conditions*(

*i*) – (

_{μ}*ii*),

_{μ}*is partially integrable on the closed invariant set £*(

*F*)

*that consists of pairwise disjoint curvelinear fibers. These fibers start from the points of the set*Σ* (

*f*)

*×*{

*a*

_{2}}

*(where*Σ* (

*f*) = Σ(

*f*) ∪ Ω

^{s}(

*f*)

*) and are graphs of C*

^{1}

*-smooth functions x*=

*x*(

*y*)

*defined on the interval I*

_{2}.

*The function*

*that realizes the partial integrability property, is C*

^{1}

*-smooth surjection, ɛ-close in the C*

^{1}

*-norm to the natural projection pr*

_{1}: Σ* (

*f*) ×

*I*

_{2}→ Σ*(

*f*).

The following statement proved in [18], is the first step of the proof of Theorem 7.

*[18] Let**Then for any δ* > 0 *there exists an ɛ-neighborhood**of the map* Φ_{0}*in the space**such that every map**obtained from* Φ_{0}*by means of the C*^{1}*-perturbation μ* = *μ*(*x, y*), *where μ satisfies conditions* (*i _{μ}*) – (

*ii*),

_{μ}*has the invariant closed set £*

^{0}(

*F*)

*that consists of pairwise disjoint curvelinear fibers. These fibers start from the points of the set*Ω(

*f*)

*×*{

*a*

_{2}},

*and are the graphs of continuous functions x*=

*x*(

*y*)

*defined on the interval I*

_{2}

*; moreover, fibers starting from the points*(

*x, a*

_{2}),

*where x is f-periodic point, are C*

^{1}

*-smooth. Every curvelinear fiber is δ-close in the C*

^{0}

*-norm to the vertical closed interval that starts from the same initial point of the set*Ω(

*f*)

*×*{

*a*

_{2}}

*just as the curvelinear fiber*.

Prove *C*^{1}-smoothness of all fibers from the set *£*^{0}(*F*).

*Let**Then for any δ* > 0 *there exists an ɛ-neighborhood**of the map* Φ_{0}*in the space**such that the closed invariant set £*^{0}(*F*) *of every map**obtained from* Φ_{0}*by means of the C*^{1}*-perturbation μ* = *μ*(*x, y*), *where μ satisfies conditions* (*i _{μ}*) – (

*ii*),

_{μ}*consists of graphs of C*

^{1}

*-smooth functions x*=

*x*(

*y*)

*defined on the interval I*

_{2}.

*In addition, every curvelinear fiber of the set £*

^{0}(

*F*)

*is ɛ-close in the C*

^{1}

*-norm to the vertical closed interval that starts from the same initial point of the set*Ω(

*f*) × {

*a*

_{2}}

*just as the curvelinear fiber.*

1. Fix a number *δ* > 0. We find a positive number *ɛ* > 0 for *δ* using the *C*^{1}- Ω-stability property of the map
*f* consists of maps such that every map is Ω-conjugate with *f* by means of the homeomorphism that is *δ*-close to the identity map of the set Ω(*f* ).

*ɛ*-neighborhood

_{0}in the space

*ii*) the inequality

_{μ}*y*∈

*I*

_{2}and means also that the map (

*f*+

*μ*) is Ω-conjugate with

*f*for every

*y*∈

*I*

_{2}by means of the homeomorphism that is

*δ*-close to the identity map of the set Ω(

*f*).

*ɛ*-neighborhoods

_{m}*m*> 1 in the space

*ɛ*-neighborhood

_{0}. Since

*y*∈

*I*

_{2}the inequality holds:

*y*∈

*I*

_{2}. Here

*ɛ*-neighborhood of the map (

_{m}*f*

^{m}○ pr_{1}) in the space

*ɛ*-neighborhood of the map

_{m}*f*in the space

^{m}*f*by means of the homeomorphisms

^{m}*δ*-close to the identity map on the nonwandering set Ω(

*f*) = Ω(

^{m}*f*) (see Lemma 3).

2. Let *f* satisfy the condition (*iii _{f}*). Denote by

*£*(

^{u}*F*) (

*£*(

^{u}*F*) ⊂

*£*

^{0}(

*F*)) the set of curvelinear fibers that start from all points of Ω

^{u}(

*f*).

*n*

_{*}such that the equality

*y*∈

*I*

_{2}.

*£*(

^{u}*F*) and Lemma 3 (see claims (3.1) – (3.3)) for every

*y*∈

*I*

_{2}we point out the least natural number

*n*

_{*}(

*y*) satisfying

*n*≥

*n*

_{*}(

*y*).

*θ*(

*y*)-neighborhood

*U*

_{θ}_{(y)}(

*£*(

^{u}*F*))(

*y*)

*×*{

*y*}) of the set (

*£*(

^{u}*F*))(

*y*)

*×*{

*y*} in

*I*such that

*n*

_{*}(

*y*) is the least natural number for which the inequality (16) holds.

^{I1}be the topological space of all closed subsets of the closed interval

*I*

_{1}with the exponential topology. By compactness of the closed intervals

*I*

_{1},

*I*

_{2}the set 2

^{I1}×

*I*

_{2}is the compact [24]. Then the closed set

^{I1}×

*I*

_{2}. Using compactness of the set

*£*(

^{u}*F*) in 2

^{I1}

*× I*

_{2}we distinguish from its infinite open cover {

*U*

_{θ}_{(y)}((

*£*(

^{u}*F*))(

*y*)

*×*{

*y*})}

_{y∈I2}the finite subcover. Let neighborhoods {

*U*

_{θ}_{(yj)}((

*£*(

^{u}*F*))(

*yj*)

*×*{

*y*})}

_{j}_{1≤j≤q}of the sets {(

*£*(

^{u}*F*))(

*y*)

_{j}*×*{

*y*}}

_{j}_{1≤ j≤q}form this finite subcover.

*£*(

^{u}*F*) consists of continuous fibers (see Proposition 8). Therefore, for every

*j*(1 ≤

*j*≤

*q*) there exists

*j*′ (

*j*′ ≠

*j,*1 ≤

*j*′ ≤

*q*) such that

*n*=

_{*}*n*

_{*}(

*y*) (1 ≤

_{j}*j*≤

*q*). Using continuity of the partial derivative

*I*we verify that the equality (14) holds.

Hence, without loss of generality we will suppose further that *n*_{*} = 1. In fact, if *n _{*}* ≠ 1 then we get over consideration of the map

*F*and use the claim (3.2) of Lemma 3.

^{n}3. Prove that every curvelinear fiber is *ɛ*-close in the *C*^{1}-norm to the vertical closed interval that starts from the same initial point (*x, a*_{2}) of the set Ω(*f* ) *×* {*a*_{2}} just as the curvelinear fiber.

*γ*

_{x0}that start from all points (

*x*

^{0},

*a*

_{2}), where

*x*

^{0}∈

*Per*(

*f*) ∩ Ω

^{u}(

*f*). By Proposition 8 every this fiber is the graph of the

*C*

^{1}-smooth implicit function

*x*=

*x*(

*y*) defined on the interval

*I*

_{2}. Moreover,

*x*=

*x*(

*y*) satisfies the initial conditions

*m*is the least period of the initial

*f*-periodic point

*x*

^{0}. By the theorem about the

*C*

^{1}-smooth implicit function and the equation (17) we have

*x, y*) ∈

*γ*

_{x0}.

Note that by the item 2 the sequence {*ɛ _{m}*}

_{m≥1}is increasing. Therefore, we construct the special

*δ*-trajectory for the real trajectory {

*F*(

^{j}*x, y*)}

_{j≥0}. Denote by

*l*the vertical closed interval that starts from the point (

_{x}*x, a*

_{2}).

*x*

^{rm}^{+i},

*y*

^{rm+i}) ∈

*l*. Since

_{xi}*z*

_{rm}_{+i}}

_{r≥0,0≤i≤m−1}is

*δ*-trajectory for the real trajectory of the point (

*x, y*). Moreover,

*j*≥ 1. Therefore, using (18) we have

*M*> 1 then the equality holds:

_{*}*m*

^{0}≥ 1 such that the inequality

*m*≥

*m*

^{0}.

*C*

^{1}-smooth functions {

*x*=

*x*(

*y*)} that start from the points {(

*x*

^{0},

*a*

_{2})}, where

*x*

^{0}∈

*Per*(

*f*) ∩ Ω

_{p}(

*f*),

*m*(

*x*

^{0}) ≥

*m*

^{0}(here

*m*(

*x*

^{0}) is the (least) period of

*x*

^{0}), is dense in itself in the

*C*

^{1}-topology. It implies, in particular, that for every

*C*

^{1}-smooth function

*x*=

*x*(

*y*) that starts from a point (

*x*

_{0},

*a*

_{2}) for

*x*

_{0}∈

*Per*(

*f*) ∩ Ω

_{p}(

*f*),

*m*(

*x*

_{0}) <

*m*

^{0}, the inequality holds

*x*

^{0}∈ Ω(

*f*) there exists the unique

*C*

^{1}-smooth function

*x*=

*x*(

*y*) :

*I*

_{2}→

*I*

_{1}with the graph that starts from the point (

*x*

^{0},

*a*

_{2}) and with the derivative satisfying the inequality (21) (see the inequality (20)). Proposition 9 is proved.

Extend the lamination *£*^{0}(*F*) up to the lamination *£*(*F*), where *£*(*F*) consists of *C*^{1}-smooth fibers that start from the points of the set Σ (*f* ) *×* {*a*_{2}}.

Use conditions (*iii _{f}*) – (

*iv*), definition of the set Δ(

_{f}*f*), Lemma 5 and Proposition 9. Then we obtain the following statement.

*Let the quotient f of the skew product of interval maps**satisfy conditions* (*iii _{f}*) – (

*iv*).

_{f}*Then for any δ*> 0

*there exists an ɛ-neighborhood*

*of the map*Φ

_{0}

*in the space*

*such that every map*

*obtained from*Φ

_{0}

*by means of the C*

^{1}

*-perturbation μ*=

*μ*(

*x, y*),

*where μ satisfies conditions*(

*i*) – (

_{μ}*ii*),

_{μ}*has the closed invariant set £*(

*F*)

*that consists of pairwise disjoint curvelinear fibers. These fibers start from the points of the set*Σ

^{*}(

*f*)

*×*{

*a*

_{2}},

*and are graphs of C*

^{1}

*-smooth functions x*=

*x*(

*y*)

*defined on the interval I*

_{2}.

*Every curvelinear fiber of the set £*(

*F*)

*is ɛ-close in the C*

^{1}

*-norm to the vertical closed interval that starts from the same initial point of the set*Σ

^{*}(

*f*)

*×*{

*a*

_{2}}

*just as the curvelinear fiber.*

*£*(

*F*) constructed in the Corollary 10 is the

*C*

^{1}-smooth lamination, i.e. the set

*C*

^{1}-smoothly on the variables

*x*

^{0},

*x, y*.

*μ*depends on

*x*and

*y*. Let

*γ*

_{x0}be the curvelinear fiber from

*£*(

*F*) that starts from an arbitrary point (

*x*

^{0},

*a*

_{2}), where

*x*

^{0}∈ Σ

^{*}(

*f*). We set

*x, y*) ∈

*γ*

_{x0}, that is

*£*(

*F*) on the set Σ

^{*}(

*f*).

*x*=

*x*

_{x}_{0}(

*y*) is the function with the graph

*γ*

_{x}_{0}.

As it follows from the equality (23) and Corollary 10, the curvelinear projection
*C*^{1}-smooth on the lamination *£*(*F*). Moreover, surjection
*ɛ*-close in the *C*^{1}-norm to the natural projection *pr*_{1} : Σ (*f* ) *× I*_{2} → Σ^{*} (*f* ).

*F*-invariance of the lamination

*£*(

*F*) and the formula (22) imply the equality

Comparison of the equalities (24) and (10) shoes that *F* is partially integrable map (see Definition 2). It completes the proof of Theorem 7.

In the end of the paper we give the example of the partially integrable map.

*Let F*(*x, y*) = (*f* (*x*) + *λx*(1 − *x*)*y*(1 − *y*), *g*(*x, y*)), *where**λ* > 0.

*C*

^{1}-smooth Ω-stable map

*f*of type ≻ 2

^{∞}from the paper [16]:

*f*(0) =

*f*(1) = 0,

*f*: [0, 1/4] → [0, 1/4] is increasing bijection,

*f*: [3/4, 1] → [0, 1/4] is decreasing bijection. Let

*M*= sup{

*f*(

*x*)} satisfy the inequality 3/4 <

*M*< 1, and the point

*x*such that the equality

_{M}*f*(

*x*) =

_{M}*M*holds, be the unique. Then the equality is valid:

*K*(

*f*) is the unique locally maximal quasiminimal set of

*f*,

*x*∈ Δ° (

_{M}*f*).

Let *λ* be so small that the function *μ*(*x, y*) = *λ x*(1 *− x*)*y*(1 *− y*) satisfies the condition (*ii _{μ}*). We have also

*μ*(0,

*y*) =

*μ*(1,

*y*) =

*μ*(

*x,*0) =

*μ*(

*x,*1) = 0 for all

*x, y*∈ [0, 1]. Hence, the condition (

*i*) is valid. It means that conditions of Theorem 7 are fulfilled, and there exists the invariant lamination

_{μ}*£*(

*F*) over the points of the set Ω(

*f*) = {0} ∪

*K*(

*f*). It implies the semiconjugacy of

*F*

_{|£}_{(F)}and

*f*

_{|}_{Ω(f )}. Therefore,

*F*is the partially integrable map (see Definition 2).

This paper is supported in the part by Ministry of Science and Education of Russia, grant No 1.3287.2017.

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## Footnotes

^{a}

A point *x* ∈ *I*_{1} ((*x, y*) ∈ *I*) is said to be *f*-nonwandering (*F*-nonwandering) point if for every its neighborhood *U*_{1}(*x*) (*U*((*x, y*)) = *U*_{1}(*x*) × *U*_{2}(*y*)) there is a natural number *n* such that the inequality *U*_{1}(*x*)∩ *f ^{n}*(

*U*

_{1}(

*x*)) ≠ ∅ (

*U*((

*x, y*)) ∩

*F*(

^{n}*U*(

*x, y*))) ≠ ∅ holds. The set of all

*f*-nonwandering (

*F*-nonwandering) points is said to be the nonwandering set of

*f*(

*F*) [22]. We use the notation Ω(

*f*) (Ω(

*F*)) for this set.

^{b}

The reducibility problem of integrable maps to skew products has been formulated by Grigorchuk to the Author during our verbal discussions (the formulation of the problem is not published) in the framework of the Conference devoted to the 70-th birthday of Professor V.M. Alexeev (Moscow, Russia, 2002).

^{c}

A quasiminimal set is the closure of an infinite recurrent trajectory (see [28]).

^{d}

It means correctness of the equality *f* (*D*(*Orb*(*x, f* ))) = *D*(*Orb*(*x, f* )).