Small C1-smooth perturbations of skew products and the partial integrability property

L.S. Efremova 1
  • 1 National Research Nizhni Novgorod State University, Moscow Institute of Physics and Technology, Nizhni Novgorod, Russia
L.S. Efremova
  • Corresponding author
  • National Research Nizhni Novgorod State University, Nizhni Novgorod, Moscow Institute of Physics and Technology, Moscow region, Dolgoprudny, Russia
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Abstract

In this paper we investigate stability of the integrability property of skew products of interval maps under small C1-smooth perturbations satisfying some conditions. We obtain here (sufficient) conditions of the partial integrability for maps under considerations. These conditions are formulated in the terms of properties of the unperturbed skew product. We give also the example of the partially integrable map.

1 Introduction

1. This paper is the direct continuation of the work [18], where stability of the Sharkovsky's order respectively small C1-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 C1-smooth maps defined below is presented.

We consider a map F of a closed rectangle I = I1 × I2 into itself, where I1, I2 are closed intervals of the straight line R1, Ik = [ak, bk] for k = 1, 2, and F satisfies the equality
F(x,y)=(f(x)+μ(x,y),g(x,y))forany(x,y)I.
Further we use the notation gx(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 I1 into itself [8], [26].

Following [18], [19] we suppose in this paper that the map (1) is C1-smooth on I, and the map f : I1I1 is so that the conditions hold:

  • (if) f (∂ I1) ⊂ ∂ I1, where (·) is the boundary of a set;
  • (iif) f is the Ω-stable in the space of C1-smooth self-maps of the interval I1 with the invariant boundary.

We suppose also that the C1-smooth function μ (of variables x and y) satisfies the boundary conditions:

  • (iμ) the equalities μ(x, a2) = μ(x, b2) = 0 are valid for every xI1; and the equalities μ(a1, y) = μ(b1, y) = 0 are valid for every yI2.

By the properties (if) and (iμ) the set ({a1} × I2) ∪ ({b1} × I2) is F-invariant. If, in addition, the inclusion gx(∂ I2) ⊂ ∂ I2 holds for all xI1 then the union of the horizontal intervals I1× {a2} and I1× {b2} is F-invariant too.

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

Let C˜ω1(I1) be the set of all C1-smooth maps of the interval I1 into itself satisfying conditions (if) – (iif). The standard C1-norm || · ||1,1 of the linear normalized space of C1-smooth maps of the interval I1 into the straight line R1 generates the C1-topology in C˜ω1(I1) . Denote by B˜1,ε1(f) elements of the base of the C1-topology in C˜ω1(I1) for every ɛ > 0 and fC˜ω1(I1) .

By the C1- Ω-stability of the map f (see the condition (iif)) for any δ > 0 there exists an ɛ-neighborhood B˜1,ε1(f) of f in the space C˜ω1(I1) such that every map from this neighborhood is Ω-conjugate with f by means of a homeomorphism which is δ-close in the C0-topology of the uniform convergence to the identity map of the nonwandering seta of the map f.

Let C˜1(I,I1) be the set of C1-smooth maps of the rectangle I into the interval I1 endowed with the standard C1-norm || · ||1,(1,1) of the linear normalized space of C1-smooth maps of the rectangle I into the straight line R1 that contains the interval I1. This norm induces the C1-topology in the space C˜1(I,I1) with the base given by the set of ɛ-balls B˜(1,1),ε1(φ) for every φC˜1(I,I1) and ɛ > 0.

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

  • (iiμ) ||μ||1,(1,1) < ɛ, where ɛ is found for δ > 0 by the property of the C1- Ω-stability of f.

The following inequality connects norms || · ||1,1 for every yI2 and || · ||1,(1,1):
||μ||1,1<||μ||1,(1,1).
Every function f of one variable can be considered as the function of two variables of the type fpr1, where pr1 : II1 is the natural projection of I on I1. Hence, by the condition (iiμ) and the inequality (2) we have (f+μ)B˜(1,1),ε1(fpr1) , and the belonging
(f+μ)B˜1,ε1(f)
holds for every yI2.
Denote by Cω1(I) the set of C1-smooth maps (1) such that the function f satisfies conditions (if) – (iif), and μ satisfies conditions (iμ) – (iiμ). Endow Cω1(I) with the standard C1-norm || · ||1 of the linear normalized space of C1-smooth maps of the rectangle I into the plane R2. The base of the C1-topology generated by this norm, is given by the system of ɛ-balls
Bε1(F)={GCω1(I):||GF||1<ε}
with the center F for every ɛ > 0 and FCω1(I) .
3. The map (1) from the space Cω1(I) is obtained by small C1-smooth perturbations (satisfying conditions (iμ) – (iiμ)) of the C1-smooth skew product of interval maps of the type
Φ0(x,y)=(f(x),gx(y)).
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 (x0, y0) ∈ I as the composition of values of various skew products of interval maps in the corresponding points:
Fn(x0,y0)=(xn,yn)=Φyn1Φy0(x0,y0),
where a skew product Φyi : II for every i (0 ≤ in − 1) is presented in the form
Φyi(x,y)=(φyi(x),gx(y)).
Here
φyi(x)=f(x)+μyi(x),andμyi(x)=μ(x,yi).
To write the equality (5) in the coordinate form we set
φy0,n(x0)=φyn1φy0(x0);g(x0,y0),n(y0)=gφyn1(xn1)gx0(y0).
Then we have:
Fn(x0,y0)=(φy0,n(x0),g(x0,y0),n(y0)).
All previous information of the item 3 means that an important role in this paper belongs to skew products of interval maps.

Denote by Tω,01(I) the space of C1-smooth skew products of interval maps with quotients satisfying conditions (if) – (iif). Endow this space with the C1-topology generated by the standard C1-norm || · ||1. The structure of the functional space Tω,01(I) and dynamical properties of skew products from this space are studied in [9] – [16].

We use also the space T˜ω,01(I) of skew products of interval maps respectively which the boundary ∂ I of the rectangle I is invariant. Then the inclusions are valid:
T˜ω,01(I)Tω,01(I)Cω1(I).
The base of the C1-topology in T˜ω,01(I) is given by the set of ɛ-balls B˜ε1(Φ) with the center Φ for all ɛ > 0 and ΦT˜ω,01(I) .
Note that by the formula (1) and the condition (iμ) every map FCω1(I) obtained from the skew products of interval maps Φ0T˜ω,01(I) possesses the property:
F(I)I.

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.)

Definition 1
[4] We say that a map G of some (open or closed) domain Π ⊂ R2 to itself is integrable if there exists a self-map ψ of an interval J of the real line R1 such that G is semiconjugate with ψ by means of a continuous surjection H˜:ΠJ , so that
H˜G=ψH˜.
Remark 1

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

Remark 2

As it follows from Definition 1 skew products of interval maps are integrable maps. Here H˜=pr1 . Moreover, every integrable map satisfying some natural conditions can be reduced to a skew productb.

Theorem 1

[4]. Let Π be a convex connected compact subset ofR2such 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 surjectionH˜:ΠJJ that is one-to-one with respect to x (here J is a closed interval ofR1) if and only if some homeomorphism reduces G to a skew product of interval maps defined in a compact planar rectangle.

Remark 3

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.

Remark 4

Point out that the existence of a continuous invariant foliation for Lorenz type maps is proven in [3], and existence of a C1-smooth invariant foliation (with C2-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.

Definition 2
We say that a map G defined on some (open or closed) domain Π of the plane R2 with values in Π is 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 R1 and a closed invariant set BJ (BJ) such that G|A is semiconjugate with ψ|B by means of a continuous surjection H˜:AB , i. e. the equality holds
H˜G|A=ψ|BH˜.

5. In this paper we investigate stability of the integrability property of skew products of interval maps under small C1-smooth perturbations satisfying conditions (iμ) – (iiμ). We obtain here (sufficient) conditions of partial integrability for maps from the space Cω1(I) (§3). These conditions are formulated in the terms of properties of the unperturbed skew product Φ0Tω,01(I) (see the formula (4)). We give also the example of the partially integrable map (1) (§3).

2 Preliminaries

This section contains the relevant definitions and results on dynamics both of continuous maps and C1-smooth Ω-stable maps of a closed interval.

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

Theorem 2
[32] If a continuous map f : I1I1contains a periodic orbit of a (least) period m > 1 then it contains also periodic orbits of every (least) period n, where n precedes m (nm) in the Sharkovsky's order:
122223229227225223292725239753.

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 {1,2,22,...,2ν}, where 0 ≤ ν < +∞; the second subspace consists of the maps of type 2, that is, maps whose periodic points have (least) periods {1,2,22,...,2i,2i+1,...}; the third subspace consists of the maps of type ≻ 2, that is, maps with periodic points that possess (least) periods outside the set {2i}i ≥0.

In this paper we consider maps from the space Cω1(I) such that f satisfies the following additional condition: (iiif) f has type ≻ 2.

The above condition (iiif) means that f demonstrates a chaotic behavior (see, e.g., [33]).

2. Formulate the properties of C1-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.

Lemma 3

[21], [27] LetfC˜ω1(I1)and satisfy the condition (iiif). Then

  • (3.1) the nonwandering set Ω(f ) is the union of a finite number of hyperbolic periodic points (that form the rarefied set Ωr(f )) and a finite number of locally maximal (i.e., maximal quasiminimalcsets in some their neighborhood) hyperbolic perfect nowhere dense sets (that form the perfect set Ωp(f )));
  • (3.2) periodic points are everywhere dense in the set Ωp(f ); moreover, for every natural number m ≥ 2 periodic points with multiple m (least) periods are everywhere dense in Ωp(f );
  • (3.3) there are numbers α = α(f ) > 0 and c = c(f ) > 1 so that for every x ∈ Ωp(f ) and n ≥ 1 the inequality |(f n(x))′| > αcn holds (that is, Ωp(f ) is the repelling hyperbolic set);
  • (3.4) the subspaceC˜ω1(I1)of maps satisfying condition (iiif) is open and everywhere dense in the containing it space of C1-smooth self-maps of the closed interval I1of type ≻ 2with the invariant boundary.

Corollary 4
[21], [27] LetfC˜ω1(I1)and satisfy the condition (iiif). Then the equality holds:
Ω(f)=Ωs(f)Ωu(f).
Here Ωs(f ) is the nonempty finite invariant set of all f-sinks. The set Ωu(f ) is invariant and equals the union of Ωp(f ) with a finite (possibly empty) set that consists of isolated sources in the set of f-periodic points.

In the set I1 \ Ω(f ) (just as in the set I1 \ Ωp(f )) the points of attraction domains of f-sinks are everywhere dense.

Point out that for the map fC˜ω1(I1) the inclusion 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.

Following [21] we construct the set which is invariant both with respect to f, and with respect to f−1, and contains the set Ωu(f) as its subset. For this goal we need the attraction domain of all f-sinks:
Δ(f)=Orb(x,f)Ωs(f)i=0+Di(Orb(x,f)),
where Orb(x, f ) is the periodic orbit of the sink x ∈ Ωs(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 ).
Immediate attraction domain 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 I1) f m-invariant intervals Df j(x) (0 ≤ jm − 1) such that every of these intervals contains the point f j(x) :
D(Orb(x,f))=j=0m1Dfj(x).

Complete f-invariance of the immediate attraction domaindD(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 (iiif) 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 I1); 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

  • (ivf) Cr(f ) ⊂ Δ° (f ), where Δ° (f) is the interior of the set Δ (f).

Lemma 5

[33] LetfC˜ω1(I1)satisfy conditions (iiif) – (ivf). Then one of the following three cases is realized for the boundary ∂ (Df j(x)) of every interval Df j(x) (0 ≤ jm − 1):

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

Define the closed set that is invariant with respect to f and f−1:
Σ(f)=I1\Δ(f).
By Lemma 5 and formula (12) we have: Ωu(f ) ⊂ Σ(f), and Ωs(f) ∩ Σ (f) = ∅.
Definition 3
[21] The map fC˜ω1(I1) is said to be Σ-stable (in the C1-topology) if for every δ > 0 there exists an ɛ-neighborhood B˜1,ε1(f) of the map fC˜ω1(I1) such that every map φB˜1,ε1(f) is Σ-conjugate to f, that is, the equality
hf|Σ(f)=φ|Σ(φ)h
holds for some homeomorphism h : Σ(f ) → Σ(ϕ). Here h is δ-close to the identity map on Σ(f ) in the C0-topology.

The following claim is the direct corollary of Definition 3.

Lemma 6

LetfC˜ω1(I1)satisfy conditions (iiif) – (ivf). Then f is Σ-stable in the C1-topology.

Remark 5

By [21] the set of maps from C˜ω1(I1) satisfying conditions (iiif) – (ivf) contains the open everywhere dense subset of C2-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.

Theorem 7

Let the quotient f of the skew product of interval mapsΦ0T˜ω,01(I)satisfy conditions (iiif) – (ivf). Then for any δ > 0 there exists an ɛ-neighborhoodBε1(Φ0)of the map Φ0in the spaceCω1(I)such that every mapFBε1(Φ0)obtained from Φ0by means of the C1-perturbation μ = μ(x, y), where μ satisfies conditions (iμ) – (iiμ), 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) × {a2} (where Σ* (f ) = Σ(f ) ∪ Ωs(f)) and are graphs of C1-smooth functions x = x(y) defined on the interval I2. The functionH˜:£(F)Σ*(f)that realizes the partial integrability property, is C1-smooth surjection, ɛ-close in the C1-norm to the natural projection pr1 : Σ* (f ) × I2 → Σ*(f).

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

Proposition 8

[18] LetΦ0T˜ω,01(I) . Then for any δ > 0 there exists an ɛ-neighborhoodBε1(Φ0)of the map Φ0in the spaceCω1(I)such that every mapFBε1(Φ0)obtained from Φ0by means of the C1-perturbation μ = μ(x, y), where μ satisfies conditions (iμ) – (iiμ), 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 ) × {a2}, and are the graphs of continuous functions x = x(y) defined on the interval I2; moreover, fibers starting from the points (x, a2), where x is f-periodic point, are C1-smooth. Every curvelinear fiber is δ-close in the C0-norm to the vertical closed interval that starts from the same initial point of the set Ω(f ) × {a2} just as the curvelinear fiber.

Prove C1-smoothness of all fibers from the set £0(F).

Proposition 9

LetΦ0T˜ω,01(I) . Then for any δ > 0 there exists an ɛ-neighborhoodBε1(Φ0)of the map Φ0in the spaceCω1(I)such that the closed invariant set £0(F) of every mapFBε1(Φ0)obtained from Φ0by means of the C1-perturbation μ = μ(x, y), where μ satisfies conditions (iμ) – (iiμ), consists of graphs of C1-smooth functions x = x(y) defined on the interval I2. In addition, every curvelinear fiber of the set £0(F) is ɛ-close in the C1-norm to the vertical closed interval that starts from the same initial point of the set Ω(f ) × {a2} just as the curvelinear fiber.

Proof

1. Fix a number δ > 0. We find a positive number ɛ > 0 for δ using the C1- Ω-stability property of the map fC˜ω1(I1) . The neighborhood B˜1,ε1(f) 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 ).

We use also the ɛ-neighborhood Bε1(Φ0) of the map Φ0 in the space Cω1(I) . Then by formulas (1), (4) and by the property (iiμ) the inequality
||FΦ0||1=||μ||1,(1,1)<ε
is valid for any map FBε1(Φ0) . It implies, in particular, correctness of the belonging (3) for any yI2 and means also that the map (f + μ) is Ω-conjugate with f for every yI2 by means of the homeomorphism that is δ-close to the identity map of the set Ω(f ).
We need ɛm-neighborhoods Bεm1(Φ0m) of iterations Φ0m for any m > 1 in the space Cω1(I) that correspond the chosen ɛ-neighborhood Bε1(Φ0) of the map Φ0. Since FBε1(Φ0) then FmBεm1(Φ0m) . Using formulas (8)(9) we obtain the inequalities
||φy,mfm||1,(1,1)||FmΦ0m||1<εm;
moreover, for every yI2 the inequality holds:
||φy,mfm||1,1<||φy,mfm||1,(1,1).
Therefore,
φy,mB˜(1,1),εm1(fmpr1)and,themoreso,φy,mB˜1,εm1(fm)
for every yI2. Here B˜(1,1),εm1(fmpr1) is the ɛm-neighborhood of the map (fm ○ pr1) in the space C˜1(I,I1) and B˜1,εm1(fm) is the ɛm-neighborhood of the map f m in the space C˜ω1(I1) . The neighborhood B˜1,εm1(fm) consists of the maps, which are Ω-conjugate to f m by means of the homeomorphisms δ-close to the identity map on the nonwandering set Ω(f m) = Ω(f ) (see Lemma 3).

2. Let f satisfy the condition (iiif). Denote by £u(F) (£u(F) ⊂ £0(F)) the set of curvelinear fibers that start from all points of Ωu(f ).

Prove that there is the universal natural number n* such that the equality
inf(x,y)£u(F)|xφy,n*(x)|=M*,whereM*>1,
holds for every yI2.
In fact, using the definition of the set £u(F) and Lemma 3 (see claims (3.1) – (3.3)) for every yI2 we point out the least natural number n* (y) satisfying
infx(£u(F))(y)|xφy,n*(y)(x)|>1.
By the claim (3.3) of Lemma 3 the inequality
infx(£u(F))(y)|xφy,n(x)|>1
is valid for every nn* (y).
Since the partial derivative xφy,n*(y)(x) is uniformly continuous on the compact
(£u(F))(y)×{y}
then by the inequality (15) there exists a θ (y)-neighborhood Uθ(y) (£u(F))(y)×{y}) of the set (£u(F))(y)×{y} in I such that
inf(x,y)Uθ(y)((£u(F))(y)×{y})|xφy,n*(y)(x)|>1.
Moreover, by the formula (13)n* (y) is the least natural number for which the inequality (16) holds.
Let 2I1 be the topological space of all closed subsets of the closed interval I1 with the exponential topology. By compactness of the closed intervals I1, I2 the set 2I1 × I2 is the compact [24]. Then the closed set
£u(F)=yI2((£u(F))(y)×{y})
is the compact in 2I1 × I2. Using compactness of the set £u(F) in 2I1× I2 we distinguish from its infinite open cover {Uθ(y) ((£u(F))(y) × {y})}yI2 the finite subcover. Let neighborhoods {Uθ(yj) ((£u(F))(yj) × {yj})}1≤jq of the sets {(£u(F))(yj) × {yj}}1≤ jq form this finite subcover.
The set £u(F) consists of continuous fibers (see Proposition 8). Therefore, for every j (1 ≤ jq) there exists j′ (j′ ≠ j, 1 ≤ j′ ≤ q) such that
Uθ(yj)((£u(F))(yj)×{yj})Uθ(yj)((£u(F))(yj)×{yj}).
Thus, using the above considerations of this item 2 we obtain from here that the equalities hold
n*(y1)==n*(yj)==n*(yq).
Set n* = n* (yj) (1 ≤ jq). Using continuity of the partial derivative xφy,n*(x) on 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 Fn and use the claim (3.2) of Lemma 3.

3. Prove that every curvelinear fiber is ɛ-close in the C1-norm to the vertical closed interval that starts from the same initial point (x, a2) of the set Ω(f ) × {a2} just as the curvelinear fiber.

3.1. Begin from the curvelinear fibers γx0 that start from all points (x0, a2), where x0Per(f ) ∩ Ωu(f ). By Proposition 8 every this fiber is the graph of the C1-smooth implicit function x = x(y) defined on the interval I2. Moreover, x = x(y) satisfies the initial conditions
x(a2)=x(b2)=x0
and the equation
φy,m(x)=x[18],
where m is the least period of the initial f-periodic point x0. By the theorem about the C1-smooth implicit function and the equation (17) we have
ddyx(y)=yφy,m(x)xφy,m(x)1
in any point (x, y) ∈ γx0.

Note that by the item 2 the sequence {ɛm}m≥1 is increasing. Therefore, we construct the special δ-trajectory for the real trajectory {Fj(x, y)}j≥0. Denote by lx the vertical closed interval that starts from the point (x, a2).

Choose the points
z0=(x0,y),z1=(x1,y1),,zi=(xi,yi),,zm1=(xm1,ym1),zrm+i=(xrm+i,yrm+i)(r0,0im1),
where
xrm+i=frm+i(x0)=fi(x0),yrm+i=g(x,y),rm+i(y)(here1im1,m2).
Then (xrm+i, yrm+i) ∈ lxi. Since
Frm+i(x,y)=(φy,rm+i(x),g(x,y),rm+i(y)),andFrm+i(x,y)γxi,
then {zrm+i}r≥0,0≤im−1 is δ-trajectory for the real trajectory of the point (x, y). Moreover,
|fj(x0)φy,j(x)|<δ.
It implies, in particular, correctness of the inequality
|yφy,j(x)|<ε
for every j ≥ 1. Therefore, using (18) we have
|ddyx(y)|<εM*m1.
Since M* > 1 then the equality holds:
limm+M*m=+.
Then there exists m0 ≥ 1 such that the inequality
|ddyx(y)|<ε(seetheinequality(19))
is valid for every mm0.
3.2. As it follows from the item 3.1, the set of C1-smooth functions {x = x(y)} that start from the points {(x0, a2)}, where x0Per(f ) ∩ Ωp(f ), m(x0) ≥ m0 (here m(x0) is the (least) period of x0), is dense in itself in the C1-topology. It implies, in particular, that for every C1-smooth function x = x(y) that starts from a point (x0, a2) for x0Per(f ) ∩ Ωp(f ), m(x0) < m0, the inequality holds
|ddyx(y)|ε.
In addition, for every nonperiodic point x0 ∈ Ω(f ) there exists the unique C1-smooth function x = x(y) : I2I1 with the graph that starts from the point (x0, a2) 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 C1-smooth fibers that start from the points of the set Σ (f ) × {a2}.

Use conditions (iiif) – (ivf), definition of the set Δ(f ), Lemma 5 and Proposition 9. Then we obtain the following statement.

Corollary 10

Let the quotient f of the skew product of interval mapsΦ0T˜ω,01(I)satisfy conditions (iiif) – (ivf). Then for any δ > 0 there exists an ɛ-neighborhoodBε1(Φ0)of the map Φ0in the spaceCω1(I)such that every mapFBε1(Φ0)obtained from Φ0by means of the C1-perturbation μ = μ(x, y), where μ satisfies conditions (iμ) – (iiμ), has the closed invariant set £(F) that consists of pairwise disjoint curvelinear fibers. These fibers start from the points of the set Σ* (f ) × {a2}, and are graphs of C1-smooth functions x = x(y) defined on the interval I2. Every curvelinear fiber of the set £(F) is ɛ-close in the C1-norm to the vertical closed interval that starts from the same initial point of the set Σ* (f ) × {a2} just as the curvelinear fiber.

Remark 6
The set £(F) constructed in the Corollary 10 is the C1-smooth lamination, i.e. the set
£(F)={x0,x,y},wherex0Σ*(f),x=x(y),
depends C1-smoothly on the variables x0, x, y.
Let conditions of Theorem 7 be fulfilled, and FBε1(Φ0) be given by the formula (1), where μ depends on x and y. Let γx0 be the curvelinear fiber from £(F) that starts from an arbitrary point (x0, a2), where x0 ∈ Σ* (f ). We set
H˜(x,y)=x0
for any point (x, y) ∈ γx0, that is H˜(γx0)=x0 , and H˜(£(F))=Σ*(f) . It means that the curvelinear projection H˜ is the surjection of the set £(F) on the set Σ* (f).
By the equality (22) we have
H˜(x,y)=pr1(x,y)xx0(y)+x0,
where x = xx0 (y) is the function with the graph γx0.
Remark 7

As it follows from the equality (23) and Corollary 10, the curvelinear projection H˜ is C1-smooth on the lamination £(F). Moreover, surjection H˜:£(F)Σ*(f) is ɛ-close in the C1-norm to the natural projection pr1 : Σ (f ) × I2 → Σ* (f ).

Remark 8
F-invariance of the lamination £(F) and the formula (22) imply the equality
H˜F|£(F)=f|Σ*(f)H˜.

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.

Example 11

Let F(x, y) = (f (x) + λx(1 − x)y(1 − y), g(x, y)), whereFCω1([0,1]2) , λ > 0.

The condition “ FCω1([0,1]2) ” implies “ fC˜ω1([0,1]) ”. We use here the model C1-smooth Ω-stable map f of type ≻ 2 from the paper [16]:
f(x)={h˜(x),ifx[0,14);9(14x)(x34)+14,ifx[14,34);h˜(1x),ifx[34,1];
where h˜ is so that 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 xM such that the equality f (xM) = M holds, be the unique. Then the equality is valid:
Ω(f)={0}K(f),
where K(f ) is the unique locally maximal quasiminimal set of f, K(f)=Ωp(f)[14,34] and xM ∈ Δ° (f ).

Let λ be so small that the function μ(x, y) = λ x(1 − x)y(1 − y) satisfies the condition (iiμ). 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 (iμ) is valid. It means that conditions of Theorem 7 are fulfilled, and there exists the invariant lamination £(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).

Acknowledgement

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

References

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    V.E. Adler. (2000), Discretizations of the Landau-Lifshits equation, TMPh, 124:1, 897–908.

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    V.E. Adler, A.I. Bobenko, Yu.B. Suris. (2009), Discrete Nonlinear Hyperbolic Equations. Classification of Integrable Cases, Funct. Anal. Appl., 43, 3–17.

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    V.S. Afraimovich, V.V. Bykov, L.P. Shilnikov. (1982), Attractive nonrough limit sets of Lorenz-attractor type. Trudy Moskovskogo Matematicheskogo Obshchestva 44, 150–212.

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    S.S. Belmesova, L.S. Efremova. (2015), On the Concept of Integrability for Discrete Dynamical Systems. Investigation of Wandering Points of Some Trace Map, Nonlinear Maps and their Applic. Springer Proc. in Math. and Statist., 112, 127–158.

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  • [5]

    V.N. Belykh. (1984), On bifurcations of saddle separatrixes of Lorenz system” (Russian), Differential Equations, 20, 1666–1674.

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    G.D. Birkhoff. (1927), Dynamical systems, Amer. Math. Soc. Colloq. Publ., 9, Amer. Math. Soc., New York.

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    S.V. Bolotin, V.V. Kozlov. (2017), Topology, singularities and integrability in Hamiltonian systems with two degrees of freedom, Izv. Math., 81, 671–687.

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    P. Brandão. (2014), On the structure of Lorenz maps, Preprint, arXiv:1402.2862.

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    L.S. Efremova. (2001), On the Concept of the Ω-Function for the Skew Product of Interval Mappings, Itogi Nauki Tekh. Ser. Sovrem. Mat. Prilozh. Temat. Obz., Vseross. Inst. Nauchn. i Tekhn. Inform. (VINITI), [English translation], Journ. Math. Sci. (N.-Y.), 105, 1779–1798, original work published 1999.

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    L.S. Efremova. (2010), Space of C1-smooth skew products of maps of an interval, TMPh, [English translation], Theoretical and Mathematical Physics, 164, 1208–1214.

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  • [11]

    L.S. Efremova. (2013), A decomposition theorem for the space of C1-smooth skew products with complicated dynamics of the quotient map, Mat.Sb., [English translation], Sb. Math., 204, 1598–1623.

    • Crossref
    • Export Citation
  • [12]

    L.S. Efremova. (2014), Remarks on the nonwandering set of skew products with a closed set of periodic points of the quotient map, Nonlinear Maps and their Applic. Springer Proc. in Math. and Statist., 57, 39–58.

    • Crossref
    • Export Citation
  • [13]

    L.S. Efremova. (2016), Multivalued functions and nonwandering set of skew products of maps of an interval with complicated dynamics of quotient map, Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, [English translation], Russian Math., 60, 77–81.

    • Crossref
    • Export Citation
  • [14]

    L.S. Efremova. (2016), Nonwandering sets of C1-smooth skew products of interval maps with complicated dynamics of quotient map, Problemy matem. analiza, Novosibirsk, [English translation], Journ. Math. Sci. (New York), 219, 86–98.

    • Crossref
    • Export Citation
  • [15]

    L.S. Efremova. (2016), Stability as a whole of a family of fibers maps and Ω-stability of C1-smooth skew products of maps of an interval, J. Phys.: Conf. Ser., 692, 012010.

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    L.S. Efremova. (2017), Dynamics of skew products of maps of an interval, Uspekhi Mat. Nauk, [English translation], Russian Math. Surveys, 72, 101–178.

    • Crossref
    • Export Citation
  • [17]

    L.S. Efremova. (2018), The Trace Map and Integrability of the Multifunctions, J. Phys.: Conf. Ser., 990, 012003.

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    L.S. Efremova. (2020), Small Perturbations of Smooth Skew Products and Sharkovsky's Theorem, JDEA, https://doi.org/10.1080/10236198.2020.1804556

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    L.S. Efremova. (2020), Periodic behavior of maps obtained by small perturbations of smooth skew products, Discontinuity, Nonlinearity, Complexity, 9(4), 519–523.

    • Crossref
    • Export Citation
  • [20]

    R.I. Grigorchuk, A. Žuk. (2001), The Lamplighter group as a group generated by a 2-state automata, and its spectrum, Geometriae Dedicata, 87, 209–244.

    • Crossref
    • Export Citation
  • [21]

    M.V. Jakobson. (1971), Smooth mappings of the circle into itself, Mat. Sb. (N.S.), [English translation], Mathematics of the USSR – Sbornik, 14, 161–185.

    • Crossref
    • Export Citation
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    A.B. Katok, B. Hasselblatt. (1995), Introduction to the modern theory of dynamical systems, Encyclopedia Math. Appl., 54, Cambridge Univ. Press, Cambridge.

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    V.V. Kozlov. (2019), Tensor invariants and integration of differential equations, Russian Math. Surveys, 74, 111–140.

    • Crossref
    • Export Citation
  • [24]

    V.V. Kozlov, D.V. Treschev. (2016), Topology of the configuration space, singularities of the potential, and polynomial integrals of equations of dynamics, Sb. Math., 207, 1435–1449.

    • Crossref
    • Export Citation
  • [25]

    K. Kuratowski. (1966), Topology, 1, Academic Press, New York-London, 1966.

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    M.I. Malkin. (1991), Rotation intervals and the dynamics of Lorenz type mappings, Selecta Math. Sovietica, 10, 265–275.

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    W. de Melo, S. van Strien. (1993), One-dimensional dynamics, A series of modern surveys in mathematics, Springer-Verlag.

  • [28]

    V.V. Nemytskii, V.V. Stepanov. (1960), Qualitative theory of differential equations, Princeton Univ. Press, Princeton, NJ, 1960, original work published 1947.

  • [29]

    C. Robinson. (1989), Homoclinic bifurcation to a transitive attractor of the Lorenz type, Nonlinearity, 495–518.

  • [30]

    C. Robinson. (1992), Homoclinic bifurcation to a transitive attractor of the Lorenz type, II, SIAM J. Math. Anal., 23, 1255–1268.

    • Crossref
    • Export Citation
  • [31]

    M. Rychlik. (1990), Lorenz attractors through Sil’nikov type bifurcations, Part I, Ergodic Theory Dynamical Systems, 10, 793–822.

    • Crossref
    • Export Citation
  • [32]

    A.N. Sharkovskii. (1964), Coexistence of cycles of a continuous map of the line into itself (Russian), Ukrain. Mat. Zh. 16, 61–71, [English translation], (1995) Bifurcation and Chaos, 5, 1263 – 1273.

  • [33]

    A.N. Sharkovsky, Yu.L. Maistrenko, and E.Yu. Romanenko. (1993), Difference equations and their applications, Naukova Dumka, Kyev, [English translation], Math. Appl., 250, Kluwer Acad. Publ., Dordrecht, original work published 1986.

  • [34]

    M.V. Shashkov, L.P. Shilnikov. (1994), On the existence of a smooth invariant foliation in Lorenz-type mappings, Differ. Equ., 30, 536–544.

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    Yu. B. Suris. (1989), Integrable mappings of the standard type, Funct. Anal. Appl., 23, 74–76.

    • Crossref
    • Export Citation
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    A.P. Veselov. (1991), Integrable maps, Russian Math. Survey, 46, 1–51.

    • Crossref
    • Export Citation

Footnotes

a

A point xI1 ((x, y) ∈ I) is said to be f-nonwandering (F-nonwandering) point if for every its neighborhood U1(x) (U((x, y)) = U1(x) × U2(y)) there is a natural number n such that the inequality U1(x)∩ f n(U1(x)) ≠ ∅ (U((x, y)) ∩ Fn(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 )).

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1]

    V.E. Adler. (2000), Discretizations of the Landau-Lifshits equation, TMPh, 124:1, 897–908.

  • [2]

    V.E. Adler, A.I. Bobenko, Yu.B. Suris. (2009), Discrete Nonlinear Hyperbolic Equations. Classification of Integrable Cases, Funct. Anal. Appl., 43, 3–17.

    • Crossref
    • Export Citation
  • [3]

    V.S. Afraimovich, V.V. Bykov, L.P. Shilnikov. (1982), Attractive nonrough limit sets of Lorenz-attractor type. Trudy Moskovskogo Matematicheskogo Obshchestva 44, 150–212.

  • [4]

    S.S. Belmesova, L.S. Efremova. (2015), On the Concept of Integrability for Discrete Dynamical Systems. Investigation of Wandering Points of Some Trace Map, Nonlinear Maps and their Applic. Springer Proc. in Math. and Statist., 112, 127–158.

    • Crossref
    • Export Citation
  • [5]

    V.N. Belykh. (1984), On bifurcations of saddle separatrixes of Lorenz system” (Russian), Differential Equations, 20, 1666–1674.

  • [6]

    G.D. Birkhoff. (1927), Dynamical systems, Amer. Math. Soc. Colloq. Publ., 9, Amer. Math. Soc., New York.

  • [7]

    S.V. Bolotin, V.V. Kozlov. (2017), Topology, singularities and integrability in Hamiltonian systems with two degrees of freedom, Izv. Math., 81, 671–687.

    • Crossref
    • Export Citation
  • [8]

    P. Brandão. (2014), On the structure of Lorenz maps, Preprint, arXiv:1402.2862.

  • [9]

    L.S. Efremova. (2001), On the Concept of the Ω-Function for the Skew Product of Interval Mappings, Itogi Nauki Tekh. Ser. Sovrem. Mat. Prilozh. Temat. Obz., Vseross. Inst. Nauchn. i Tekhn. Inform. (VINITI), [English translation], Journ. Math. Sci. (N.-Y.), 105, 1779–1798, original work published 1999.

    • Crossref
    • Export Citation
  • [10]

    L.S. Efremova. (2010), Space of C1-smooth skew products of maps of an interval, TMPh, [English translation], Theoretical and Mathematical Physics, 164, 1208–1214.

    • Crossref
    • Export Citation
  • [11]

    L.S. Efremova. (2013), A decomposition theorem for the space of C1-smooth skew products with complicated dynamics of the quotient map, Mat.Sb., [English translation], Sb. Math., 204, 1598–1623.

    • Crossref
    • Export Citation
  • [12]

    L.S. Efremova. (2014), Remarks on the nonwandering set of skew products with a closed set of periodic points of the quotient map, Nonlinear Maps and their Applic. Springer Proc. in Math. and Statist., 57, 39–58.

    • Crossref
    • Export Citation
  • [13]

    L.S. Efremova. (2016), Multivalued functions and nonwandering set of skew products of maps of an interval with complicated dynamics of quotient map, Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, [English translation], Russian Math., 60, 77–81.

    • Crossref
    • Export Citation
  • [14]

    L.S. Efremova. (2016), Nonwandering sets of C1-smooth skew products of interval maps with complicated dynamics of quotient map, Problemy matem. analiza, Novosibirsk, [English translation], Journ. Math. Sci. (New York), 219, 86–98.

    • Crossref
    • Export Citation
  • [15]

    L.S. Efremova. (2016), Stability as a whole of a family of fibers maps and Ω-stability of C1-smooth skew products of maps of an interval, J. Phys.: Conf. Ser., 692, 012010.

  • [16]

    L.S. Efremova. (2017), Dynamics of skew products of maps of an interval, Uspekhi Mat. Nauk, [English translation], Russian Math. Surveys, 72, 101–178.

    • Crossref
    • Export Citation
  • [17]

    L.S. Efremova. (2018), The Trace Map and Integrability of the Multifunctions, J. Phys.: Conf. Ser., 990, 012003.

  • [18]

    L.S. Efremova. (2020), Small Perturbations of Smooth Skew Products and Sharkovsky's Theorem, JDEA, https://doi.org/10.1080/10236198.2020.1804556

  • [19]

    L.S. Efremova. (2020), Periodic behavior of maps obtained by small perturbations of smooth skew products, Discontinuity, Nonlinearity, Complexity, 9(4), 519–523.

    • Crossref
    • Export Citation
  • [20]

    R.I. Grigorchuk, A. Žuk. (2001), The Lamplighter group as a group generated by a 2-state automata, and its spectrum, Geometriae Dedicata, 87, 209–244.

    • Crossref
    • Export Citation
  • [21]

    M.V. Jakobson. (1971), Smooth mappings of the circle into itself, Mat. Sb. (N.S.), [English translation], Mathematics of the USSR – Sbornik, 14, 161–185.

    • Crossref
    • Export Citation
  • [22]

    A.B. Katok, B. Hasselblatt. (1995), Introduction to the modern theory of dynamical systems, Encyclopedia Math. Appl., 54, Cambridge Univ. Press, Cambridge.

  • [23]

    V.V. Kozlov. (2019), Tensor invariants and integration of differential equations, Russian Math. Surveys, 74, 111–140.

    • Crossref
    • Export Citation
  • [24]

    V.V. Kozlov, D.V. Treschev. (2016), Topology of the configuration space, singularities of the potential, and polynomial integrals of equations of dynamics, Sb. Math., 207, 1435–1449.

    • Crossref
    • Export Citation
  • [25]

    K. Kuratowski. (1966), Topology, 1, Academic Press, New York-London, 1966.

  • [26]

    M.I. Malkin. (1991), Rotation intervals and the dynamics of Lorenz type mappings, Selecta Math. Sovietica, 10, 265–275.

  • [27]

    W. de Melo, S. van Strien. (1993), One-dimensional dynamics, A series of modern surveys in mathematics, Springer-Verlag.

  • [28]

    V.V. Nemytskii, V.V. Stepanov. (1960), Qualitative theory of differential equations, Princeton Univ. Press, Princeton, NJ, 1960, original work published 1947.

  • [29]

    C. Robinson. (1989), Homoclinic bifurcation to a transitive attractor of the Lorenz type, Nonlinearity, 495–518.

  • [30]

    C. Robinson. (1992), Homoclinic bifurcation to a transitive attractor of the Lorenz type, II, SIAM J. Math. Anal., 23, 1255–1268.

    • Crossref
    • Export Citation
  • [31]

    M. Rychlik. (1990), Lorenz attractors through Sil’nikov type bifurcations, Part I, Ergodic Theory Dynamical Systems, 10, 793–822.

    • Crossref
    • Export Citation
  • [32]

    A.N. Sharkovskii. (1964), Coexistence of cycles of a continuous map of the line into itself (Russian), Ukrain. Mat. Zh. 16, 61–71, [English translation], (1995) Bifurcation and Chaos, 5, 1263 – 1273.

  • [33]

    A.N. Sharkovsky, Yu.L. Maistrenko, and E.Yu. Romanenko. (1993), Difference equations and their applications, Naukova Dumka, Kyev, [English translation], Math. Appl., 250, Kluwer Acad. Publ., Dordrecht, original work published 1986.

  • [34]

    M.V. Shashkov, L.P. Shilnikov. (1994), On the existence of a smooth invariant foliation in Lorenz-type mappings, Differ. Equ., 30, 536–544.

  • [35]

    Yu. B. Suris. (1989), Integrable mappings of the standard type, Funct. Anal. Appl., 23, 74–76.

    • Crossref
    • Export Citation
  • [36]

    A.P. Veselov. (1991), Integrable maps, Russian Math. Survey, 46, 1–51.

    • Crossref
    • Export Citation
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