# Onξ(s)-Quadratic Stochastic Operators on Two-Dimensional Simplex and Their Behavior

Authors: Farrukh Mukhamedov; Mansoor Saburov; Izzat Qaralleh
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

A quadratic stochastic operator (in short QSO) is usually used to present the time evolution of differing species in biology. Some quadratic stochastic operators have been studied by Lotka and Volterra. The general problem in the nonlinear operator theory is to study the behavior of operators. This problem was not fully finished even for quadratic stochastic operators which are the simplest nonlinear operators. To study this problem, several classes of QSO were investigated. We studyξ(s)-QSO defined on 2D simplex. We first classify ξ(s)-QSO into 20 nonconjugate classes. Further, we investigate the dynamics of three classes of such operators.


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## Body

## 1. Introduction

The history of quadratic stochastic operators can be traced back to Bernstein’s work [1]. The quadratic stochastic operator was considered an important source of analysis for the study of dynamical properties and modelings in various fields such as biology [1–7], physics [8, 9], economics, and mathematics [3, 6, 10, 11].One of such systems which relates to the population genetics is given by a quadratic stochastic operator [1]. A quadratic stochastic operator (in short QSO) is usually used to present the time evolution of species in biology, which arises as follows. Consider a population consisting of m species (or traits) 1,2,…,m. We denote a set of all species (traits) by I={1,2,…,m}. Let x(0)=(x1(0),…,xm(0)) be a probability distribution of species at an initial state and let  Pij,k be a probability that individuals in the ith and jth species (traits) interbreed to produce an individual from kth species (trait). Then, a probability distribution x(1)=(x1(1),…,xm(1)) of the species (traits) in the first generation can be found as a total probability; that is,
(1)xk(1)=∑i,j=1m‍Pij,kxi(0)xj(0),k=1,m¯.
This means that the association x(0)→x(1) defines a mapping V called the evolution operator. The population evolves by starting from an arbitrary state x(0), then passing to the state x(1)=V(x(0)) (the first generation), then to the state x(2)=V(x(1))=V(V(x(0)))=V(2)(x(0)) (the second generation), and so on. Therefore, the evolution states of the population system are described by the following discrete dynamical system:
(2)x(0),x(1)=V(x(0)),x(2)=V(2)(x(0)),x(3)=V(3)(x(0))⋯.In other words, a QSO describes a distribution of the next generation if the distribution of the current generation was given. The fascinating applications of QSO to population genetics were given in [6].In [12], it was given a long self-contained exposition of the recent achievements and open problems in the theory of the QSO. The main problem in the nonlinear operator theory is to study the behavior of nonlinear operators. This problem was not fully finished even in the class of QSO (the QSO is the simplest nonlinear operator). The difficulty of the problem depends on the given cubic matrix (Pijk)i,j,k=1m. An asymptotic behavior of the QSO even on the small dimensional simplex is complicated [11, 13–16]. In order to solve this problem, many researchers always introduced a certain class of QSO and studied their behavior, for example, Volterra-QSO [11, 17–20], permutated Volterra-QSO [21, 22], Quasi-Volterra-QSO [23], ℓ-Volterra-QSO [24, 25], non-Volterra-QSO [13, 15], strictly non-Volterra-QSO [26], F-QSO [27], and non-Volterra operators generated by product measure [28–30]. However, all these classes together would not cover a set of all QSO. Therefore, there are many classes of QSO which were not studied yet. Recently, in the papers [31, 32], a new class of QSO was introduced. This class was called a ξ(s)-QSO. In this paper, we are going to continue the study of ξ(s)-QSO. This class of operators depends on a partition of the coupled index set (the coupled trait set) Pm={(i,j):i<j}⊂I×I. In case of two-dimensional simplex (m=3), the coupled index set (the coupled trait set) P3 has five possible partitions. The dynamics of ξ(s)-QSO corresponding to the point partition (the maximal partition) of P3 have been investigated in [31, 32]. In the present paper, we are going to describe and classify such operators generated by other three partitions. Further, we also investigate the dynamics of three classes of such operators.The paper is organized as follows. In Section2, we give some preliminary definitions. In Section 3, we discuss the classification of ξ(s)-QSO related to |ξ|=2. It turns out that some obtained operators are ℓ-Volterra-QSO (see [24, 25]) and permuted ℓ-Volterra-QSO. The dynamics of ℓ-Volterra-QSO are not fully studied yet. In [24, 25], some particular cases have been investigated, which do not cover our operators. Therefore, in further sections, we study dynamics of ℓ-Volterra-QSO and permuted ℓ-Volterra-QSO. In Section 4, we study the behavior of ℓ-Volterra-QSO V13 taken from class K1. In Section 5, we study the behavior of a permuted ℓ-Volterra-QSO V4 taken from class K4. Note that V4 is a permutation of V13. In Section 6, we study the behavior of a permuted Volterra-QSO V28 taken from class K19. In the last section, we just highlight the dynamics of Volterra-QSO V25 taken from class K17 which was already studied in [17–19].


## 2. Preliminaries

Recall that a quadratic stochastic operator (QSO) is a mapping of the simplex:(3)Sm-1={∑i=1m‍x=(x1,…,xm)∈ℝm:∑i=1m‍xi=1,xi≥0,i=1,m¯}
into itself, of the form:
(4)xk′=∑i,j=1m‍Pij,kxixj,k=1,m¯,
where V(x)=x′=(x1′,…,xm′) and Pij,k is a coefficient of heredity, which satisfies the following conditions:
(5)Pij,k≥0,Pij,k=Pji,k,∑k=1m‍Pij,k=1.
Thus, each quadratic stochastic operator V:Sm-1→Sm-1 can be uniquely defined by a cubic matrix 𝒫=(Pijk)i,j,k=1m with conditions (5).We denote sets of fixed points andk-periodic points of V:Sm-1→Sm-1 by Fix(V) and Perk(V), respectively. Due to Brouwer’s fixed point theorem, one always has that Fix(V)≠∅ for any QSO V. For a given point x(0)∈Sm-1, a trajectory {x(n)}n=0∞ of V:Sm-1→Sm-1 starting from x(0) is defined by x(n+1)=V(x(n)). By ωV(x(0)), we denote a set of omega limiting points of the trajectory {x(n)}n=0∞. Since {x(n)}n=0∞⊂Sm-1 and Sm-1 is compact, one has that ωV(x(0))≠∅. Obviously, if ωV(x(0)) consists of a single point, then the trajectory converges and a limiting point is a fixed point of V:Sm-1→Sm-1.Recall that a Volterra-QSO is defined by (4) and (5) and the additional assumption:
(6)Pij,k=0ifk∉{i,j}.The biological treatment of condition (6) is clear: the offspring repeats the genotype (trait) of one of its parents.One can see that a Volterra-QSO has the following form:(7)xk′=xk(1+∑i=1m‍akixi),k∈I,
where
(8)aki=2Pik,k-1fori≠k,aii=0,i∈I.
Moreover,
(9)aki=-aik,|aki|≤1.This kind of operators is intensively studied in [11, 17–20, 33]. Note that this operator is a discretization of the Lotka-Volterra model [5, 7] which models an interacting competing species in the population system. Such a model has received considerable attention in the fields of biology, ecology, and mathematics (see, e.g., [2, 3, 7, 8]).In [24], a notion of ℓ-Volterra-QSO, which generalizes a notion of Volterra-QSO, has been introduced. Let us recall it here.In order to introduce a new class of QSO, we need some auxiliary notations.We fixℓ∈I and assume that elements Pij,k of the matrix (Pij,k)i,j,k=1m satisfy
(10)Pij,k=0ifk∉{i,j}foranyk∈{1,…,ℓ},i,j∈I,Pi0j0,k>0forsome(i0,j0),i0≠k,j0≠k,k∈{ℓ+1,…,m}.For any fixedℓ∈I, the QSO defined by (4), (5), and (10) is called ℓ-Volterra-QSO.Remark 1.
Here, we stress the following points.(1)
Note that anℓ-Volterra-QSO is a Volterra-QSO if and only if ℓ=m.(2)
It is known [17] that there is not a periodic trajectory for Volterra-QSO. However, there are such trajectories for ℓ-Volterra-QSO [24].By following [25], take k∈{1,…,ℓ}; then, Pkk,i=0 for i≠k and
(11)1=∑i=1m‍Pkk,i=Pkk,k+∑i=ℓ+1m‍Pkk,i.
By using Pij,k=Pji,k and denoting aki=2Pik,k-1, k≠i,  akk=Pkk,k-1, one then gets
(12)V:{xk′=xk(1+∑i=1m‍akixi)ifk=1,ℓ¯,xk′=xk(1+∑i=1m‍akixi)+∑i,j=1i≠k,j≠km‍Pij,kxixjifk=ℓ+1,m¯.ℓ-Volterra-QSO.Note that(13)akk∈[-1,0],|aki|≤1,aki+aik=2(Pik,i+Pik,k)-2≤0,i,k∈I.We call that an operatorV is permutedℓ-Volterra-QSO if there is a permutation τ of the set I and an ℓ-Volterra-QSO V0 such that (V(x))τ(k)=(V0(x))k for any k∈I. In other words, V can be represented as follows:
(14)Vτ:{xτ(k)′=xk(1+∑i=1m‍akixi)ifk=1,ℓ¯,xτ(k)′=xk(1+∑i=1m‍akixi)+∑i,j=1i≠k,j≠km‍Pij,kxixjifk=ℓ+1,m¯.We remark that ifℓ=m, then a permuted ℓ-Volterra-QSO becomes a permuted Volterra-QSO. Some properties of such operators were studied in [19, 34]. The dynamics of certain class of permuted Volterra-QSO have been investigated in [32]. Note that, in [24, 25], a class of ℓ-Volterra-QSO has been studied. An asymptotic behavior of permuted ℓ-Volterra-QSO has not been investigated yet. Some particular cases have been considered in [31, 32].In this paper, we are going to introduce a new class of QSO which containℓ-Volterra-QSO and permuted ℓ-Volterra-QSO as a particular case.Note that each elementx∈Sm-1 is a probability distribution of the set I={1,…,m}. Let x=(x1,…,xm) and y=(y1,…,ym) be vectors taken from Sm-1. We say that xis equivalent toy if xk=0⇔yk=0. We denote this relation by x~y.Letsupp(x)={i:xi≠0} be a support of x∈Sm-1. We say that xis singular toy and denote by x⊥y if supp(x)∩supp(y)=∅. Note that if x,y∈Sm-1, then x⊥y if and only if (x,y)=0; here, (·,·) stands for a standard inner product in ℝm.We denote sets of coupled indexes by(15)Pm={(i,j):i<j}⊂I×I,Δm={(i,i):i∈I}⊂I×I.
For a given pair (i,j)∈Pm∪Δm, we set a vector ℙij=(Pij,1,…,Pij,m). It is clear due to the condition (5) that ℙij∈Sm-1.Letξ1={Ai}i=1N and ξ2={Bi}i=1M be some fixed partitions of Pm and Δm, respectively; that is, Ai∩Aj=∅, Bi∩Bj=∅, and ⋃i=1NAi=Pm, ⋃i=1MBi=Δm, where N,M≤m.Definition 2.
A quadratic stochastic operatorV:Sm-1→Sm-1 given by (4) and (5) is called a ξ(as)-QSO with respect to the partitions ξ1,ξ2 (where the letter “as” stands for absolutely continuous-singular) if the following conditions are satisfied:  (i)
for eachk∈{1,…,N} and any (i,j), (u,v)∈Ak, one has that ℙij~ℙuv;(ii)
for anyk≠ℓ, k,ℓ∈{1,…,N} and any (i,j)∈Ak and (u,v)∈Aℓ, one has that ℙij⊥ℙuv;(iii)
for eachd∈{1,…,M} and any (i,i), (j,j)∈Bd, one has that ℙii~ℙjj;(iv)
for anys≠h, s,h∈{1,…,M} and any (u,u)∈Bs, and (v,v)∈Bh one has that ℙuu⊥ℙvv.Remark 3.
Ifξ2 is the point partition, that is, ξ2={{(1,1)},…{(m,m)}}, then we call the corresponding QSO by ξ(s)-QSO (where the letter “s” stands for singularity) since in this case every two different vectors ℙii and ℙjj are singular. If ξ2 is the trivial, that is, ξ2={Δm}, then we call the corresponding QSO by ξ(a)-QSO (where the letter “a” stands for absolute continuous) since in this case every two vectors ℙii and ℙjj are equivalent. We note that some classes of ξ(a)-QSO have been studied in [35]. In the present paper, we restrict ourselves to the ξ(s)-case. Note that, in general, the class of ξ(as)-QSO will be studied elsewhere in the future.Remark 4.
For theξ(s)-QSO, that is, in the case ξ2={{(1,1)},…{(m,m)}}, the condition (iii) of Definition 2 is trivial and the condition (iv) means that there is a permutation π of the set I={1,…,m} such that ℙii=eπ(i) for any i=1,m¯ where ek=(0,…,0,1︸k,0,…,0),   k=1,m¯, are vertices of the simplex Sm-1.A Biological Interpretation of a
ξ
(
s
)
-QSO. We treat I={1,…,m} as a set of all possible traits of the population system. A coefficient Pij,k is a probability that parents in the ith and jth traits interbreed to produce a child from the kth trait. The condition Pij,k=Pji,k means that the gender of parents do not influence having a child from the kth trait. In this sense, Pm∪Δm is a set of all possible coupled traits of parents. A vector ℙij=(Pij,1,…,Pij,m) is a possible distribution of children in a family while parents are carrying traits from the ith and jth types. A biological meaning of a ξ(s)-QSO is as follows: a set Pm of all differently coupled traits of parents is splitted into N groups A1,…,AN (here N is less than the number m of traits) such that the chance (probability) of having a child from any trait in two different families whose parents’ coupled traits belong to the same group Ak is simultaneously either positive or zero (the condition (i) of Definition 2); meanwhile, two families whose parents’ coupled traits belong to two different groups Ak and Al cannot have a child from the same trait, simultaneously (condition (ii) of Definition 2). Moreover, the parents who are sharing the same type of traits can have a child from only one type of traits (condition (iv) of Definition 2 and Remark 4).


## 3. Classification ofξ(s)-QSO on 2D Simplex

In this section, we are going to studyξ(s)-QSO in two-dimensional simplex; that is, m=3. In this case, we have the following possible partitions of P3:
(16)ξ1={{(1,2)},{(1,3)},{(2,3)}},|ξ1|=3,ξ2={{(2,3)},{(1,2),(1,3)}},|ξ2|=2,ξ3={{(1,3)},{(1,2),(2,3)}},|ξ3|=2,ξ4={{(1,2)},{(1,3),(2,3)}},|ξ4|=2,ξ5={(1,2),(1,3),(2,3)},|ξ5|=1.We note that, in [31, 32],  ξ(s)-QSO related to the partition ξ1 which is the maximal partition of P3 has been investigated. In this paper, we are aiming to study ξ(s)-QSO related to the partitions ξ2, ξ3, and ξ4. We shall show that these three classes of ξ(s)-QSO are conjugate to each other. Therefore, it is enough to study ξ(s)-QSO related to the partition ξ2. A class of ξ(s)-QSO related to the partition ξ5 will be studied in elsewhere in the future.Let us recall that two operatorsV1 and V2 are called (topologically or linearly) conjugate if there is a permutation matrix P such that P-1V1P=V2. Let π be a permutation of the set I={1,…,m}. For any vector x, we define π(x)=(xπ(1),…,xπ(m)). It is easy to check that if π is a permutation of the set I corresponding to the given permutation matrix P, then one has that Px=π(x). Therefore, two operators V1 and V2 are conjugate if and only if π-1V1π=V2 for some permutation π. Throughout this paper, we shall consider “conjugate operators” in this sense. We say that two classes K1 and K2 of operators are conjugate if every operator taken from K1 is conjugate to some operator taken from K2 and vice versa.Proposition 5.
A class of allξ(s)-QSO corresponding to partition ξ3 (or ξ4) is conjugate to a class of all ξ(s)-QSO corresponding to partition ξ2.Proof.
We show that two classes of allξ(s)-QSO corresponding to partitions ξ2 and ξ3 are conjugate to each other. Analogously, one can show that two classes of all ξ(s)-QSO corresponding to partitions ξ2 and ξ4 are conjugate to each other as well.
Assume that an operatorV:S2→S2 given by
(17)V:xk′=∑i,j=13‍Pij,kxixj,k=1,2,3,
is a ξ(s)-QSO corresponding to partition ξ3={{(1,3)},{(1,2),(2,3)}}. This means that the coefficients (Pij,k)i,j,k=13 of V satisfy the following three conditions: (i) ℙ12~ℙ23, (ii) ℙ13⊥ℙ12,ℙ13⊥ℙ23, and (iii) ℙ11⊥ℙ22⊥ℙ33 where ℙij=(Pij,1,Pij,2,Pij,3).
We consider the following operator:Vπ=π-1Vπ, where π=(123213). It is clear that Vπ is conjugate to V, where
(18)Vπ:xk′=∑i,j=13‍Pij,kπxixj,k=1,2,3
such that Pij,kπ=Pπ(i)π(j),π(k) for any i,j,k=1,2,3; equivalently, ℙijπ=πℙπ(i)π(j) (in a vector form) for any i,j=1,2,3. Now, we are going to show that Vπ is a ξ(s)-QSO corresponding to ξ2={{(2,3)},{(1,2),(1,3)}}. In order to show it we have to check three conditions.(i)
ℙ12π~ℙ13π. Indeed, since ℙ12π=πℙ12,ℙ13π=πℙ23,  ℙ12~ℙ23, one has ℙ12π~ℙ13π.(ii)
ℙ12π⊥ℙ23π,ℙ13π⊥ℙ23π. Indeed, since ℙ12π=πℙ12, ℙ23π=πℙ13, and ℙ12⊥ℙ13, we obtain that ℙ12π⊥ℙ23π. In the same manner, we can get that ℙ13π⊥ℙ23π.(iii)
ℙ11π⊥ℙ22π⊥ℙ33π. Indeed, since ℙ11π=πℙ22,   ℙ22π=πℙ11,   ℙ33π=πℙ33, and ℙ11⊥ℙ22⊥ℙ33, we have that ℙ11π⊥ℙ22π⊥ℙ33π.
This shows that anyξ(s)-QSO taken from the class corresponding to partition ξ3 is conjugate to some ξ(s)-QSO taken from the class corresponding to partition ξ2. Analogously, we can show that any ξ(s)-QSO V taken from the class corresponding to partition ξ2 is conjugate to a ξ(s)-QSO Vπ=π-1Vπ which belongs to the class corresponding to partition ξ3, where π is the same permutation as given above. This completes the proof.Therefore, it is enough to study a class of allξ(s)-QSO corresponding to the partition ξ2. Now, we shall consider some subclass of a class of all ξ(s)-QSO corresponding to partition ξ2 by choosing coefficients (Pij,k)i,j,k=13 in special forms where a∈[0,1] (see Table 1).Table 1
(a)
Case
ℙ
12
ℙ
13
ℙ
23
I
1
(
a
,
1
-
a
,
0
)
(
a
,
1
-
a
,
0
)
(
0,0
,
1
)
I
2
(
0
,
a
,
1
-
a
)
(
0
,
a
,
1
-
a
)
(
1,0
,
0
)
I
3
(
a
,
0,1
-
a
)
(
a
,
0,1
-
a
)
(
0,1
,
0
)
I
4
(
0,0
,
1
)
(
0,0
,
1
)
(
a
,
1
-
a
,
0
)
I
5
(
1,0
,
0
)
(
1,0
,
0
)
(
0
,
a
,
1
-
a
)
I
6
(
0,1
,
0
)
(
0,1
,
0
)
(
a
,
0,1
-
a
)
(b)
Case
ℙ
11
ℙ
22
ℙ
33
I
I
1
(
1,0
,
0
)
(
0,1
,
0
)
(
0,0
,
1
)
I
I
2
(
0,1
,
0
)
(
1,0
,
0
)
(
0,0
,
1
)
I
I
3
(
0,0
,
1
)
(
0,1
,
0
)
(
1,0
,
0
)
I
I
4
(
1,0
,
0
)
(
0,0
,
1
)
(
0,1
,
0
)
I
I
5
(
0,0
,
1
)
(
1,0
,
0
)
(
0,1
,
0
)
I
I
6
(
0,1
,
0
)
(
0,0
,
1
)
(
1,0
,
0
)The choices of the cases(Ii,IIj), where i,j=1,6¯, will give 36 operators from the class of ξ(s)-QSO corresponding to partition ξ2. Finally, we obtain 36 parametric operators which are defined as follows:
(19)V1:{x1′=x12+2ax1(1-x1),x2′=x22+2(1-a)x1(1-x1),x3′=x32+2x2x3,V2:{x1′=x22+2ax1(1-x1),x2′=x12+2(1-a)x1(1-x1),x3′=x32+2x2x3,V3:{x1′=x32+2ax1(1-x1),x2′=x22+2(1-a)x1(1-x1),x3′=x12+2x2x3,V4:{x1′=x12+2ax1(1-x1),x2′=x32+2(1-a)x1(1-x1),x3′=x22+2x2x3,V5:{x1′=x22+2ax1(1-x1),x2′=x32+2(1-a)x1(1-x1),x3′=x12+2x2x3,V6:{x1′=x32+2ax1(1-x1),x2′=x12+2(1-a)x1(1-x1),x3′=x22+2x2x3,V7:{x1′=x12+2x2x3,x2′=x22+2ax1(1-x1),x3′=x32+2(1-a)x1(1-x1),V8:{x1′=x22+2x2x3,x2′=x12+2ax1(1-x1),x3′=x32+2(1-a)x1(1-x1),V9:{x1′=x32+2x2x3,x2′=x22+2ax1(1-x1),x3′=x12+2(1-a)x1(1-x1),V10:{x1′=x12+2x2x3,x2′=x32+2ax1(1-x1),x3′=x22+2(1-a)x1(1-x1),V11:{x1′=x22+2x2x3,x2′=x32+2ax1(1-x1),x3′=x12+2(1-a)x1(1-x1),V12:{x1′=x32+2x2x3,x2′=x12+2ax1(1-x1),x3′=x22+2(1-a)x1(1-x1),V13:{x1′=x12+2ax1(1-x1),x2′=x22+2x2x3,x3′=x32+2(1-a)x1(1-x1),V14:{x1′=x22+2ax1(1-x1),x2′=x12+2x2x3,x3′=x32+2(1-a)x1(1-x1),V15:{x1′=x32+2ax1(1-x1),x2′=x22+2x2x3,x3′=x12+2(1-a)x1(1-x1),V16:{x1′=x12+2ax1(1-x1),x2′=x32+2x2x3,x3′=x22+2(1-a)x1(1-x1),V17:{x1′=x22+2ax1(1-x1),x2′=x32+2x2x3,x3′=x12+2(1-a)x(1-x),V18:{x1′=x32+2ax1(1-x1),x2′=x12+2x2x3,x3′=x22+2(1-a)x1(1-x1),V19:{x1′=x12+2ax2x3,x2′=x22+2(1-a)x2x3,x3′=x32+2x1(1-x1),V20:{x1′=x22+2ax2x3,x2′=x12+2(1-a)x2x3,x3′=x32+2x1(1-x1),V21:{x1′=x32+2ax2x3,x2′=x22+2(1-a)x2x3,x3′=x12+2x1(1-x1),V22:{x1′=x12+2ax2x3,x2′=x32+2(1-a)x2x3,x3′=x22+2x1(1-x1),V23:{x1′=x22+2ax2x3,x2′=x32+2(1-a)x2x3,x3′=x12+2x1(1-x1),V24:{x1′=x32+2ax2x3,x2′=x12+2(1-a)x2x3,x3′=x22+2x1(1-x1),V25:{x1′=x12+2x1(1-x1),x2′=x22+2ax2x3,x3′=x32+2(1-a)x2x3,V26:{x1′=x22+2x1(1-x1),x2′=x12+2ax2x3,x3′=x32+2(1-a)x2x3,V27:{x1′=x32+2x1(1-x1),x2′=x22+2ax2x3,x3′=x12+2(1-a)x2x3,V28:{x1′=x12+2x1(1-x1),x2′=x32+2ax2x3,x3′=x22+2(1-a)x2x3,V29:{x1′=x22+2x1(1-x1),x2′=x32+2ax2x3,x3′=x12+2(1-a)x2x3,V30:{x1′=x32+2x1(1-x1),x2′=x12+2ax2x3,x3′=x22+2(1-a)x2x3,V31:{x1′=x12+2ax2x3,x2′=x22+2x1(1-x1),x3′=x32+2(1-a)x2x3,V32:{x1′=x22+2ax2x3,x2′=x12+2x1(1-x1),x3′=x32+2(1-a)x2x3,V33:{x1′=x32+2ax2x3,x2′=x22+2x1(1-x1),x3′=x12+2(1-a)x2x3,V34:{x1′=x12+2ax2x3,x2′=x32+2x1(1-x1),x3′=x22+2(1-a)x2x3,V35:{x1′=x22+2ax2x3,x2′=x32+2x1(1-x1),x3′=x12+2(1-a)x2x3,V36:{x1′=x32+2ax2x3,x2′=x12+2x1(1-x1),x3′=x22+2(1-a)x2x3.Theorem 6.
All 36 operators from the class ofξ(s)-QSO corresponding to partition ξ2 defined as above are classified into 20 nonconjugate classes:
(20)K1={V1,V13},K2={V2,V15},K3={V3,V14},K4={V4,V16},K5={V5,V18},K6={V6,V17},K7={V7},K8={V8,V9},K9={V10},K10={V11,V12},K11={V19,V31},K12={V20,V33},K13={V21,V32},K14={V22,V34},K15={V23,V36},K16={V24,V35},K17={V25},K18={V26,V27},K19={V28},K20={V29,V30}.Proof.
It is easy to check that partitionξ2={{(2,3)},{(1,2),(1,3)}} is invariant under only one permutation π=(123132). The proof of the theorem can be easily insured with respect to this permutation. It is straightforward.The main problem is to investigate the dynamics of these classes of operators. In what follows, we are going to study three classesK1, K4, and K19. From the list, one can conclude that these three classes of operators are either ℓ-Volterra-QSO or permuted ℓ-Volterra-QSO. The class K17 was already studied in [17, 18]. The rest of the classes of operators would be studied in elsewhere in the future.


## 4. Dynamics ofξ(s)-QSO from Class K1

In this section we are going to study dynamics ofξ(s)-QSO from class K1.We need some auxiliary facts about properties of the functionfa:[0,1]→[0,1] given by
(21)fa(x)=x2+2ax(1-x),
where a∈[0,1]. If a=1/2, then the function becomes the identity mapping. Therefore, we shall consider only the case of a≠1/2.Proposition 7.
Letfa:[0,1]→[0,1] be a function given by (21) where a≠1/2. Then, the following statements hold true.(i)
One has thatFix(fa)={0,1}.(ii)
The functionfais increasing.(iii)
One has that(a-(1/2))(fa(x)-x)>0for anyx∈(0,1).(iv)
One has thatωfa(x0)={{0}if0≤a<1/2{1}if1/2<a≤1for anyx0∈(0,1).Proof.
Letfa:[0,1]→[0,1] be a function given by (21) where a≠1/2.(i)
In order to find fixed points of the functionfa, we should solve the following equation: x2+2ax(1-x)=x. It follows from the last equation that (1-2a)(x2-x)=0. Since a≠1/2, we get that x=0 or x=1. Therefore, Fix(fa)={0,1}.(ii)
Sincefa′(x)=2x(1-a)+2a(1-x)≥0, the function fa is increasing.(iii)
Sincefa(x)-x=(1-2a)(x2-x), we may get that  (a-(1/2))(fa(x)-x)>0.(iv)
Let0≤a<1/2 and x0∈(0,1). Due to (iii), we have that fa(x0)<x0. Since fa is increasing, we obtain that fa(n+1)(x0)<fa(n)(x0) for any n∈ℕ. This means that {fa(n)(x0)}n=1∞ is a bounded decreasing sequence. Consequently, it converges to some point x*, and x* should be a fixed point; that is, x*=0. This means that wfa(x0)={0}. Similarly, one can show that if 1/2<a≤1, then ωfa(x0)={1} for any x0∈(0,1). This completes the proof.Now, we are going to study dynamics of aξ(s)-QSO V13:S2→S2 taken from K1:
(22)V13:{x1′=x12+2x1a(1-x1)x2′=x22+2x2x3x3′=x32+2(1-a)x1(1-x1),
where 0≤a≤1. This operator is an ℓ-Volterra-QSO, and its behavior was not studied in [24, 25].Lete1, e2, and e3 be the vertices of the simplex S2 and let Γi={x∈S2:xi=0} be an edge of the simplex S2, where i=1,2,3. Let S1≤3={x∈S2:x1≤x3}, S1≥3={x∈S2:x1≥x3}, and l13={x∈S2:x1=x3}.Theorem 8.
LetV13:S2→S2 be a ξ(s)-QSO given by (22) and let x(0)=(x1(0),x2(0),x3(0))∉Fix(V13) be an initial point. Then the following statements hold true. (i)
One has that(23)Fix(V13)={{e1,e2,e3}ifa≠12,Γ2∪l13ifa=12.(ii)
If0≤a<1/2, then
(24)ωV13(x(0))={{e2}ifx2(0)≠0,{e3}ifx2(0)=0.(iii)
If1/2<a≤1, then
(25)ωV13(x(0))={{e1}ifx1(0)≠0,{e2}ifx1(0)=0.(iv)
Ifa=1/2, then
(26)ωV13(x(0))={{(x1(0),0,1-x1(0))}ifx1(0)>12,{(x1(0),1-2x1(0),x1(0))}ifx1(0)≤12.Proof.
LetV13:S2→S2 be a ξ(s)-QSO given by (22), let  x(0)=(x1(0),x2(0),x3(0))∉Fix(V13) be an initial point, and  let  {x(n)}n=0∞ be a trajectory of  V13 starting from the point x(0).
(
i
) In order to find fixed points of (22), we should solve the following system of equations:
(27)x1=x12+2x1a(1-x1),x2=x22+2x2x3,x3=x32+2(1-a)x1(1-x1).
We shall separately consider two cases: a=1/2 and a≠1/2.
Leta≠1/2. From the first equation of (27), we get that x1=0 and x1=1 (see Proposition 7(i)). It follows from the second equation of (27) that if x1=0, then x2=0, x3=1 or x2=1, x3=0 and if x1=1, then x2=x3=0. This means that Fix(V13)={e1,e2,e3}.
Leta=1/2. The first equation of (27) takes the form x1=x1. From the second equation of (27), we get that x2(x1-x3)=0. This yields that x2=0 or x1=x3. In both cases, the third equation of (27) holds true. Therefore, we have that Fix(V13)=Γ2∪l13.
(
ii
)  Let 0≤a<1/2. It is clear that x1(n)=fa(n)(x1(0)). Therefore, due to Proposition 7(iv), we have that limn→∞x1(n)=0. Hence, ωV13(x(0))⊂Γ1={x∈S2:x1=0}. Now, we shall separately consider two cases: x2(0)=0 and x2(0)≠0.
Letx2(0)=0. In this case, x2(n)=0 and limn→∞x3(n)=1. Hence, ωV13(x(0))={e3}.
Letx2(0)≠0. We need the following result.
Claim. One has that V13(S1≤3)⊂S1≤3. Moreover, for any x(0)∉Fix(V13), there exists n0 (depending on x(0)) such that {x(n)}n=n0∞⊂S1≤3.
Proof of Claim. If x1≤x3, then x1′=x12+2ax1(1-x1)≤x32+2(1-a)x1(1-x1)=x3′. This means that V13(S1≤3)⊂S1≤3.
Letx(0)∉S1≤3, and suppose that all elements of the trajectory belong to set S2∖S1≤3; that is, {x(n)}n=0∞⊂S2∖S1≤3. It follows from x1(n)→0 and x3(n)<x1(n) that x3(n)→0. This with ∑i=13xi(n)=1 implies that x2(n)→1. On the other hand, we have that x2(n+1)=x2(n)(1-(x1(n)-x3(n)))≤x2(n). It yields that {x2(n)}n=0∞ is decreasing; hence, it converges to some point x*<1. This is a contradiction. This completes the proof of Claim.
Due to Claim, there existsn0 such that x1(n)≤x3(n) for all n≥n0. Therefore, x2(n+1)=x2(n)(1-(x1(n)-x3(n)))≥x2(n), and {x2(n)}n=n0∞ is an increasing sequence which converges to x2*. This yields that x(n)=(x1(n),x2(n),x3(n)) converges to (0,x2*,1-x2*), where x2*>0. We know that (0,x2*,1-x2*) should be a fixed point. Consequently, (0,x2*,1-x2*)=(0,1,0) and ωV13(x(0))={e2}.
(
iii
) Let 1/2<a≤1. Due to Proposition 7(iv), we have that limn→∞x1(n)=1, whenever 0<x1(0)<1. Therefore, if x1(0)≠0, then ωV13(x(0))⊂{e1}. Since ωV13(x(0)) is not empty, we obtain that ωV13(x(0))={e1}. Let x1(0)=0, then x1(n)=0 for all n∈ℕ. Moreover, we have that x3(n+1)=(x3(n))2 and limn→∞x3(n)=0. This means that limn→∞x2(n)=1. Therefore, if x1(0)=0, then ωV13(x(0))={e2}.
(
iv
) Let a=1/2 and x1(0)>1/2. Then x1(n)=x1(0)>1/2 for any n∈ℕ. Since x3(n)=1-x1(n)-x2(n), one gets that x3(n)<1/2<x1(n). This implies that {x2(n)}n=0∞ is decreasing, and hence it converges to x2*. Consequently, (x1(n),x2(n),x3(n))→(x1(0),x2*,x3*). We know that (x1(0),x2*,x3*) should be a fixed point. Since x1(0)>1/2≥x3(*), we find that x2*=0 and x3*=1-x1(0). This means that ωV13(x(0))={(x1(0),0,1-x1(0))}.
Letx1(0)≤1/2. Then x1(n)=x1(0)≤1/2 for any n∈ℕ. Since x3′-x1′=x32-x12, we have that V(S1≤3)⊂S1≤3 and V(S1≥3)⊂S1≥3. If x(0)=(x1(0),x2(0),x3(0))∈S1≤3, then x(n)=(x1(n),x2(n),x3(n))∈S1≤3. This yields that {x2(n)}n=0∞ is decreasing, and hence it converges to x2*. Therefore, (x1(n),x2(n),x3(n)) converges to (x1(0),x2(*),x3(*)). Since (x1(0),x2*,x3*)∈S1≤3 is a fixed point, we have that x2(*)=1-2x1(0) and x3(*)=x1(0). In the similar manner, one may have that if x(0)∈S1≥3, then ωV13(x(0))={(x1(0),1-2x1(0),x1(0))}.
This completes the proof.


## 5. Dynamics ofξ(s)-QSO from Class K4

We are going to study dynamics of aξ(s)-QSO V4:S2→S2 taken from K4:
(28)V4:{x1′=x12+2ax1(1-x1),x2′=x32+2(1-a)x1(1-x1),x3′=x22+2x2x3,
where 0≤a≤1. One can immediately see that this operator is a permuted ℓ-Volterra-QSO. As we mentioned, the behavior of such kinds of operators is not studied yet. It is worth mentioning that V4 is a permutation of V13.Let(29)B(b)=3-2b-4b2-8b+52,b∈[0,1],C±(c)=1-2c±4c2-8c+12,c∈[0,2-32].It is clear that(30)0≤B(b)=2(1-b)1+2(1-b)+1+4(1-b)2≤1,0≤1-2c±(1-2c)2-4c2=C±(c)≤1-2c+4c2+4c+12=1
for any b∈[0,1] and c∈[0,(2-3)/2]⊂[0,1/2].Theorem 9.
LetV4:S2→S2 be a ξ(s)-QSO given by (28) and let  x(0)=(x1(0),x2(0),x3(0))∉Fix(V4)∪Per2(V4) be an initial point. Then, the following statements hold true.(i)
One has that(31)Fix(V4)={{e1,(0,3-52,-1+52)}ifa≠12,{b,B(b),1-B(b)}b∈[0,1]ifa=12.(ii)
One has that(32)Per2(V4)={{e2,e3}ifa≠12,{c,C±(c),1-C±(c)}c∈[0,(2-3)/2)ifa=12.(iii)
If1/2<a≤1, then
(33)ωV4(x(0))={{e1}ifx1(0)≠0,{e2,e3}ifx1(0)=0.(iv)
Ifa=1/2, then
(34)ωV4(x(0))={{(x1(0),C±(x1(0)),1-C±(x1(0)))}ifx1(0)∈[0,2-32),{(x1(0),B(x1(0)),1-B(x1(0)))}ifx1(0)∈[2-32,1].Proof.
Letx(0)=(x1(0),x2(0),x3(0))∉Fix(V4)∪Per2(V4) be an initial point and let  {x(n)}n=0∞ be a trajectory of V4 starting from the point x(0).
(
i
) In order to find fixed points of V4, we have to solve the following system:
(35)x1=x12+2ax1(1-x1),x2=x32+2(1-a)x1(1-x1),x3=x22+2x2x3.
Leta≠1/2. From the first equation of the system (35), one can find that x1=0 or x1=1 (see Proposition 7(i)). If x1=1, then x2=x3=0. If x1=0, then the second equation of the system (35) becomes as follows:
(36)x22-3x2+1=0.
So, the solutions of this quadratic equation are x2±=(3±5)/2. We can verify that the only solution x2=(3-5)/2 belongs to [0,1]. Therefore, one has x3=(-1+5)/2. Hence, Fix(V4)={e1,(0,(3-5)/2,(-1+5)/2)} whenever a≠1/2.
Leta=1/2. The system (35) then takes the following form
(37)x1=x1,x2=x32+x1(1-x1),x3=x22+2x2x3.
So, by letting x1=b (any b∈[0,1]), the second equation of the system (37) can be written as follows
(38)x22-(3-2b)x2+(1-b)=0.
The solutions of the last equation are x2±=(3-2b±4b2-8b+5)/2. One can check that the only solution x2=(3-2b-4b2-8b+5)/2=B(b) belongs to [0,1]. Therefore, one has that Fix(V4)={(b,B(b),1-B(b))}b∈[0,1] whenever a=1/2.
(ii) Leta≠1/2. Now, we are going to show that the operator V4 given by (28) does not have any order periodic points in the set S2∖Γ1, where Γ1={x∈S2:x1=0}. In fact, since the function fa(x)=x2+2ax(1-x) is increasing (due to Proposition 7(ii)), the first coordinate of V4 increases along the iteration of V4 in the set S2∖Γ1. This means that V4 does not have any order periodic points in set S2∖Γ1. Therefore, it is enough to find periodic points of V4 in Γ1. In this case, in order to find 2-periodic points, we have to solve the following system of equations:
(39)x1=0,x2=(1-(1-x2)2)2,x3=1-(1-x32)2.
The solutions of the second equation of the last system are 0,1,(3±5)/2. Hence, we have that Per(V4)={e2,e3} whenever a≠1/2.
Leta=1/2. In order to find 2-periodic points of V4, we should solve the following system of equations:
(40)x1=x1,x2=x22(1-x1+x3)2+x1(1-x1),x3=(x32+x1(1-x1))(1-x1+x2(1-x1+x3)).
By letting x1=c, where c∈[0,1], the second equation of system (40) reduces to the following equation:
(41)x22(2-2c-x2)2+c(1-c)=x2.
One can easily check that the solutions of the last equation which belong to [0,1] are only B(c) and C±(c) whenever c∈[0,(2-3)/2) and B(c) whenever c∈[(2-3)/2,1]. Consequently, we get that Per2(V4)={(c,C±(c),1-C±(c))}c∈[0,(2-3)/2).
(iii) Let1/2<a≤1 and x1(0)=0. In this case, the second coordinate of V4 has the form x2′=h(x2), where h(x2)=(1-x2)2. It is clear that function h is decreasing on [0,1]. This yields that function h(2) is increasing on [0,1], As we already discussed in (i) and (ii) that Fix(h)∩[0,1]={(3-5)/2} and Fix(h(2))∩[0,1]={0,(3-5)/2,1}. This means that the sets [0,(3-5)/2] and [(3-5)/2,1] are invariant function h(2). We immediately find (see the above discussion (ii)) that h(2)(x2)>x2 whenever x2>(3-5)/2 and h(2)(x2)<x2 whenever x2<(3-5)/2. Consequently, one has that if x2(0)∈[0,(3-5)/2), then ωh(2)(x2(0))={0} and if x2(0)∈((3-5)/2,1], then ωh(2)(x2(0))={1}. On the other hand, we have that
(42)V4(n)(x(0))={(0,h(2k)(x2(0)),1-h(2k)(x2(0)))ifn=2k,(0,h(2k)(h(x2(0))),1-h(2k)(h(x2(0))))ifn=2k+1.
Therefore, we obtain that ωV4(x(0))={e2,e3} if x1(0)=0.
Let1/2<a≤1 and 0<x1(0)<1. In this case, it is clear that x1(n)=fa(n)(x1(0)). Therefore, due to Proposition 7(iv), we have that limn→∞x1(n)=1. This means that ωV4(x(0))⊂{e1}. Since ωV4(x(0))≠∅, one has that ωV4(x(0))={e1}.
(
iv
) Let a=1/2. Then the operator V4 takes the following form:
(43)x1′=x1,x2′=x32+x1(1-x1),x3′=x2(1-x1+x3).
It is clear that Lc={x∈S2:x1=c} is invariant under V4 where c∈[0,1]. Therefore, we shall study the dynamics of V4 over Lc.
Letx1(0)=c∈[0,(2-3)/2) be a fixed number. Let us consider function hc:[0,1-c]→[0,1-c],
(44)hc(x2)=(1-c-x2)2+c(1-c).
One can show that the function hc is decreasing on [0,1-c]. This yields that the function hc(2) is increasing. It follows from the discussion presented above (see (ii)) that Fix(hc)∩[0,1-c]={B(c)} and Fix(hc(2))∩[0,1-c]={B(c),C±(c)} where C-(c)<B(c)<C+(c). Moreover, one has that
(45)hc[0,C-(c)]⊂[C+(c),1-c],hc[C-(c),B(c)]⊂[B(c),C+(c)],hc[B(c),C+(c)]⊂[C-(c),B(c)],hc[C+(c),1-c]⊂[0,C-(c)],
Therefore, the sets [0,C-(c)], [C-(c),B(c)], [B(c),C+(c)], and [C+(c),1-c] are invariant under function hc(2). Simple calculations show that
(46)hc(2)(x2)>x2,∀x2∈[0,C-(c))∪(B(c),C+(c)),hc(2)(x2)<x2,∀x2∈(C-(c),B(c))∪(C+(c),1-c].
Consequently, we get that
(47)ωhc(2)(x2(0))={C-(c)},∀x2(0)∈[0,B(c)),ωhc(2)(x2(0))={C+(c)},∀x2(0)∈(B(c),1-c].
On the other hand, we have that(48)V4(n)(x(0))={(c,hc(2k)(x2(0)),1-hc(2k)(x2(0)))ifn=2k,(c,hc(2k)(hc(x2(0))),1-hc(2k)(hc(x2(0))))ifn=2k+1.
Therefore, we obtain that
(49)ωV4(x(0))={(c,C±(c),1-C±(c))}ifx1(0)=c∈[0,2-32).
In the same manner, one can show that ωV4(x(0))={(c,B(c),1-B(c))} whenever x1(0)=c∈[(2-3)/2,1].
This completes the proof.


## 6. Dynamics ofξ(s)-QSO from Class K19

We are going to study dynamics of aξ(s)-QSO V28:S2→S2 taken from K19:
(50)V28:{x1′=x12+2x1(1-x1),x2′=x32+2ax2x3,x3′=x22+2(1-a)x2x3,
where 0≤a≤1. One can see that this operator is a permuted Volterra-QSO. The behavior of this operator was not studied in [21, 22, 34]. It is worth mentioning that V28 is a permutation of V25.LetA=(3-2a-4+(2a-1)2)/2(1-2a) for any a≠1/2. Then 0≤A≤1. In fact, one has that
(51)0≤A=22+(1-2a)+4+(2a-1)2≤1.Theorem 10.
LetV28:S2→S2 be a ξ(s)-QSO given by (50) and let  x(0)=(x1(0),x2(0),x3(0))∉Fix(V28)∪Per2(V28) be an initial point. Then the following statements hold true.(i)
One has that(52)Fix(V28)={{e1,(0,A,1-A)}ifa≠12,{e1,(0,12,12)}ifa=12.(ii)
One has that(53)Per2(V28)={{e2,e3}ifa≠12,Γ1∖{(0,12,12)}ifa=12.(iii)
Ifa≠1/2, then
(54)ωV28(x(0))={{e2,e3}ifx1(0)=0,{e1}ifx1(0)≠0.(iv)
Ifa=1/2, then ωV28(x(0))={e1}.Proof.
Letx(0)=(x1(0),x2(0),x3(0))∉Fix(V28)∪Per2(V28) be an initial point and let {x(n)}n=0∞ be a trajectory of V28 starting from point x(0).
(
i
) In order to find fixed points of V28, we need to solve the following system of equations:
(55)x1=x12+2x1(1-x1),x2=x32+2ax2x3,x3=x22+2(1-a)x2x3.
From the first equation of (55), one can find that x1=0 or x1=1. If x1=1, then x2=x3=0. If x1=0, then x2+x3=1. So, the second equation of (55) becomes as follows:
(56)(1-x2)2+2ax2(1-x2)=x2.
Leta≠1/2. Then, one can find that solutions of (56) are x2±=(3-2a±4+(2a-1)2)/2(1-2a). We can verify that the only solution which lies in the interval [0,1] is x2=(3-2a-4+(2a-1)2)/2(1-2a)=A. Therefore, we have x3=1-A=(-1-2a+4+(2a-1)2)/2(1-2a). Hence, Fix(V28)={e1,(0,A,1-A)} whenever a≠1/2.
Leta=1/2. Then, (56) has a solution x2=1/2. This yields that x3=1/2. Therefore, one has that Fix(V28)={e1,(0,1/2,1/2)} whenever a=1/2.
(
ii
) It is clear that V28 does not have any order periodic points in S2∖Γ1 (see Proposition 7(ii)), where Γ1={x∈S2:x1=0}. So, any order periodic points of V28 lie on Γ1 (if any). In order to find 2-periodic points of V28, we have to solve the equation V282(x)=x with the condition x1=0. Then, by taking into account x2+x3=1, we may get the following equation:
(57)[x22+2(1-a)x2(1-x2)]2+2a[(1-x2)2+2ax2(1-x2)]×[x22+2(1-a)x2(1-x2)]=x2.
Leta≠1/2. Then, the last equation has the solutions {0,1,±A}. So, 2-periodic points of V28 are only e2=(0,1,0) and e3=(0,0,1).
Leta=1/2. Then, the equation given above becomes an identity x2=x2. This means that all points of the edge Γ1 except (0,1/2,1/2) are 2-periodic points.
(
iii
) Let a≠1/2. It is clear that the edge Γ1 is invariant under V28. We want to study the behavior of V28 over this line. In this case, V28∣Γ1 takes the following form:
(58)V28∣Γ1:{x1′=0,x2′=x32+2ax2x3,x3′=x22+2(1-a)x2x3.
Let us consider the functionga(x2)=(1-x2)2+2ax2(1-x2), where a≠1/2. One can easily check that ga is decreasing on [0,1]. This yields that ga(2) is increasing on [0,1], as we already discussed that Fix(ga)∩[0,1]={A} and Fix(ga(2))∩[0,1]={0,A,1}. This means that sets [0,A] and [A,1] are invariant under function ga(2). We immediately find (see the above discussion (ii)) that ga(2)(x2)>x2 whenever x2>A and ga(2)(x2)<x2 whenever x2<A. Consequently, one has that if x2(0)∈[0,A), then ωga(2)(x2(0))={0} and if x2(0)∈(A,1], then ωga(2)(x2(0))={1}. On the other hand, we have that
(59)V28(n)(x(0))={(0,ga(2k)(x2(0)),1-ga(2k)(x2(0)))ifn=2k,(0,ga(2k)(ga(x2(0))),1-ga(2k)(ga(x2(0))))ifn=2k+1.
Therefore, we obtain that ωV28(x(0))={e2,e3} if x1(0)=0.
Letx1(0)≠0. It is clear that x1(n)=f1(n)(x1(0)). Therefore, due to Proposition 7(iv), we have that limn→∞x1(n)=1. This means that ωV28(x(0))⊂{e1}. Since ωV28(x(0)) is not empty, we obtain that ωV28(x(0))={e1}.
(
iv
) Let a=1/2. Since x(0)∉Fix(V28)⋃‍Per2(V28), we have that x1(0)>0. Then, due to Proposition 7(iv), we again have that x1(n)=f1(n)(x1(0)) and limn→∞x1(n)=1. Since ωV28(x(0)) is not empty, we obtain that ωV28(x(0))={e1}.
This completes the proof.


## 7. Dynamics ofξ(s)-QSO from the Class K17

We are going to highlight the dynamics of aξ(s)-QSO V25:S2→S2 taken from K17:
(60)V25:{x1′=x12+2x1(1-x1),x2′=x22+2ax2x3,x3′=x32+2(1-a)x2x3,
where 0≤a≤1. One can immediately see that operator (60) is a Volterra-QSO. The dynamics of such kinds of operators have been studied in [17–19]. By means of the results of the mentioned papers, one can formulate the following.Theorem 11.
LetV25:S2→S2 be a ξ(s)-QSO given by (60) and let x(0)=(x1(0),x2(0),x3(0))∉Fix(V25) be an initial point. Then, the following statements hold true.(i)
One has that(61)Fix(V25)={{e1,e2,e3}ifa≠12,Γ1ifa=12.(ii)
If0≤a<1/2, then
(62)ωV25(x(0))={{e3}ifx1(0)=0,{e1}ifx1(0)≠0.(iii)
If1/2<a≤1, then
(63)ωV25(x(0))={{e2}ifx1(0)=0,{e1}ifx1(0)≠0.(iv)
Ifa=1/2, then ωV25(x(0))={e1}.



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*Source: 942038-2013-11-10.xml*