On 𝜉^(𝑠)-Quadratic Stochastic Operators on Two-Dimensional Simplex and Their Behavior

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 study 𝜉^(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.

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⁾,…,𝑥_𝑚⁽⁰⁾) be a probability distribution of species at an initial state and let 𝑃_{𝑖𝑗,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⁾,…,𝑥_𝑚⁽¹⁾) of the species (traits) in the first generation can be found as a total probability; that is, 𝑥_𝑘⁽¹⁾=^𝑚_𝑖,𝑗=1𝑃_{𝑖𝑗,𝑘}𝑥_𝑖⁽⁰⁾𝑥_𝑗⁽⁰⁾,𝑘=1,𝑚. (1) 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: 𝑥⁽⁰⁾,𝑥⁽¹⁾=𝑉𝑥⁽⁰⁾,𝑥⁽²⁾=𝑉⁽²⁾𝑥⁽⁰⁾,𝑥⁽³⁾=𝑉⁽³⁾𝑥⁽⁰⁾⋯. (2)

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 (𝑃_𝑖𝑗𝑘)^𝑚_{𝑖,𝑗,𝑘=1}. 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) 𝐏_𝑚={(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 Section 2, 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: 𝑆^𝑚−1=𝑥=𝑥₁,…,𝑥_𝑚∈ℝ^𝑚∶^𝑚_𝑖=1𝑥_𝑖=1,𝑥_𝑖≥0,𝑖=1,𝑚 (3) into itself, of the form: 𝑥^′_𝑘=^𝑚_𝑖,𝑗=1𝑃_{𝑖𝑗,𝑘}𝑥_𝑖𝑥_𝑗,𝑘=1,𝑚, (4) where 𝑉(𝑥)=𝑥^′=(𝑥^′₁,…,𝑥^′_𝑚) and 𝑃_{𝑖𝑗,𝑘} is a coefficient of heredity, which satisfies the following conditions: 𝑃_{𝑖𝑗,𝑘}≥0,𝑃_{𝑖𝑗,𝑘}=𝑃_{𝑗𝑖,𝑘},_𝑚𝑘=1𝑃_{𝑖𝑗,𝑘}=1. (5) Thus, each quadratic stochastic operator 𝑉∶𝑆^𝑚−1→𝑆^𝑚−1 can be uniquely defined by a cubic matrix 𝒫=(𝑃_𝑖𝑗𝑘)^𝑚_{𝑖,𝑗,𝑘=1} with conditions (5).

We denote sets of fixed points and 𝑘-periodic points of 𝑉∶𝑆^𝑚−1→𝑆^𝑚−1 by Fix(𝑉) and Per_𝑘(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⁾∈𝑆^𝑚−1, a trajectory {𝑥^(𝑛)}^∞_𝑛=0 of 𝑉∶𝑆^𝑚−1→𝑆^𝑚−1 starting from 𝑥⁽⁰⁾ is defined by 𝑥^(𝑛+1)=𝑉(𝑥^(𝑛)). By 𝜔_𝑉(𝑥⁽⁰⁾), we denote a set of omega limiting points of the trajectory {𝑥^(𝑛)}^∞_𝑛=0. Since {𝑥^(𝑛)}^∞_𝑛=0⊂𝑆^𝑚−1 and 𝑆^𝑚−1 is compact, one has that 𝜔_𝑉(𝑥⁽⁰⁾)≠∅. Obviously, if 𝜔_𝑉(𝑥⁽⁰⁾) consists of a single point, then the trajectory converges and a limiting point is a fixed point of 𝑉∶𝑆^𝑚−1→𝑆^𝑚−1.

Recall that a Volterra-QSO is defined by (4) and (5) and the additional assumption: 𝑃_{𝑖𝑗,𝑘}=0if𝑘∉𝑖,𝑗. (6)

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: 𝑥^′_𝑘=𝑥_𝑘1+^𝑚_𝑖=1𝑎_𝑘𝑖𝑥_𝑖,𝑘∈𝐼, (7) where 𝑎_𝑘𝑖=2𝑃_{𝑖𝑘,𝑘}−1for𝑖≠𝑘,𝑎_𝑖𝑖=0,𝑖∈𝐼. (8) Moreover, 𝑎_𝑘𝑖=−𝑎_𝑖𝑘,𝑎_𝑘𝑖≤1. (9)

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 fix ℓ∈𝐼 and assume that elements 𝑃_{𝑖𝑗,𝑘} of the matrix (𝑃_{𝑖𝑗,𝑘})^𝑚_{𝑖,𝑗,𝑘=1} satisfy 𝑃_{𝑖𝑗,𝑘}=0if𝑘∉𝑖,𝑗forany{}𝑃𝑘∈1,…,ℓ,𝑖,𝑗∈𝐼,_{𝑖0𝑗0,𝑘}>0forsome𝑖₀,𝑗₀,𝑖₀≠𝑘,𝑗₀≠𝑘,𝑘∈{ℓ+1,…,𝑚}. (10)

For any fixed ℓ∈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 an ℓ-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, 𝑃_{𝑘𝑘,𝑖}=0 for 𝑖≠𝑘 and 1=^𝑚_𝑖=1𝑃_{𝑘𝑘,𝑖}=𝑃_{𝑘𝑘,𝑘}+^𝑚_𝑖=ℓ+1𝑃_{𝑘𝑘,𝑖}. (11) By using 𝑃_{𝑖𝑗,𝑘}=𝑃_{𝑗𝑖,𝑘} and denoting 𝑎_𝑘𝑖=2𝑃_{𝑖𝑘,𝑘}−1, 𝑘≠𝑖,  𝑎_𝑘𝑘=𝑃_{𝑘𝑘,𝑘}−1, one then gets 𝑥𝑉∶^′𝑘=𝑥_𝑘1+∑^𝑚_𝑖=1𝑎_𝑘𝑖𝑥_𝑖if𝑘=𝑥1,ℓ,_′𝑘=𝑥_𝑘1+∑^𝑚_𝑖=1𝑎_𝑘𝑖𝑥_𝑖+∑^𝑚_{𝑖,𝑗=1𝑖≠𝑘,𝑗≠𝑘}𝑃_{𝑖𝑗,𝑘}𝑥_𝑖𝑥_𝑗if𝑘=ℓ+1,𝑚. (12)

ℓ-Volterra-QSO.

Note that 𝑎_𝑘𝑘∈[−1,0],𝑎_𝑘𝑖𝑎≤1,_𝑘𝑖+𝑎_𝑖𝑘=2𝑃_{𝑖𝑘,𝑖}+𝑃_{𝑖𝑘,𝑘}−2≤0,𝑖,𝑘∈𝐼. (13)

We call that an operator 𝑉 is permuted ℓ-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: 𝑉_𝜏∶𝑥^′_𝜏(𝑘)=𝑥_𝑘1+∑^𝑚_𝑖=1𝑎_𝑘𝑖𝑥_𝑖if𝑘=𝑥1,ℓ,_{′𝜏(𝑘)}=𝑥_𝑘1+∑^𝑚_𝑖=1𝑎_𝑘𝑖𝑥_𝑖+∑^𝑚_{𝑖,𝑗=1𝑖≠𝑘,𝑗≠𝑘}𝑃_{𝑖𝑗,𝑘}𝑥_𝑖𝑥_𝑗if𝑘=ℓ+1,𝑚. (14)

We remark that if ℓ=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 contain ℓ-Volterra-QSO and permuted ℓ-Volterra-QSO as a particular case.

Note that each element 𝑥∈𝑆^𝑚−1 is a probability distribution of the set 𝐼={1,…,𝑚}. Let 𝑥=(𝑥₁,…,𝑥_𝑚) and 𝑦=(𝑦₁,…,𝑦_𝑚) be vectors taken from 𝑆^𝑚−1. We say that 𝑥 is equivalent to 𝑦 if 𝑥_𝑘=0⇔𝑦_𝑘=0. We denote this relation by 𝑥∼𝑦.

Let supp(𝑥)={𝑖∶𝑥_𝑖≠0} be a support of 𝑥∈𝑆^𝑚−1. We say that 𝑥 is singular to 𝑦 and denote by 𝑥⟂𝑦 if supp(𝑥)∩supp(𝑦)=∅. Note that if 𝑥,𝑦∈𝑆^𝑚−1, then 𝑥⟂𝑦 if and only if (𝑥,𝑦)=0; here, (⋅,⋅) stands for a standard inner product in ℝ^𝑚.

We denote sets of coupled indexes by 𝐏_𝑚=Δ𝑖,𝑗∶𝑖<𝑗⊂𝐼×𝐼,_𝑚={(𝑖,𝑖)∶𝑖∈𝐼}⊂𝐼×𝐼. (15) For a given pair (𝑖,𝑗)∈𝐏_𝑚∪Δ_𝑚, we set a vector ℙ_𝑖𝑗=(𝑃_𝑖𝑗,1,…,𝑃_{𝑖𝑗,𝑚}). It is clear due to the condition (5) that ℙ_𝑖𝑗∈𝑆^𝑚−1.

Let 𝜉₁={𝐴_𝑖}^𝑁_𝑖=1 and 𝜉₂={𝐵_𝑖}^𝑀_𝑖=1 be some fixed partitions of 𝐏_𝑚 and Δ_𝑚, respectively; that is, 𝐴_𝑖∩𝐴_𝑗=∅, 𝐵_𝑖∩𝐵_𝑗=∅, and ⋃^𝑁_𝑖=1𝐴_𝑖=𝐏_𝑚, ⋃^𝑀_𝑖=1𝐵_𝑖=Δ_𝑚, where 𝑁,𝑀≤𝑚.

Definition 2. A quadratic stochastic operator 𝑉∶𝑆^𝑚−1→𝑆^𝑚−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 each 𝑘∈{1,…,𝑁} and any (𝑖,𝑗), (𝑢,𝑣)∈𝐴_𝑘, one has that ℙ_𝑖𝑗∼ℙ_𝑢𝑣;(ii)for any 𝑘≠ℓ, 𝑘,ℓ∈{1,…,𝑁} and any (𝑖,𝑗)∈𝐴_𝑘 and (𝑢,𝑣)∈𝐴_ℓ, one has that ℙ_𝑖𝑗⟂ℙ_𝑢𝑣;(iii)for each 𝑑∈{1,…,𝑀} and any (𝑖,𝑖), (𝑗,𝑗)∈𝐵_𝑑, one has that ℙ_𝑖𝑖∼ℙ_𝑗𝑗;(iv)for any 𝑠≠ℎ, 𝑠,ℎ∈{1,…,𝑀} and any (𝑢,𝑢)∈𝐵_𝑠, and (𝑣,𝑣)∈𝐵_ℎ one has that ℙ_𝑢𝑢⟂ℙ_𝑣𝑣.

Remark 3. If 𝜉₂ 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 ℙ_𝑖𝑖 and ℙ_𝑗𝑗 are singular. If 𝜉₂ is the trivial, that is, 𝜉₂={Δ_𝑚}, 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 ℙ_𝑖𝑖 and ℙ_𝑗𝑗 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 the 𝜉^(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 ℙ_𝑖𝑖=𝑒_𝜋(𝑖) for any 𝑖=1,𝑚 where 𝑒_𝑘=(0,…,0,1|{z}_𝑘,0,…,0),   𝑘=1,𝑚, are vertices of the simplex 𝑆^𝑚−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 𝑃_{𝑖𝑗,𝑘} is a probability that parents in the 𝑖th and 𝑗th traits interbreed to produce a child from the 𝑘th trait. The condition 𝑃_{𝑖𝑗,𝑘}=𝑃_{𝑗𝑖,𝑘} means that the gender of parents do not influence having a child from the 𝑘th trait. In this sense, 𝐏_𝑚∪Δ_𝑚 is a set of all possible coupled traits of parents. A vector ℙ_𝑖𝑗=(𝑃_𝑖𝑗,1,…,𝑃_{𝑖𝑗,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 𝐏_𝑚 of all differently coupled traits of parents is splitted into 𝑁 groups 𝐴₁,…,𝐴_𝑁 (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 𝐴_𝑘 is simultaneously either positive or zero (the condition (i) of Definition 2); meanwhile, two families whose parents’ coupled traits belong to two different groups 𝐴_𝑘 and 𝐴_𝑙 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 of 𝜉^(𝑠)-QSO on 2D Simplex

In this section, we are going to study 𝜉^(𝑠)-QSO in two-dimensional simplex; that is, 𝑚=3. In this case, we have the following possible partitions of 𝐏₃: 𝜉₁={{(1,2)},{(1,3)},{(2,3)}},𝜉₁𝜉=3,₂={{(2,3)},{(1,2),(1,3)}},𝜉₂𝜉=2,₃={{(1,3)},{(1,2),(2,3)}},𝜉₃𝜉=2,₄={{(1,2)},{(1,3),(2,3)}},𝜉₄𝜉=2,₅={(1,2),(1,3),(2,3)},𝜉₅=1. (16)

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 operators 𝑉₁ 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)=(𝑥_𝜋(1),…,𝑥_𝜋(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 all 𝜉^(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 all 𝜉^(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 operator 𝑉∶𝑆²→𝑆² given by 𝑉∶𝑥^′_𝑘=³_𝑖,𝑗=1𝑃_{𝑖𝑗,𝑘}𝑥_𝑖𝑥_𝑗,𝑘=1,2,3, (17) is a 𝜉^(𝑠)-QSO corresponding to partition 𝜉₃={{(1,3)},{(1,2),(2,3)}}. This means that the coefficients (𝑃_{𝑖𝑗,𝑘})³_{𝑖,𝑗,𝑘=1} of 𝑉 satisfy the following three conditions: (i) ℙ₁₂∼ℙ₂₃, (ii) ℙ₁₃⟂ℙ₁₂,ℙ₁₃⟂ℙ₂₃, and (iii) ℙ₁₁⟂ℙ₂₂⟂ℙ₃₃ where ℙ_𝑖𝑗=(𝑃_𝑖𝑗,1,𝑃_𝑖𝑗,2,𝑃_𝑖𝑗,3).
We consider the following operator: 𝑉_𝜋=𝜋⁻¹𝑉𝜋, where 𝜋=¹²³₂₁₃. It is clear that 𝑉_𝜋 is conjugate to 𝑉, where 𝑉_𝜋∶𝑥^′_𝑘=³_𝑖,𝑗=1𝑃^𝜋_{𝑖𝑗,𝑘}𝑥_𝑖𝑥_𝑗,𝑘=1,2,3 (18) such that 𝑃^𝜋_{𝑖𝑗,𝑘}=𝑃_{𝜋(𝑖)𝜋(𝑗),𝜋(𝑘)} for any 𝑖,𝑗,𝑘=1,2,3; equivalently, ℙ^𝜋_𝑖𝑗=𝜋ℙ_{𝜋(𝑖)𝜋(𝑗)} (in a vector form) for any 𝑖,𝑗=1,2,3. Now, we are going to show that 𝑉_𝜋 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)ℙ^𝜋₁₂∼ℙ^𝜋₁₃. Indeed, since ℙ^𝜋₁₂=𝜋ℙ₁₂,ℙ^𝜋₁₃=𝜋ℙ₂₃,  ℙ₁₂∼ℙ₂₃, one has ℙ^𝜋₁₂∼ℙ^𝜋₁₃.(ii)ℙ^𝜋₁₂⟂ℙ^𝜋₂₃,ℙ^𝜋₁₃⟂ℙ^𝜋₂₃. Indeed, since ℙ^𝜋₁₂=𝜋ℙ₁₂, ℙ^𝜋₂₃=𝜋ℙ₁₃, and ℙ₁₂⟂ℙ₁₃, we obtain that ℙ^𝜋₁₂⟂ℙ^𝜋₂₃. In the same manner, we can get that ℙ^𝜋₁₃⟂ℙ^𝜋₂₃.(iii)ℙ^𝜋₁₁⟂ℙ^𝜋₂₂⟂ℙ^𝜋₃₃. Indeed, since ℙ^𝜋₁₁=𝜋ℙ₂₂,   ℙ^𝜋₂₂=𝜋ℙ₁₁,   ℙ^𝜋₃₃=𝜋ℙ₃₃, and ℙ₁₁⟂ℙ₂₂⟂ℙ₃₃, we have that ℙ^𝜋₁₁⟂ℙ^𝜋₂₂⟂ℙ^𝜋₃₃.
This shows that any 𝜉^(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 𝑉_𝜋=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 all 𝜉^(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 (𝑃_{𝑖𝑗,𝑘})³_{𝑖,𝑗,𝑘=1} in special forms where 𝑎∈[0,1] (see Table 1).

Table 1 

The choices of the cases (𝐈_𝑖,𝐈𝐈_𝑗), 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: 𝑉₁∶𝑥^′₁=𝑥²₁+2𝑎𝑥₁1−𝑥₁,𝑥^′₂=𝑥²₂+2(1−𝑎)𝑥₁1−𝑥₁,𝑥^′₃=𝑥²₃+2𝑥₂𝑥₃,𝑉₂∶𝑥^′₁=𝑥²₂+2𝑎𝑥₁1−𝑥₁,𝑥^′₂=𝑥²₁+2(1−𝑎)𝑥₁1−𝑥₁,𝑥^′₃=𝑥²₃+2𝑥₂𝑥₃,𝑉₃∶𝑥^′₁=𝑥²₃+2𝑎𝑥₁1−𝑥₁,𝑥^′₂=𝑥²₂+2(1−𝑎)𝑥₁1−𝑥₁,𝑥^′₃=𝑥²₁+2𝑥₂𝑥₃,𝑉₄∶𝑥^′₁=𝑥²₁+2𝑎𝑥₁1−𝑥₁,𝑥^′₂=𝑥²₃+2(1−𝑎)𝑥₁1−𝑥₁,𝑥^′₃=𝑥²₂+2𝑥₂𝑥₃,𝑉₅∶𝑥^′₁=𝑥²₂+2𝑎𝑥₁1−𝑥₁,𝑥^′₂=𝑥²₃+2(1−𝑎)𝑥₁1−𝑥₁,𝑥^′₃=𝑥²₁+2𝑥₂𝑥₃,𝑉₆∶𝑥^′₁=𝑥²₃+2𝑎𝑥₁1−𝑥₁,𝑥^′₂=𝑥²₁+2(1−𝑎)𝑥₁1−𝑥₁,𝑥^′₃=𝑥²₂+2𝑥₂𝑥₃,𝑉₇∶𝑥^′₁=𝑥²₁+2𝑥₂𝑥₃,𝑥^′₂=𝑥²₂+2𝑎𝑥₁1−𝑥₁,𝑥^′₃=𝑥²₃+2(1−𝑎)𝑥₁1−𝑥₁,𝑉₈∶𝑥^′₁=𝑥²₂+2𝑥₂𝑥₃,𝑥^′₂=𝑥²₁+2𝑎𝑥₁1−𝑥₁,𝑥^′₃=𝑥²₃+2(1−𝑎)𝑥₁1−𝑥₁,𝑉₉∶𝑥^′₁=𝑥²₃+2𝑥₂𝑥₃,𝑥^′₂=𝑥²₂+2𝑎𝑥₁1−𝑥₁,𝑥^′₃=𝑥²₁+2(1−𝑎)𝑥₁1−𝑥₁,𝑉₁₀∶𝑥^′₁=𝑥²₁+2𝑥₂𝑥₃,𝑥^′₂=𝑥²₃+2𝑎𝑥₁1−𝑥₁,𝑥^′₃=𝑥²₂+2(1−𝑎)𝑥₁1−𝑥₁,𝑉₁₁∶𝑥^′₁=𝑥²₂+2𝑥₂𝑥₃,𝑥^′₂=𝑥²₃+2𝑎𝑥₁1−𝑥₁,𝑥^′₃=𝑥²₁+2(1−𝑎)𝑥₁1−𝑥₁,𝑉₁₂∶𝑥^′₁=𝑥²₃+2𝑥₂𝑥₃,𝑥^′₂=𝑥²₁+2𝑎𝑥₁1−𝑥₁,𝑥^′₃=𝑥²₂+2(1−𝑎)𝑥₁1−𝑥₁,𝑉₁₃∶𝑥^′₁=𝑥²₁+2𝑎𝑥₁1−𝑥₁,𝑥^′₂=𝑥²₂+2𝑥₂𝑥₃,𝑥^′₃=𝑥²₃+2(1−𝑎)𝑥₁1−𝑥₁,𝑉₁₄∶𝑥^′₁=𝑥²₂+2𝑎𝑥₁1−𝑥₁,𝑥^′₂=𝑥²₁+2𝑥₂𝑥₃,𝑥^′₃=𝑥²₃+2(1−𝑎)𝑥₁1−𝑥₁,𝑉₁₅∶𝑥^′₁=𝑥²₃+2𝑎𝑥₁1−𝑥₁,𝑥^′₂=𝑥²₂+2𝑥₂𝑥₃,𝑥^′₃=𝑥²₁+2(1−𝑎)𝑥₁1−𝑥₁,𝑉₁₆∶𝑥^′₁=𝑥²₁+2𝑎𝑥₁1−𝑥₁,𝑥^′₂=𝑥²₃+2𝑥₂𝑥₃,𝑥^′₃=𝑥²₂+2(1−𝑎)𝑥₁1−𝑥₁,𝑉₁₇∶𝑥^′₁=𝑥²₂+2𝑎𝑥₁1−𝑥₁,𝑥^′₂=𝑥²₃+2𝑥₂𝑥₃,𝑥^′₃=𝑥²₁()𝑥(),𝑉+21−𝑎1−𝑥₁₈∶𝑥^′₁=𝑥²₃+2𝑎𝑥₁1−𝑥₁,𝑥^′₂=𝑥²₁+2𝑥₂𝑥₃,𝑥^′₃=𝑥²₂+2(1−𝑎)𝑥₁1−𝑥₁,𝑉₁₉∶𝑥^′₁=𝑥²₁+2𝑎𝑥₂𝑥₃,𝑥^′₂=𝑥²₂+2(1−𝑎)𝑥₂𝑥₃,𝑥^′₃=𝑥²₃+2𝑥₁1−𝑥₁,𝑉₂₀∶𝑥^′₁=𝑥²₂+2𝑎𝑥₂𝑥₃,𝑥^′₂=𝑥²₁+2(1−𝑎)𝑥₂𝑥₃,𝑥^′₃=𝑥²₃+2𝑥₁1−𝑥₁,𝑉₂₁∶𝑥^′₁=𝑥²₃+2𝑎𝑥₂𝑥₃,𝑥^′₂=𝑥²₂+2(1−𝑎)𝑥₂𝑥₃,𝑥^′₃=𝑥²₁+2𝑥₁1−𝑥₁,𝑉₂₂∶𝑥^′₁=𝑥²₁+2𝑎𝑥₂𝑥₃,𝑥^′₂=𝑥²₃+2(1−𝑎)𝑥₂𝑥₃,𝑥^′₃=𝑥²₂+2𝑥₁1−𝑥₁,𝑉₂₃∶𝑥^′₁=𝑥²₂+2𝑎𝑥₂𝑥₃,𝑥^′₂=𝑥²₃+2(1−𝑎)𝑥₂𝑥₃,𝑥^′₃=𝑥²₁+2𝑥₁1−𝑥₁,𝑉₂₄∶𝑥^′₁=𝑥²₃+2𝑎𝑥₂𝑥₃,𝑥^′₂=𝑥²₁+2(1−𝑎)𝑥₂𝑥₃,𝑥^′₃=𝑥²₂+2𝑥₁1−𝑥₁,𝑉₂₅∶𝑥^′₁=𝑥²₁+2𝑥₁1−𝑥₁,𝑥^′₂=𝑥²₂+2𝑎𝑥₂𝑥₃,𝑥^′₃=𝑥²₃+2(1−𝑎)𝑥₂𝑥₃,𝑉₂₆∶𝑥^′₁=𝑥²₂+2𝑥₁1−𝑥₁,𝑥^′₂=𝑥²₁+2𝑎𝑥₂𝑥₃,𝑥^′₃=𝑥²₃+2(1−𝑎)𝑥₂𝑥₃,𝑉₂₇∶𝑥^′₁=𝑥²₃+2𝑥₁1−𝑥₁,𝑥^′₂=𝑥²₂+2𝑎𝑥₂𝑥₃,𝑥^′₃=𝑥²₁+2(1−𝑎)𝑥₂𝑥₃,𝑉₂₈∶𝑥^′₁=𝑥²₁+2𝑥₁1−𝑥₁,𝑥^′₂=𝑥²₃+2𝑎𝑥₂𝑥₃,𝑥^′₃=𝑥²₂+2(1−𝑎)𝑥₂𝑥₃,𝑉₂₉∶𝑥^′₁=𝑥²₂+2𝑥₁1−𝑥₁,𝑥^′₂=𝑥²₃+2𝑎𝑥₂𝑥₃,𝑥^′₃=𝑥²₁+2(1−𝑎)𝑥₂𝑥₃,𝑉₃₀∶𝑥^′₁=𝑥²₃+2𝑥₁1−𝑥₁,𝑥^′₂=𝑥²₁+2𝑎𝑥₂𝑥₃,𝑥^′₃=𝑥²₂+2(1−𝑎)𝑥₂𝑥₃,𝑉₃₁∶𝑥^′₁=𝑥²₁+2𝑎𝑥₂𝑥₃,𝑥^′₂=𝑥²₂+2𝑥₁1−𝑥₁,𝑥^′₃=𝑥²₃+2(1−𝑎)𝑥₂𝑥₃,𝑉₃₂∶𝑥^′₁=𝑥²₂+2𝑎𝑥₂𝑥₃,𝑥^′₂=𝑥²₁+2𝑥₁1−𝑥₁,𝑥^′₃=𝑥²₃+2(1−𝑎)𝑥₂𝑥₃,𝑉₃₃∶𝑥^′₁=𝑥²₃+2𝑎𝑥₂𝑥₃,𝑥^′₂=𝑥²₂+2𝑥₁1−𝑥₁,𝑥^′₃=𝑥²₁+2(1−𝑎)𝑥₂𝑥₃,𝑉₃₄∶𝑥^′₁=𝑥²₁+2𝑎𝑥₂𝑥₃,𝑥^′₂=𝑥²₃+2𝑥₁1−𝑥₁,𝑥^′₃=𝑥²₂+2(1−𝑎)𝑥₂𝑥₃,𝑉₃₅∶𝑥^′₁=𝑥²₂+2𝑎𝑥₂𝑥₃,𝑥^′₂=𝑥²₃+2𝑥₁1−𝑥₁,𝑥^′₃=𝑥²₁+2(1−𝑎)𝑥₂𝑥₃,𝑉₃₆∶𝑥^′₁=𝑥²₃+2𝑎𝑥₂𝑥₃,𝑥^′₂=𝑥²₁+2𝑥₁1−𝑥₁,𝑥^′₃=𝑥²₂+2(1−𝑎)𝑥₂𝑥₃. (19)

Theorem 6. All 36 operators from the class of 𝜉^(𝑠)-QSO corresponding to partition 𝜉₂ defined as above are classified into 20 nonconjugate classes: 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. (20)

Proof. It is easy to check that partition 𝜉₂={{(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 classes 𝐾₁, 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 of 𝜉^(𝑠)-QSO from Class 𝐾₁

In this section we are going to study dynamics of 𝜉^(𝑠)-QSO from class 𝐾₁.

We need some auxiliary facts about properties of the function 𝑓_𝑎∶[0,1]→[0,1] given by 𝑓_𝑎(𝑥)=𝑥²+2𝑎𝑥(1−𝑥), (21) 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. Let 𝑓_𝑎∶[0,1]→[0,1] be a function given by (21) where 𝑎≠1/2. Then, the following statements hold true.(i)One has that Fix(𝑓_𝑎)={0,1}.(ii)The function 𝑓_𝑎 is increasing.(iii)One has that (𝑎−(1/2))(𝑓_𝑎(𝑥)−𝑥)>0 for any 𝑥∈(0,1).(iv)One has that 𝜔_𝑓𝑎(𝑥₀)=^{{0}𝑖𝑓0≤𝑎<1/2}_{{1}𝑖𝑓1/2<𝑎≤1} for any 𝑥₀∈(0,1).

Proof. Let 𝑓_𝑎∶[0,1]→[0,1] be a function given by (21) where 𝑎≠1/2.(i)In order to find fixed points of the function 𝑓_𝑎, 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(𝑓_𝑎)={0,1}.(ii)Since 𝑓^′_𝑎(𝑥)=2𝑥(1−𝑎)+2𝑎(1−𝑥)≥0, the function 𝑓_𝑎 is increasing. (iii)Since 𝑓_𝑎(𝑥)−𝑥=(1−2𝑎)(𝑥²−𝑥), we may get that (𝑎−(1/2))(𝑓_𝑎(𝑥)−𝑥)>0. (iv)Let 0≤𝑎<1/2 and 𝑥₀∈(0,1). Due to (iii), we have that 𝑓_𝑎(𝑥₀)<𝑥₀. Since 𝑓_𝑎 is increasing, we obtain that 𝑓_𝑎^(𝑛+1)(𝑥₀)<𝑓_𝑎^(𝑛)(𝑥₀) for any 𝑛∈ℕ. This means that {𝑓_𝑎^(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 𝑤_𝑓𝑎(𝑥₀)={0}. Similarly, one can show that if 1/2<𝑎≤1, then 𝜔_𝑓𝑎(𝑥₀)={1} for any 𝑥₀∈(0,1). This completes the proof.

Now, we are going to study dynamics of a 𝜉^(𝑠)-QSO 𝑉₁₃∶𝑆²→𝑆² taken from 𝐾₁: 𝑉₁₃∶𝑥^′₁=𝑥²₁+2𝑥₁𝑎1−𝑥₁𝑥^′₂=𝑥²₂+2𝑥₂𝑥₃𝑥^′₃=𝑥²₃+2(1−𝑎)𝑥₁1−𝑥₁, (22) where 0≤𝑎≤1. This operator is an ℓ-Volterra-QSO, and its behavior was not studied in [24, 25].

Let 𝑒₁, 𝑒₂, and 𝑒₃ be the vertices of the simplex 𝑆² and let Γ_𝑖={𝑥∈𝑆²∶𝑥_𝑖=0} be an edge of the simplex 𝑆², where 𝑖=1,2,3. Let 𝑆_1≤3={𝑥∈𝑆²∶𝑥₁≤𝑥₃}, 𝑆_1≥3={𝑥∈𝑆²∶𝑥₁≥𝑥₃}, and 𝑙₁₃={𝑥∈𝑆²∶𝑥₁=𝑥₃}.

Theorem 8. Let 𝑉₁₃∶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 Fix𝑉₁₃=𝑒₁,𝑒₂,𝑒₃1𝑖𝑓𝑎≠2,Γ₂∪𝑙₁₃1𝑖𝑓𝑎=2. (23)(ii) If 0≤𝑎<1/2, then 𝜔_𝑉13𝑥⁽⁰⁾=𝑒₂𝑖𝑓𝑥₂⁽⁰⁾𝑒≠0,₃𝑖𝑓𝑥₂⁽⁰⁾=0. (24)(iii) If 1/2<𝑎≤1, then 𝜔_𝑉13𝑥⁽⁰⁾=𝑒₁𝑖𝑓𝑥₁⁽⁰⁾𝑒≠0,₂𝑖𝑓𝑥₁⁽⁰⁾=0. (25)(iv) If 𝑎=1/2, then 𝜔_𝑉13𝑥⁽⁰⁾=𝑥₁⁽⁰⁾,0,1−𝑥₁⁽⁰⁾𝑖𝑓𝑥₁⁽⁰⁾>12,𝑥₁⁽⁰⁾,1−2𝑥₁⁽⁰⁾,𝑥₁⁽⁰⁾𝑖𝑓𝑥₁⁽⁰⁾≤12. (26)

Proof. Let 𝑉₁₃∶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: 𝑥₁=𝑥²₁+2𝑥₁𝑎1−𝑥₁,𝑥₂=𝑥²₂+2𝑥₂𝑥₃,𝑥₃=𝑥²₃+2(1−𝑎)𝑥₁1−𝑥₁. (27) We shall separately consider two cases: 𝑎=1/2 and 𝑎≠1/2.
Let 𝑎≠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}.
Let 𝑎=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)=𝑓_𝑎^(𝑛)(𝑥₁⁽⁰⁾). Therefore, due to Proposition 7(iv), we have that lim_𝑛→∞𝑥₁^(𝑛)=0. Hence, 𝜔_𝑉13(𝑥⁽⁰⁾)⊂Γ₁={𝑥∈𝑆²∶𝑥₁=0}. Now, we shall separately consider two cases: 𝑥₂⁽⁰⁾=0 and 𝑥₂⁽⁰⁾≠0.
Let 𝑥₂⁽⁰⁾=0. In this case, 𝑥₂^(𝑛)=0 and lim_𝑛→∞𝑥₃^(𝑛)=1. Hence, 𝜔_𝑉13(𝑥⁽⁰⁾)={𝑒₃}.
Let 𝑥₂⁽⁰⁾≠0. We need the following result.
Claim. One has that 𝑉₁₃(𝑆_1≤3)⊂𝑆_1≤3. Moreover, for any 𝑥⁽⁰⁾∉Fix(𝑉₁₃), there exists 𝑛₀ (depending on 𝑥⁽⁰⁾) such that {𝑥^(𝑛)}^∞_𝑛=𝑛0⊂𝑆_1≤3.
Proof of Claim. If 𝑥₁≤𝑥₃, then 𝑥^′₁=𝑥²₁+2𝑎𝑥₁(1−𝑥₁)≤𝑥²₃+2(1−𝑎)𝑥₁(1−𝑥₁)=𝑥^′₃. This means that 𝑉₁₃(𝑆_1≤3)⊂𝑆_1≤3.
Let 𝑥⁽⁰⁾∉𝑆_1≤3, and suppose that all elements of the trajectory belong to set 𝑆²⧵𝑆_1≤3; that is, {𝑥^(𝑛)}^∞_𝑛=0⊂𝑆²⧵𝑆_1≤3. It follows from 𝑥₁^(𝑛)→0 and 𝑥₃^(𝑛)<𝑥₁^(𝑛) that 𝑥₃^(𝑛)→0. This with ∑³_𝑖=1𝑥_𝑖^(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 exists 𝑛₀ 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 𝑥^∗₂. This yields that 𝑥^(𝑛)=(𝑥₁^(𝑛),𝑥₂^(𝑛),𝑥₃^(𝑛)) converges to (0,𝑥^∗₂,1−𝑥^∗₂), where 𝑥^∗₂>0. We know that (0,𝑥^∗₂,1−𝑥^∗₂) should be a fixed point. Consequently, (0,𝑥^∗₂,1−𝑥^∗₂)=(0,1,0) and 𝜔_𝑉13(𝑥⁽⁰⁾)={𝑒₂}.
(iii) Let 1/2<𝑎≤1. Due to Proposition 7(iv), we have that lim_𝑛→∞𝑥₁^(𝑛)=1, whenever 0<𝑥₁⁽⁰⁾<1. Therefore, if 𝑥₁⁽⁰⁾≠0, then 𝜔_𝑉13(𝑥⁽⁰⁾)⊂{𝑒₁}. Since 𝜔_𝑉13(𝑥⁽⁰⁾) is not empty, we obtain that 𝜔_𝑉13(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 lim_𝑛→∞𝑥₃^(𝑛)=0. This means that lim_𝑛→∞𝑥₂^(𝑛)=1. Therefore, if 𝑥₁⁽⁰⁾=0, then 𝜔_𝑉13(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 𝑥^∗₂. Consequently, (𝑥₁^(𝑛),𝑥₂^(𝑛),𝑥₃^(𝑛))→(𝑥₁⁽⁰⁾,𝑥^∗₂,𝑥^∗₃). We know that (𝑥₁⁽⁰⁾,𝑥^∗₂,𝑥^∗₃) should be a fixed point. Since 𝑥₁⁽⁰⁾>1/2≥𝑥₃^(∗), we find that 𝑥^∗₂=0 and 𝑥^∗₃=1−𝑥₁⁽⁰⁾. This means that 𝜔_𝑉13(𝑥⁽⁰⁾)={(𝑥₁⁽⁰⁾,0,1−𝑥₁⁽⁰⁾)}.
Let 𝑥₁⁽⁰⁾≤1/2. Then 𝑥₁^(𝑛)=𝑥₁⁽⁰⁾≤1/2 for any 𝑛∈ℕ. Since 𝑥^′₃−𝑥^′₁=𝑥²₃−𝑥²₁, we have that 𝑉(𝑆_1≤3)⊂𝑆_1≤3 and 𝑉(𝑆_1≥3)⊂𝑆_1≥3. If 𝑥⁽⁰⁾=(𝑥₁⁽⁰⁾,𝑥₂⁽⁰⁾,𝑥₃⁽⁰⁾)∈𝑆_1≤3, then 𝑥^(𝑛)=(𝑥₁^(𝑛),𝑥₂^(𝑛),𝑥₃^(𝑛))∈𝑆_1≤3. This yields that {𝑥₂^(𝑛)}^∞_𝑛=0 is decreasing, and hence it converges to 𝑥^∗₂. Therefore, (𝑥₁^(𝑛),𝑥₂^(𝑛),𝑥₃^(𝑛)) converges to (𝑥₁⁽⁰⁾,𝑥₂^(∗),𝑥₃^(∗)). Since (𝑥₁⁽⁰⁾,𝑥^∗₂,𝑥^∗₃)∈𝑆_1≤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⁾∈𝑆_1≥3, then 𝜔_𝑉13(𝑥⁽⁰⁾)={(𝑥₁⁽⁰⁾,1−2𝑥₁⁽⁰⁾,𝑥₁⁽⁰⁾)}.
This completes the proof.

5. Dynamics of 𝜉^(𝑠)-QSO from Class 𝐾₄

We are going to study dynamics of a 𝜉^(𝑠)-QSO 𝑉₄∶𝑆²→𝑆² taken from 𝐾₄: 𝑉₄∶𝑥^′₁=𝑥²₁+2𝑎𝑥₁1−𝑥₁,𝑥^′₂=𝑥²₃+2(1−𝑎)𝑥₁1−𝑥₁,𝑥^′₃=𝑥²₂+2𝑥₂𝑥₃, (28) 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 𝐵(𝑏)=3−2𝑏−4𝑏²−8𝑏+52[],𝐶,𝑏∈0,1_±(𝑐)=1−2𝑐±√4𝑐²−8𝑐+12√,𝑐∈0,2−32. (29)

It is clear that (𝑏)=2()0≤𝐵1−𝑏1+2(1−𝑏)+1+4(1−𝑏)²≤1,0≤1−2𝑐±(1−2𝑐)²−4𝑐2=𝐶_±(𝑐)≤1−2𝑐+√4𝑐²+4𝑐+12=1 (30) for any 𝑏∈[0,1] and 𝑐∈[0,(2−√3)/2]⊂[0,1/2].

Theorem 9. Let 𝑉₄∶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 Fix𝑉₄=𝑒₁,√0,3−52,−1+√521𝑖𝑓𝑎≠2,{𝑏,𝐵(𝑏),1−𝐵(𝑏)}_𝑏∈[0,1]1𝑖𝑓𝑎=2. (31)(ii) One has that Per₂𝑉₄=𝑒₂,𝑒₃1𝑖𝑓𝑎≠2,𝑐,𝐶_±(𝑐),1−𝐶_±(𝑐)_{√𝑐∈[0,(2−3)/2)}1𝑖𝑓𝑎=2. (32)(iii) If 1/2<𝑎≤1, then 𝜔_𝑉4𝑥⁽⁰⁾=𝑒₁𝑖𝑓𝑥₁⁽⁰⁾𝑒≠0,₂,𝑒₃𝑖𝑓𝑥₁⁽⁰⁾=0. (33)(iv) If 𝑎=1/2, then 𝜔_𝑉4𝑥⁽⁰⁾=𝑥₁⁽⁰⁾,𝐶_±𝑥₁⁽⁰⁾,1−𝐶_±𝑥₁⁽⁰⁾𝑖𝑓𝑥₁⁽⁰⁾∈√0,2−32,𝑥₁⁽⁰⁾,𝐵𝑥₁⁽⁰⁾,1−𝐵𝑥₁⁽⁰⁾𝑖𝑓𝑥₁⁽⁰⁾∈2−√32,1. (34)

Proof. Let 𝑥⁽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: 𝑥₁=𝑥²₁+2𝑎𝑥₁1−𝑥₁,𝑥₂=𝑥²₃+2(1−𝑎)𝑥₁1−𝑥₁,𝑥₃=𝑥²₂+2𝑥₂𝑥₃. (35)
Let 𝑎≠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: 𝑥²₂−3𝑥₂+1=0. (36) So, the solutions of this quadratic equation are 𝑥^±₂=(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.
Let 𝑎=1/2. The system (35) then takes the following form 𝑥₁=𝑥₁,𝑥₂=𝑥²₃+𝑥₁1−𝑥₁,𝑥₃=𝑥²₂+2𝑥₂𝑥₃. (37) So, by letting 𝑥₁=𝑏 (any 𝑏∈[0,1]), the second equation of the system (37) can be written as follows 𝑥²₂−(3−2𝑏)𝑥₂+(1−𝑏)=0. (38) The solutions of the last equation are 𝑥^±₂=(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) Let 𝑎≠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 𝑓_𝑎(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: 𝑥₁𝑥=0,₂=1−(1−𝑥₂)²²,𝑥₃=1−1−𝑥²₃². (39) 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.
Let 𝑎=1/2. In order to find 2-periodic points of 𝑉₄, we should solve the following system of equations: 𝑥₁=𝑥₁,𝑥₂=𝑥²₂1−𝑥₁+𝑥₃²+𝑥₁1−𝑥₁,𝑥₃=𝑥²₃+𝑥₁1−𝑥₁1−𝑥₁+𝑥₂1−𝑥₁+𝑥₃. (40) By letting 𝑥₁=𝑐, where 𝑐∈[0,1], the second equation of system (40) reduces to the following equation: 𝑥²₂2−2𝑐−𝑥₂²+𝑐(1−𝑐)=𝑥₂. (41) 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) Let 1/2<𝑎≤1 and 𝑥₁⁽⁰⁾=0. In this case, the second coordinate of 𝑉₄ has the form 𝑥^′₂=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 𝜔_ℎ⁽²⁾(𝑥₂⁽⁰⁾)={0} and if 𝑥₂⁽⁰⁾∈((3−√5)/2,1], then 𝜔_ℎ⁽²⁾(𝑥₂⁽⁰⁾)={1}. On the other hand, we have that 𝑉₄^(𝑛)𝑥⁽⁰⁾=0,ℎ^(2𝑘)𝑥₂⁽⁰⁾,1−ℎ^(2𝑘)𝑥₂⁽⁰⁾if𝑛=2𝑘,0,ℎ^(2𝑘)ℎ𝑥₂⁽⁰⁾,1−ℎ^(2𝑘)ℎ𝑥₂⁽⁰⁾if𝑛=2𝑘+1. (42) Therefore, we obtain that 𝜔_𝑉4(𝑥⁽⁰⁾)={𝑒₂,𝑒₃} if 𝑥₁⁽⁰⁾=0.
Let 1/2<𝑎≤1 and 0<𝑥₁⁽⁰⁾<1. In this case, it is clear that 𝑥₁^(𝑛)=𝑓_𝑎^(𝑛)(𝑥₁⁽⁰⁾). Therefore, due to Proposition 7(iv), we have that lim_𝑛→∞𝑥₁^(𝑛)=1. This means that 𝜔_𝑉4(𝑥⁽⁰⁾)⊂{𝑒₁}. Since 𝜔_𝑉4(𝑥⁽⁰⁾)≠∅, one has that 𝜔_𝑉4(𝑥⁽⁰⁾)={𝑒₁}.
(iv) Let 𝑎=1/2. Then the operator 𝑉₄ takes the following form: 𝑥^′₁=𝑥₁,𝑥^′₂=𝑥²₃+𝑥₁1−𝑥₁,𝑥^′₃=𝑥₂1−𝑥₁+𝑥₃. (43) It is clear that 𝐿_𝑐={𝑥∈𝑆²∶𝑥₁=𝑐} is invariant under 𝑉₄ where 𝑐∈[0,1]. Therefore, we shall study the dynamics of 𝑉₄ over 𝐿_𝑐.
Let 𝑥₁⁽⁰⁾=𝑐∈[0,(2−√3)/2) be a fixed number. Let us consider function ℎ_𝑐∶[0,1−𝑐]→[0,1−𝑐], ℎ_𝑐𝑥₂=1−𝑐−𝑥₂²+𝑐(1−𝑐). (44) One can show that the function ℎ_𝑐 is decreasing on [0,1−𝑐]. This yields that the function ℎ_𝑐⁽²⁾ is increasing. It follows from the discussion presented above (see (ii)) that Fix(ℎ_𝑐)∩[0,1−𝑐]={𝐵(𝑐)} and Fix(ℎ_𝑐⁽²⁾)∩[0,1−𝑐]={𝐵(𝑐),𝐶_±(𝑐)} where 𝐶₋(𝑐)<𝐵(𝑐)<𝐶₊(𝑐). Moreover, one has that ℎ_𝑐0,𝐶₋(𝑐)⊂𝐶₊(𝑐),ℎ,1−𝑐_𝑐𝐶₋(𝑐),𝐵(𝑐)⊂𝐵(𝑐),𝐶₊(𝑐,ℎ_𝑐)𝐵(𝑐),𝐶₊(𝑐)⊂𝐶₋(𝑐)(𝑐,ℎ,𝐵)_𝑐𝐶₊(𝑐),1−𝑐⊂0,𝐶₋(𝑐), (45) Therefore, the sets [0,𝐶₋(𝑐)], [𝐶₋(𝑐),𝐵(𝑐)], [𝐵(𝑐),𝐶₊(𝑐)], and [𝐶₊(𝑐),1−𝑐] are invariant under function ℎ_𝑐⁽²⁾. Simple calculations show that ℎ_𝑐⁽²⁾𝑥₂>𝑥₂,∀𝑥₂∈0,𝐶₋(𝑐)∪𝐵(𝑐),𝐶₊(𝑐),ℎ_𝑐⁽²⁾𝑥₂<𝑥₂,∀𝑥₂∈𝐶₋(𝑐),𝐵(𝑐)∪𝐶₊(𝑐),1−𝑐. (46) Consequently, we get that 𝜔_ℎ𝑐⁽²⁾𝑥₂⁽⁰⁾=𝐶₋(𝑐),∀𝑥₂⁽⁰⁾∈[(𝑐,𝜔0,𝐵))_ℎ𝑐(2)𝑥₂⁽⁰⁾=𝐶₊(𝑐),∀𝑥₂⁽⁰⁾∈(𝐵(𝑐),1−𝑐]. (47)
On the other hand, we have that 𝑉₄^(𝑛)𝑥⁽⁰⁾=𝑐,ℎ_𝑐^(2𝑘)𝑥₂⁽⁰⁾,1−ℎ_𝑐^(2𝑘)𝑥₂⁽⁰⁾if𝑛=2𝑘,𝑐,ℎ_𝑐^(2𝑘)ℎ_𝑐𝑥₂⁽⁰⁾,1−ℎ_𝑐^(2𝑘)ℎ_𝑐𝑥₂⁽⁰⁾if𝑛=2𝑘+1. (48) Therefore, we obtain that 𝜔_𝑉4𝑥⁽⁰⁾=𝑐,𝐶_±(𝑐),1−𝐶_±(𝑐)if𝑥₁⁽⁰⁾√=𝑐∈0,2−32. (49) In the same manner, one can show that 𝜔_𝑉4(𝑥⁽⁰⁾)={(𝑐,𝐵(𝑐),1−𝐵(𝑐))} whenever 𝑥₁⁽⁰⁾=𝑐∈[(2−√3)/2,1].
This completes the proof.

6. Dynamics of 𝜉^(𝑠)-QSO from Class 𝐾₁₉

We are going to study dynamics of a 𝜉^(𝑠)-QSO 𝑉₂₈∶𝑆²→𝑆² taken from 𝐾₁₉: 𝑉₂₈∶𝑥^′₁=𝑥²₁+2𝑥₁1−𝑥₁,𝑥^′₂=𝑥²₃+2𝑎𝑥₂𝑥₃,𝑥^′₃=𝑥²₂+2(1−𝑎)𝑥₂𝑥₃, (50) 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.

Let 𝐴=(3−2𝑎−4+(2𝑎−1)²)/2(1−2𝑎) for any 𝑎≠1/2. Then 0≤𝐴≤1. In fact, one has that 20≤𝐴=2+(1−2𝑎)+4+(2𝑎−1)²≤1. (51)

Theorem 10. Let 𝑉₂₈∶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 Fix𝑉₂₈=𝑒₁,(10,𝐴,1−𝐴)𝑖𝑓𝑎≠2,𝑒₁,10,2,121𝑖𝑓𝑎=2. (52)(ii) One has that Per₂𝑉₂₈=𝑒₂,𝑒₃1𝑖𝑓𝑎≠2,Γ₁⧵10,2,121𝑖𝑓𝑎=2. (53)(iii) If 𝑎≠1/2, then 𝜔_𝑉28𝑥⁽⁰⁾=𝑒₂,𝑒₃𝑖𝑓𝑥₁⁽⁰⁾𝑒=0,₁𝑖𝑓𝑥₁⁽⁰⁾≠0. (54)(iv) If 𝑎=1/2, then 𝜔_𝑉28(𝑥⁽⁰⁾)={𝑒₁}.

Proof. Let 𝑥⁽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: 𝑥₁=𝑥²₁+2𝑥₁1−𝑥₁,𝑥₂=𝑥²₃+2𝑎𝑥₂𝑥₃,𝑥₃=𝑥²₂+2(1−𝑎)𝑥₂𝑥₃. (55)
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: 1−𝑥₂²+2𝑎𝑥₂1−𝑥₂=𝑥₂. (56)
Let 𝑎≠1/2. Then, one can find that solutions of (56) are 𝑥^±₂=(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.
Let 𝑎=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 𝑉²₂₈(𝑥)=𝑥 with the condition 𝑥₁=0. Then, by taking into account 𝑥₂+𝑥₃=1, we may get the following equation: 𝑥²₂+2(1−𝑎)𝑥₂1−𝑥₂²+2𝑎1−𝑥₂²+2𝑎𝑥₂1−𝑥₂×𝑥²₂+2(1−𝑎)𝑥₂1−𝑥₂=𝑥₂. (57)
Let 𝑎≠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).
Let 𝑎=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: 𝑉₂₈∣_Γ1∶𝑥^′₁𝑥=0,_′2=𝑥²₃+2𝑎𝑥₂𝑥₃,𝑥^′₃=𝑥²₂+2(1−𝑎)𝑥₂𝑥₃. (58)
Let us consider the function 𝑔_𝑎(𝑥₂)=(1−𝑥₂)²+2𝑎𝑥₂(1−𝑥₂), where 𝑎≠1/2. One can easily check that 𝑔_𝑎 is decreasing on [0,1]. This yields that 𝑔_𝑎⁽²⁾ is increasing on [0,1], as we already discussed that Fix(𝑔_𝑎)∩[0,1]={𝐴} and Fix(𝑔_𝑎⁽²⁾)∩[0,1]={0,𝐴,1}. This means that sets [0,𝐴] and [𝐴,1] are invariant under function 𝑔_𝑎⁽²⁾. We immediately find (see the above discussion (ii)) that 𝑔_𝑎⁽²⁾(𝑥₂)>𝑥₂ whenever 𝑥₂>𝐴 and 𝑔_𝑎⁽²⁾(𝑥₂)<𝑥₂ whenever 𝑥₂<𝐴. Consequently, one has that if 𝑥₂⁽⁰⁾∈[0,𝐴), then 𝜔_𝑔𝑎⁽²⁾(𝑥₂⁽⁰⁾)={0} and if 𝑥₂⁽⁰⁾∈(𝐴,1], then 𝜔_𝑔𝑎⁽²⁾(𝑥₂⁽⁰⁾)={1}. On the other hand, we have that 𝑉^(𝑛)₂₈𝑥⁽⁰⁾=0,𝑔_𝑎^(2𝑘)𝑥₂⁽⁰⁾,1−𝑔_𝑎^(2𝑘)𝑥₂⁽⁰⁾if𝑛=2𝑘,0,𝑔_𝑎^(2𝑘)𝑔_𝑎𝑥₂⁽⁰⁾,1−𝑔_𝑎^(2𝑘)𝑔_𝑎𝑥₂⁽⁰⁾if𝑛=2𝑘+1. (59) Therefore, we obtain that 𝜔_𝑉28(𝑥⁽⁰⁾)={𝑒₂,𝑒₃} if 𝑥₁⁽⁰⁾=0.
Let 𝑥₁⁽⁰⁾≠0. It is clear that 𝑥₁^(𝑛)=𝑓₁^(𝑛)(𝑥₁⁽⁰⁾). Therefore, due to Proposition 7(iv), we have that lim_𝑛→∞𝑥₁^(𝑛)=1. This means that 𝜔_𝑉28(𝑥⁽⁰⁾)⊂{𝑒₁}. Since 𝜔_𝑉28(𝑥⁽⁰⁾) is not empty, we obtain that 𝜔_𝑉28(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 lim_𝑛→∞𝑥₁^(𝑛)=1. Since 𝜔_𝑉28(𝑥⁽⁰⁾) is not empty, we obtain that 𝜔_𝑉28(𝑥⁽⁰⁾)={𝑒₁}.
This completes the proof.

7. Dynamics of 𝜉^(𝑠)-QSO from the Class 𝐾₁₇

We are going to highlight the dynamics of a 𝜉^(𝑠)-QSO 𝑉₂₅∶𝑆²→𝑆² taken from 𝐾₁₇: 𝑉₂₅∶𝑥^′₁=𝑥²₁+2𝑥₁1−𝑥₁,𝑥^′₂=𝑥²₂+2𝑎𝑥₂𝑥₃,𝑥^′₃=𝑥²₃+2(1−𝑎)𝑥₂𝑥₃, (60) 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. Let 𝑉₂₅∶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 Fix𝑉₂₅=𝑒₁,𝑒₂,𝑒₃1𝑖𝑓𝑎≠2,Γ₁1𝑖𝑓𝑎=2. (61)(ii) If 0≤𝑎<1/2, then 𝜔_𝑉25𝑥⁽⁰⁾=𝑒₃𝑖𝑓𝑥₁⁽⁰⁾𝑒=0,₁𝑖𝑓𝑥₁⁽⁰⁾≠0. (62)(iii) If 1/2<𝑎≤1, then 𝜔_𝑉25𝑥⁽⁰⁾=𝑒₂𝑖𝑓𝑥₁⁽⁰⁾𝑒=0,₁𝑖𝑓𝑥₁⁽⁰⁾≠0. (63)(iv) If 𝑎=1/2, then 𝜔_𝑉25(𝑥⁽⁰⁾)={𝑒₁}.

Acknowledgments

The authors acknowledge the MOHE Grant ERGS13-024-0057 and the IIUM Grant EDW B 13-019-0904 for the financial support.