An investigation on the recurrent function of the Pythagorean fuzzy cellular topological dynamical system

Abstract

This study is about Pythagorean fuzzy cellular topological dynamical system which is generated using Pythagorean fuzzy cellular space. A dynamical system receives input for a certain function and performs an iterative procedure for that same function. A continuous function can be employed in a topological dynamical system and the same function is iterated again and again. As it is an iterative process, there is a perspective that the views of every individual may be ambiguous or imprecise. To overcome this uncertainty, Pythagorean fuzzy sets are employed in dynamical system. Fixing a boundary on the Pythagorean fuzzy dynamical system culminates in a Pythagorean fuzzy cellular topological dynamical system. It is shown that the Pythagorean fuzzy cellular space is compact, normal and homeomorphic and it is called the Pythagorean fuzzy cellular topological dynamical system. The Pythagorean fuzzy sets in the Pythagorean fuzzy cellular topological dynamical system are iterated under the action of Pythagorean fuzzy cellular continuous map. Then, the Pythagorean fuzzy orbit* set is obtained. Additionally, it is discussed that the stipulated dynamical system is topologically transitive and different aspects of topological transitivity are investigated.

1. Introduction

Topology, the study of properties of space that are preserved under continuous deformations, plays a significant role in the field of fuzzy topological dynamical systems. The integration of fuzzy set theory with topology allows for a more flexible and nuanced representation of dynamic systems, particularly when dealing with uncertainties and imprecisions. Dynamical systems may be identified in numerous fields of research, notably algebra, biology, and the differential equations of classical mechanics in physics. For instance, enter a number into a scientific calculator and then continue repeatedly pressing one of the function keys to perform calculations. The term “discrete dynamical system” pertains to this iterative approach. Understanding the asymptotic or final behaviour of an iterative process is the fundamental objective of dynamical systems. Apart from iterated functions, there are numerous other dynamical systems. The effectiveness of a system on individual dynamics is related to points in the phase space, whereas its effectiveness on collective dynamics relates to subsets in the phase space. The graph of moving objects in two dimensions is referred to as the phase space in this context. This may also be used to fuzzy set dynamics (the system's influence on functions from the phase space to the interval [0, 1]). The simplest dynamical system is generated by taking the iterations of a single continuous map ($f:X\to X$) of a small (X) space into themselves. The set $\{x,f(x),f^2(x,...)\}$ is called the orbit of $x\in X$. A Pythagorean fuzzy topological dynamical system is an extension of the traditional topological dynamical systems framework that incorporates Pythagorean fuzzy sets. Pythagorean fuzzy sets are a generalization of classical fuzzy sets, where each element is assigned both a membership degree and a non-membership degree, such that the sum of the squares of the membership and non-membership degrees equals one. Incorporating Pythagorean fuzzy sets to topological dynamical systems enhances the capacity to represent and analyse dynamic systems in the presence of uncertainty and vagueness. In order to have better results in topological dynamical system Pythagorean fuzzy sets are considered. Here, the properties of the fuzzy topological dynamical system using Pythagorean fuzzy cellular is investigated. Cellular spaces can be defined in all the n-dimensional spaces [17]. There are various perspectives on cellular spaces, which leads to automata [15]. Here, the cellular spaces is defined using Pythagorean fuzzy sets, and they are called Pythagorean fuzzy cellular spaces. The Pythagorean fuzzy cellular space is compact, normal, and homeomorphism and Pythagorean fuzzy sets act as Pythagorean fuzzy cellular in the system. Periodic orbit is one of the distinctive features of topological dynamical systems, describes a specific sort of solution for a dynamical system, specifically one that repeats itself throughout time. Then, Pythagorean fuzzy cellular orbit is defined. The Pythagorean fuzzy cellular orbit is iterated and finding whether it is periodic. Again the property of dynamical system, transitivity is defined. Transitivity is the absence of a suitable closed invariant subset of X with a non-empty interior. Pythagorean fuzzy cellular topological transitivity is defined and properties are discussed. Hence, the defined Pythagorean fuzzy cellular topological dynamical system defined here is periodic, invariant and topologically transitive. Fig. 1 explains the outline of the study.

Figure 1.

Framework of the study.

2. Literature review

This section explores a detailed literature review on Pythagorean fuzzy cellular space (Table 1) and Pythagorean fuzzy cellular topological dynamical systems (Table 2, Table 3).

Table 1.

Literature review of Pythagorean fuzzy cellular space.

Author	Year	Source	Findings
Zadeh [18]	1965	Fuzzy sets	Definition of fuzzy sets, algebraic operation of fuzzy sets, convexity, boundedness is discussed
Chang.C.L [19]	1968	Fuzzy topological space	Definition of fuzzy topological space and basic concepts like open set, closed set, neighbourhood, interior set continuity and compactness
Merzerich [14]	1980	Cellular spaces	Definition of fuzzy sets, algebraic operation of fuzzy sets, convexity, boundedness
Coker.D [20]	1980	Intuitionistic fuzzy topological space	Definition of intuitionistic fuzzy topological space, its continuity, compactness, connectedness, Hausdorff space is discussed.
Leiderman [13]	2016	Lattices of homomorphisms and prolie group	Definition and properties of cellular spaces is discussed
Olgun M et al [15]	2019	Pythagorean fuzzy topological space	Pythagorean fuzzy topological space and its continuity is discussed.

Table 2.

Literature review of Pythagorean fuzzy cellular topological dynamical system.

Author	Year	Source	Findings
Bauer,Sigmund [2]	1975	Topological dynamics of transformations induced on the space of probability measures	Continuous transformation of a metric space is investigated.
Alseda et al [1]	1999	A splitting theorem for transitive maps	Continuous transitive maps are used here on locally connected compact metric spaces.
Kolyada et al [9]	2004	On minimality of non autonomous dynamical system	This study gave topological entropy for the dynamical system.
Kolyada [10]	2004	A survey of some aspects of dynamical system	Detailed study of dynamical system was examined.
Alessandro Fedeli [7]	2006	On discontinuous transitive maps	Detailed study of scenarios in which a dense orbit might exist for a discontinuous transitive map.
Kaki [8]	2013	Introduction to weakly transitive maps on topological spaces	Introduction to weakly transitive maps on topological spaces
Bilokopytov [3]	2014	Transitive maps on topological spaces	Properties of the topological dynamical system is discussed.
Lan et al [12]	2012	Some chaotic properties of fuzzy dynamical system	Dynamical system was extended to fuzzy sets and analysed the discrete fuzzy dynamical systems. Here metric space is taken and continuous mapping was defined on the metric space.

Table 3.

Literature review of Pythagorean fuzzy cellular topological dynamical system.

Author	Year	Source	Findings
Eisner et al [5]	2015	Operator theoretic aspects of ergodic theory	Dynamical system is introduced, which is composed of the sets X and ϕ:X → X.
Kaki [8]	2016	New types of lambda transitive and weakly lambda mixing sets	A new class of chaotic maps on locally compact Hausdorff spaces was explored.
Lan [11]	2016	Chaos in non-autonomous discrete fuzzy dynamical system	Chaotic properties of fuzzy dynamical system is discussed.
Xinxing et al [16]	2018	Topological dynamics of Zadeh's extension on upper semi continuous fuzzy sets	Topological dynamics of specific fuzzy set is investigated.
Devaney [4]	2018	An introduction to chaotic dynamical system	Topological conjugacy of the dynamical system is explained.
Martinez [17]	2021	Chaos in fuzzy dynamical system	The dynamics of (continuous and linear) operators on metrizable topological vector spaces (linear dynamics) is discussed.

3. Main contribution and motivation of the study

The dynamical system employs a variety of strategies. The key objective of the article is to establish a Pythagorean fuzzy cellular topological dynamical system in which ambiguity occurs in any system. Pythagorean fuzzy sets, which are cellular in nature and play a crucial part in this dynamical system, are employed in Pythagorean fuzzy topological dynamical systems. This environment is viewed as a Pythagorean fuzzy cellular topological dynamical system. The characteristics of the Pythagorean fuzzy cellular topological dynamical system are addressed in this work. The main contribution is listed as follows:

• –This paper introduces the concept of Pythagorean fuzzy cellular topological dynamical system, which is novel and represents an extension of existing concepts in fuzzy topological dynamical systems.
• –Pythagorean fuzzy cellular space is established. The continuity, compactness and normality of this space are defined, which paves the way to define Pythagorean fuzzy cellular topological dynamical system.
• –Pythagorean fuzzy cellular topological dynamical system is defined, which enhances the study of the topological dynamical system in a theoretical way. Pythagorean fuzzy cellular topological dynamical system involved in the study depicts how sets in a space evolve over time under the action of a continuous map.
• –This paper defines the Pythagorean fuzzy cellular orbit set in the Pythagorean fuzzy cellular topological dynamical system to demonstrate that it is topologically transitive. This topological transitivity is one of the most fundamental characteristic of any fuzzy topological dynamical system.

4. Structure of the paper

The paper is organised as follows:. Section 5, discusses the preliminary work for the study. Section 6, Pythagorean cellular space with examples, and Pythagorean fuzzy cellular normal. Section 7, defines the Pythagorean fuzzy cellular topological dynamical system with an example and investigates the periodicity and orbit of the defined topological dynamical system. Section 8, gives a detailed study of the topological transitivity of the Pythagorean fuzzy cellular topological dynamical system.

5. Preliminaries

This part introduces abbreviations and their expansions, as well as outlines the study's core principles.

PFS (Pythagorean fuzzy set), Pythagorean fuzzy topological space (PFTS), Pythagorean fuzzy interior fuzzy closure (PFC), Pythagorean fuzzy cellular ($P{F}_{cel}$)), PF(X)-collection of PFSs, Pythagorean fuzzy cellular orbit${}^{⁎}set(P{F}_{cel}{\mathcal{O}}^{⁎}S)$, Pythagorean fuzzy cellular topological dynamical system ($P{F}_{cel}$TDS)

Definition 5.1

[16] A Pythagorean fuzzy set (PFS) R of $X\ne 0$ is a pair $({\mu }_R,{\nu }_R)$ where ${\mu }_R$ and ${\nu }_R$ are fuzzy sets of X such that ${{\mu }_R}^2\left(x\right)+{{\nu }_R}^2\left(x\right)={r_R}^2\left(x\right)$ for any $xϵX$ where the fuzzy sets ${\mu }_R,{\nu }_R,r_R$ are the membership value, non-membership value, the strength of commitment at a point respectively.

Definition 5.2

[16] Let $X\ne \varnothing$ and τ be a family of PFS. If

• –$0_X,1_Xϵ\tau$
• –$A_{i}iϵI\subset \tau$, we have $\bigcup {A_i}_{iϵI}{A}_iϵ\tau$ where I is an arbitrary index set.
• –$A_1,A_2ϵ\tau$, we have $A_1\bigcap A_2ϵ\tau$, where $0_X=(0,1)$ and $1_X=(1,0)$, then $(X,\tau )$ is called a Pythagorean fuzzy topological space (PFTS).

Definition 5.3

[16] Let $S=({\mu }_S,{\nu }_S)$ and $R=({\mu }_R,{\nu }_R)$ be any two PFSs of a set X. Then,

• –$R\bigcup S=(max({\mu }_R,{\mu }_S),min({\nu }_R,{\nu }_S))$
• –$R\bigcap S=(min({\mu }_R,{\mu }_S),max({\nu }_R,{\nu }_S))$
• –$R^c=({\nu }_R,{\mu }_R)$
• –$R\subset S$ or $S\supset R$ if ${\mu }_R\le {\mu }_S$ and ${\nu }_R\ge {\nu }_S$.

Definition 5.4

[16] Let X be a PFTS and $R=({\mu }_R,{\nu }_R)$ be a PFS in X. Then the PFI and PFC are defined by,

• –$int(R)$= ⋃{$G|G$ is a PFOS in $\mathfrak{X}$ and $G\subseteq R$}
• –$cl(R)$= ⋂{$K|K$ is a PFCS in $\mathfrak{X}$ and$R\subseteq K$}

Definition 5.5

[16] Let A and U be any two PFSs in a PFTS. Then U is said to be a neighbourhood of A if there exists an open Pythagorean fuzzy subset E such that $A\subset E\subset U$.

Definition 5.6

[14] If every family of $G_{\delta }$-sets in X has a countable subfamily λ such that ∪λ is dense in ∪γ, then the topological space X is said to be ω-cellular, or, in symbols, $ce{l}_{\omega }(X)\le \omega$.

Definition 5.7

[6] A pair $(K;\varphi )$ is a topological dynamical system if K is a non-empty compact space and $\varphi :K\to K$ is continuous. If ϕ is surjective, then a topological system $(K;\varphi )$ is surjective; otherwise, the system is invertible, or a homeomorphism.

6. Pythagorean fuzzy cellular space ($P{F}_{cel}space$)

Throughout this paper Pythagorean fuzzy sets are denoted as $PF(X)$

This section defines $P{F}_{cel}$ space, $P{F}_{cel}$ compact, $P{F}_{cel}$ normal space and $P{F}_{cel}$ homeomorphism.

Definition 6.1

Pythagorean fuzzy cellular (resp., $P{F}_{cel}$) is defined as, if for every family $\mathrm{\Upsilon }=\{{\varsigma }_i\in PF(X);{\varsigma }_i^{\prime }s$ are PFS, and $i\in I$ } there exists a countable family $\mathrm{\Psi }=\{{\varpi }_i\in PF(X);{\varpi }_i^{\prime }s$ are PFS and $i\in I$ } such that $\mathrm{\Upsilon }\subseteq \mathrm{\Psi }$ and $cl(\cup {\varsigma }_i)$ = $\cup {\varpi }_i$ for every $i\in I$ where I is an index set.

A Pythagorean fuzzy topology ${\tau }_p$ on X is said to be Pythagorean fuzzy cellular space if it satisfies the condition of $P{F}_{cel}$ cellular defined above then $(X,{\tau }_{p_{cel}})$ is called $P{F}_{cel}$ space. Then the ordered pair $(X,{\tau }_{p_{cel}})$ is called $P{F}_{cel}$ space. Every member of $P{F}_{cel}$ space is called $P{F}_{cel}$ open set and its complement is $P{F}_{cel}$ closed set.

Example 6.2

Let $X=\{m,n\}{\varrho }_j\in PF(X)$ where $j=1,2,3,...8$,

$${\varrho }_1(m)=(0.2,0.8),{\varrho }_1(n)=(0.3,0.7){\varrho }_2(m)=(0.3,0.7),{\varrho }_2(n)=(0.4,0.7){\varrho }_3(m)=(0.4,0.7),{\varrho }_3(n)=(0.4,0.5){\varrho }_4(m)=(0.4,0.8),{\varrho }_4(n)=(0.3,0.5){\varrho }_5(m)=(0.8,0.2),{\varrho }_5(n)=(0.7,0.3){\varrho }_6(m)=(0.7,0.3),{\varrho }_6(n)=(0.7,0.4){\varrho }_7(m)=(0.7,0.4),{\varrho }_7(n)=(0.5,0.4){\varrho }_8(m)=(0.8,0.4),{\varrho }_8(n)=(0.5,0.3)$$

$\tau =\{0_X,1_X,{\varrho }_1,{\varrho }_2,{\varrho }_3,...,{\varrho }_8\}$ is a Pythagorean fuzzy topology on X. $\mathrm{\Upsilon }=\{{\varrho }_1,{\varrho }_2\}\subseteq \mathrm{\Psi }=\{{\varrho }_1,{\varrho }_2,{\varrho }_3,{\varrho }_4\}$. Here for every family ϒ there exists a countable family Ψ satisfies the Pythagorean fuzzy cellular condition. Hence $(X,{\tau }_{p_{cel}})$ is the Pythagorean fuzzy cellular space.

Definition 6.3

Let $(X,{\tau }_{p_{cel}})$ be a $P{F}_{cel}$ space and $Y\subset X$ and ${\mathcal{X}}_Y$ be the characteristic function of Y. Then ${\tau }_{p_{cel}}(Y)=\{\varrho \cap {\mathcal{X}}_Y,\varrho \in {\tau }_{p_{cel}}\}$ is a $P{F}_{cel}$ subspace. $(Y,{\tau }_{p_{cel}}(Y))$ is called the $P{F}_{cel}$ subspace of $(X,{\tau }_{p_{cel}})$. If ${\mathcal{X}}_Y$ is $P{F}_{cel}$ open set then $(Y,{\tau }_{p_{cel}}(Y))$ is called the $P{F}_{cel}$ open subspace of $(X,{\tau }_{p_{cel}})$.

Definition 6.4

Pythagorean fuzzy cellular closure ($P{F}_{cel}cl$) and Pythagorean fuzzy cellular interior ($P{F}_{cel}int$) of a PFS is defined by,

$$P{F}_{cel}cl(\xi )=\cap \{\eta :\xi \le \eta ;\eta \text{is}P{F}_{cel}\text{closed in}(X,{\tau }_{p_{cel}})\}P{F}_{cel}int(\xi )=\cup \{\sigma :\sigma \le \xi ;\sigma \text{is}P{F}_{cel}\text{open in}(X,{\tau }_{p_{cel}})\}$$

Definition 6.5

Let $(Y,{\tau }_{p_{cel}})$ and $(Z,{\sigma }_{p_{cel}})$ be any two $P{F}_{cel}$ spaces. A function $\mathfrak{p}:(Y,{\tau }_{p_{cel}})\to (Z,{\sigma }_{p_{cel}})$ is called $P{F}_{cel}$ continuous if and only if the inverse image of each $P{F}_{cel}$ open set in $(Z,{\sigma }_{p_{cel}})$ is $P{F}_{cel}$ open set in $(Y,{\tau }_{p_{cel}})$.

Definition 6.6

Let $(X,{\tau }_{p_{cel}})$ be a $P{F}_{cel}$ space. A collection ϱ of the subsets of $(X,{\tau }_{p_{cel}})$ is said to be $P{F}_{cel}$ cover of $(X,{\tau }_{p_{cel}})$ if the union of the elements of ϱ is equal $1_X$ ($\cup {\varrho }_i=1_X$).

Definition 6.7

Let $(X,{\tau }_{p_{cel}})$ be the $P{F}_{cel}$ space. The $P{F}_{cel}$ subcover of the $P{F}_{cel}$ cover ϱ is a subcollection of ϱ which is a $P{F}_{cel}$ cover.

Definition 6.8

A $P{F}_{cel}$ space $(X,{\tau }_{p_{cel}})$ is Pythagorean fuzzy cellular compact if every cover of $(X,{\tau }_{p_{cel}})$ by $P{F}_{cel}$ open set in $(X,{\tau }_{p_{cel}})$ contains the finite sub cover. That is if ${\varrho }_i$ is $P{F}_{cel}$ open in $(X,{\tau }_{p_{cel}})$, for every $i\in I$ where I is an index set and $\cup {\varrho }_i$ = $1_X$, then there are finitely many indices $i_1,i_2,...i_n\in I$ such that $\cup {\varrho }_{ij}=1_X$

Definition 6.9

A $P{F}_{cel}$ space $(X,{\tau }_{p_{cel}})$ is called $P{F}_{cel}$ normal space if for any $P{F}_{cel}$ closed sets $\mathfrak{S}$ and a $P{F}_{cel}$ open set $\mathfrak{T}$ in $(X,{\tau }_{p_{cel}})$ such that $\mathfrak{S}\subseteq \mathfrak{T}$, there exists a $P{F}_{cel}$ open set $\mathfrak{V}$ such that $\mathfrak{S}\subseteq \mathfrak{V}\subseteq P{F}_{cel}(\mathfrak{V})\subseteq \mathfrak{T}$.

Example 6.10

Let $X=\{m,n\}$ ${\varrho }_j\in PF(X)$ where $j=1,2,3,...8$,

$${\varrho }_1(m)=(0.2,0.8),{\varrho }_1(n)=(0.3,0.7){\varrho }_2(m)=(0.3,0.7),{\varrho }_2(n)=(0.4,0.7){\varrho }_3(m)=(0.4,0.7),{\varrho }_3(n)=(0.4,0.5){\varrho }_4(m)=(0.4,0.8),{\varrho }_4(n)=(0.3,0.5){\varrho }_5(m)=(0.8,0.2),{\varrho }_5(n)=(0.7,0.3){\varrho }_6(m)=(0.7,0.3),{\varrho }_6(n)=(0.7,0.4){\varrho }_7(m)=(0.7,0.4),{\varrho }_7(n)=(0.5,0.4){\varrho }_8(m)=(0.8,0.4),{\varrho }_8(n)=(0.5,0.3)$$

Clearly, $\tau =\{0_X,1_X,{\varrho }_1,{\varrho }_2,{\varrho }_3,...,{\varrho }_8\}$ is a Pythagorean fuzzy topology on X. $(X,{\tau }_{p_{cel}})$ is the $P{F}_{cel}$ space. Let the $P{F}_{cel}$ closed set ${\varrho }_5^c\subseteq {\varrho }_7$ there exists a $P{F}_{cel}$ open set ${\varrho }_3$ such that $cl({\varrho }_3)\subseteq {\varrho }_7$. Therefore $(X,{\tau }_{p_{cel}})$ is $P{F}_{cel}$ normal space.

Definition 6.11

A Pythagorean fuzzy cellular mapping $\mathfrak{p}:(X,{\tau }_{p_{cel}})(Y,{\tau }_{p_{cel}})$ is called Pythagorean fuzzy cellular homeomorphism if $\mathfrak{p}$ and ${\mathfrak{p}}^{-1}$ are $P{F}_{cel}$ continuous.

7. Pythagorean fuzzy cellular topological dynamical system ($P{F}_{cel}$TDS)

In this section of $P{F}_{cel}$ TDS is defined and it is verified that this system is periodic, invariant and existence of orbit for this system. Topological transitivity is defined for this system. Properties of $P{F}_{cel}TDS$ is discussed.

Definition 7.1

A $P{F}_{cel}$ space $(X,{\tau }_{p_{cel}})$ is called $P{F}_{cel}TDS$ $(X,{\tau }_{p_{cel}},{\tau }_D)$ if it is

i) $P{F}_{cel}$ compact and $P{F}_{cel}$ normal space.

ii) $P{F}_{cel}$ homeomorphism.

Example 7.2

Let $X=\{m,n\}$ ${\varrho }_j\in PF(X)$ where $j=1,2,3,...8$,

$${\varrho }_1(m)=(0.2,0.8),{\varrho }_1(n)=(0.3,0.7){\varrho }_2(m)=(0.3,0.7),{\varrho }_2(n)=(0.4,0.7){\varrho }_3(m)=(0.4,0.7),{\varrho }_3(n)=(0.4,0.5){\varrho }_4(m)=(0.4,0.8),{\varrho }_4(n)=(0.3,0.5){\varrho }_5(m)=(0.8,0.2),{\varrho }_5(n)=(0.7,0.3){\varrho }_6(m)=(0.7,0.3),{\varrho }_6(n)=(0.7,0.4){\varrho }_7(m)=(0.7,0.4),{\varrho }_7(n)=(0.5,0.4){\varrho }_8(m)=(0.8,0.4),{\varrho }_8(n)=(0.5,0.3)$$

$\tau =\{0_X,1_X,{\varrho }_1,{\varrho }_2,{\varrho }_3,...,{\varrho }_8\}$ is a Pythagorean fuzzy topology on X. $(X,{\tau }_{p_{cel}})$ is the $P{F}_{cel}$ space. Then $P{F}_{cel}$ space is a compact and $P{F}_{cel}$ normal space and $\mathfrak{p}:(X,{\tau }_{p_{cel}})\to (X,{\tau }_{p_{cel}})$ is a $P{F}_{cel}$ homeomorphism since ${\mathfrak{p}}^{-1}$ is a $P{F}_{cel}$ continuous. Thus, $(X,{\tau }_{p_{cel}},{\tau }_D)$ is a $P{F}_{cel}TDS$.

Definition 7.3

Let $(X,{\tau }_{p_{cel}})$ be a non-empty set and let $\mathfrak{p}:(X,{\tau }_{p_{cel}})\to (X,{\tau }_{p_{cel}})$ be any function in $(X,{\tau }_{p_{cel}})$. Then the $P{F}_{cel}$ orbit*($P{F}_{cel}{\mathcal{O}}^{⁎}$) of ℧ under the function $\mathfrak{p}$ is $P{F}_{cel}=\{\mho ,\mathfrak{p}(\mho ),{\mathfrak{p}}^2(\mho ,...)\}$.

The $P{F}_{cel}$ orbit* set $P{F}_{cel}{\mathcal{O}}^{⁎}(\mho )$ of ℧ under $\mathfrak{p}$ is defined by, $P{F}_{cel}{\mathcal{O}}^{⁎}(\mho )=\{\mho \bigcap \mathfrak{p}(\mho )\bigcap {\mathfrak{p}}^2(\mho ,...)\}$.

Definition 7.4

Let $(X,{\tau }_{p_{cel}})$ be a $P{F}_{cel}$ space. Let $\mathfrak{p}:(X,{\tau }_{p_{cel}})\to (X,{\tau }_{p_{cel}})$ be any function. The $P{F}_{cel}{\mathcal{O}}^{⁎}$ under the function $\mathfrak{p}$ in $P{F}_{cel}$ space is called $P{F}_{cel}$ open set. Its complement is called $P{F}_{cel}$ closed set.

Definition 7.5

Let $(X,{\tau }_{p_{cel}})$ be a $P{F}_{cel}$ space. Let $\mathfrak{p}:(X,{\tau }_{p_{cel}})\to (X,{\tau }_{p_{cel}})$ be any function. The $P{F}_{cel}{\mathcal{O}}^{⁎}$s then $P{F}_{cel}{\mathcal{O}}^{⁎}$ interior and closure of α is defined as,

$$P{F}_{cel}int(\alpha )=\cup \{\alpha :\omega \subseteq \alpha ;\omega \text{is a}P{F}_{cel}\text{open set}\}.P{F}_{cel}{\mathcal{O}}^{⁎}\text{closure of}\alpha \text{is defined as},P{F}_{cel}cl(\alpha )=\cap \{\alpha :\alpha \subseteq \delta ;\delta \text{is a}P{F}_{cel}\text{closed set}\}.$$

Definition 7.6

Let $(X,{\tau }_{p_{cel}})$ be a $P{F}_{cel}$ space under the mapping p if the elements in the $P{F}_{cel}$ space is both open $P{F}_{cel}$ open set and $P{F}_{cel}$ closed set. Then the elements are called $P{F}_{cel}$ clopen sets.

Definition 7.7

Let $(X,{\tau }_{p_{cel}})$ and $(Y,{\sigma }_{p_{cel}})$ be a $P{F}_{cel}$ space. Let $\mathfrak{p}:(X,{\tau }_{p_{cel}})\to (X,{\tau }_{p_{cel}})$ by any function. A function $\mathfrak{g}:(X,{\tau }_{p_{cel}})\to (Y,{\sigma }_{p_{cel}})$ is defined to be $P{F}_{cel}{\mathcal{O}}^{⁎}$ continuous if the inverse image of every $P{F}_{cel}$ open set in $(Y,{\sigma }_{p_{cel}})$ is $P{F}_{cel}$ orbit* open set under $\mathfrak{p}$ in $(X,{\tau }_{p_{cel}})$.

Proposition 7.8

Let$(X,{\tau }_{p_{cel}})$be a$P{F}_{cel}$space under bijective mapping. If ℧ is a $P{F}_{cel}{\mathcal{O}}^{⁎}$ open set under the mapping $\mathfrak{p}$, then $\mathfrak{p}(\mho )=\mho$.

Proof

Let $(X,{\tau }_{p_{cel}})$ be a $P{F}_{cel}$ space and $\mathfrak{p}$ be any mapping. There are three cases.

Case 1. If $\mathfrak{p}(x_i)=x_j$, $x_i,x_j\in X$ and $i\ne j$ for every $i,j\in I$. Suppose $X=\{x_1,x_2\}$, $\mathfrak{p}:(X,{\tau }_{p_{cel}})\to (X,{\tau }_{p_{cel}})$ defined as $p(x_1)=x_2$, $p(x_2)=x_1$. Let ℧ be a $P{F}_{cel}$ open set under the mapping $\mathfrak{p}$. Then there exists $P{F}_{cel}{\mathcal{O}}^{⁎}\varrho \in PF(X)$ such that $P{F}_{cel}{\mathcal{O}}^{⁎}(\varrho )=\{\varrho \cap \mathfrak{p}\cap (\varrho ){\mathfrak{p}}^2(\varrho ),...\}=\mho$.

Let $\varrho =\{(m_1,n_1),(m_2,n_2)\}$ where $x_1,x_2\in X,m_1,n_1,m_2,n_2\in PF(X)$. $\mathfrak{p}(\varrho )=\{(m_2,n_2),(m_1,n_1)\}$, ${\mathfrak{p}}^2(\varrho )=\{(m_1,n_1),(m_2,n_2)\}...$

$P{F}_{cel}(\varrho )=\{[(min(m_1,m_2,m_1...),max(n_1,n_2,n_1)...]$,

$[min(m_2,m_1,m_2...),max(n_2,n_1,n_2...)]\}=\mho$

Hence, by definition $\mathfrak{p}(\mho )=\mho$.

Case 2. If $\mathfrak{p}(x_i)=x_j$, $x_i,x_j\in X$ and $i=j$ for some $i,j\in I$. In this case least number of elements must be three elements. If $X=\{x_1,x_2,x_3\}$ the mapping $p:(X,{\tau }_{p_{cel}})\to (X,{\tau }_{p_{cel}})$ can be defined as $\mathfrak{p}(x_1)=x_1$,$\mathfrak{p}(x_2)=x_3$, $\mathfrak{p}(x_3)=x_2$. Let ℧ be a $P{F}_{cel}{\mathcal{O}}^{⁎}$ open set under the mapping $\mathfrak{p}$. Then there exists a $P{F}_{cel},\varrho \in PF(X)$ such that $P{F}_{cel}{\mathcal{O}}^{⁎}(\varrho )=\{\varrho \cap \mathfrak{p}(\varrho ){\mathfrak{p}}^2(\varrho ),...\}=\mho$.

Let $\varrho =\{(m_1,n_1),(m_2,n_2)\}$ where $x_i\in X{m}_i,n_i\in PF(X),i=1,2,3$. Then by the definition of $\mathfrak{p}$, we get, $\mathfrak{p}(\varrho )=\{(m_1,n_1),(m_3,n_3),(m_2,n_2)\}$

Q${\mathfrak{p}}^2(\varrho )=\{(m_1,n_1),(m_2,n_2),(m_3,n_3)\}$

$P{F}_{cel}{\mathcal{O}}^{⁎}(\varrho )=\{(m_1,n_1)[(min(m_3,n_2,...),max(n_3,n_2,...)]$,

$[min(m_2,m_3,...),max(n_2,n_3...)]\}=\mho$

Hence, by definition $\mathfrak{p}(\mho )=\mho$.

Case 3. If $\mathfrak{p}$ is the identity mapping the proof is obvious. □

Proposition 7.9

Let$(X,{\tau }_{p_{cel}})$be a$P{F}_{cel}$space under constant mapping. If ℧ is a $P{F}_{cel}{\mathcal{O}}^{⁎}$ open set under the mapping $\mathfrak{p}$, then $\mathfrak{p}(\mho )=\mho$.

Proof

Let $(X,{\tau }_{p_{cel}})$ be a $P{F}_{cel}$ space and let ℧ be a $P{F}_{cel}$ orbit open set under the mapping $\mathfrak{p}$. The there exists $\varrho =\{(m_i,n_i);m_i,n_i\in PF(X),i\in I\}$ such that $P{F}_{cel}{\mathcal{O}}^{⁎}(\varrho )=\mho$. Since $\mathfrak{p}$ is a constant mapping, this implies there exists a fixed element $x_k\in X$ such that $p(x_i)=x_k$ for all $x_i\in X$ and $i\in I$. Now forming the definition of $\mathfrak{p}(\varrho )$ for all $x_i\in X$. we have

$$\mathfrak{p}(\varrho )(x_j)=\{\begin{array}{cc}\bigcup _{\mathfrak{p}(x_i)=x_j}\varrho (x_i),\bigcap _{\mathfrak{p}(x_i)=x_j}\varrho (x_i)\hfill & if{\mathfrak{p}}^{-1}(x_i)\ne 0_X\hfill \\ 0\hfill & otherwise\hfill \end{array}$$

Thus,

$$\mathfrak{p}(\varrho )(x_j)=\{\begin{array}{cc}\underset{i\in I}{\max }\varrho (x_i),\underset{i\in I}{\min }\varrho (x_i)\hfill & if{x}_j=x_k\hfill \\ 0\hfill & if{x}_j\ne x_k\hfill \end{array}$$

Therefore,

$$\mathfrak{p}(\varrho )=\{\underset{i\in I}{\max }\varrho (x_i),\underset{i\in I}{\min }\varrho (x_i)\}.$$

This means $\mathfrak{p}(\varrho )$ is a $P{F}_{cel}$ point in $(X,{\tau }_{p_{cel}})$ with support $x_k$, degree $\underset{i\in I}{\min }\varrho (x_i)$ and degree $\underset{i\in I}{\max }\varrho (x_i)$.

By the same way we have ${\mathfrak{p}}^2(\varrho )$ = $\{\underset{i\in I}{\max }\varrho (x_i),\underset{i\in I}{\min }\varrho (x_i)\}$,

$${\mathfrak{p}}^3(\varrho )=\{\underset{i\in I}{\max }\varrho (x_i),\underset{i\in I}{\min }\varrho (x_i)\},\varrho \{(m_1,n_1),(m_2,n_2),...,(m_k,n_k)\},\mathfrak{p}(\varrho )=\{(0,0)(0,0),...max\varrho (x_i),min\varrho (x_j)\}{\mathfrak{p}}^2(\varrho )=\{(0,0)(0,0),...max\varrho (x_i),min\varrho (x_j)\},{\mathfrak{p}}^3(\varrho )=\{(0,0)(0,0),...max\varrho (x_i),min\varrho (x_j)\}$$

Hence, from the definition of $\mathfrak{p}$, we get $\mathfrak{p}(\mho )=\mho$. □

Remark 7.10

Let $(X,{\tau }_{p_{cel}})$ be a $P{F}_{cel}$ space and $\mathfrak{p}:(X,{\tau }_{p_{cel}})\to (X,{\tau }_{p_{cel}})$ be any mapping such that either $\mathfrak{p}$ is a bijective mapping or $\mathfrak{p}$ is a constant mapping ℧ is $P{F}_{cel}{\mathcal{O}}^{⁎}$ open set under the mapping $\mathfrak{p}$ then $\mathfrak{p}(\mho )=\mho$.

Proposition 7.11

Let$(X,{\tau }_{p_{cel}})$be a$P{F}_{cel}$space and$\mathfrak{p}:(X,{\tau }_{p_{cel}})\to (X,{\tau }_{p_{cel}})$be a mapping. If ℧ is a $P{F}_{cel}{\mathcal{O}}^{⁎}$ open set under the mapping $\mathfrak{p}$, then $\mathfrak{p}(\mho )=\mho$.

Proof

The proof follows from the above remark. From the remark we have $\mathfrak{p}(\mho )=\mho$ this implies ${\mathfrak{p}}^2(\mho )=\mathfrak{p}(\mathfrak{p}(\mho ))=\mho$,${\mathfrak{p}}^3(\mho )=\mathfrak{p}({\mathfrak{p}}^2(\mho ))=\mho$...Hence $\mathfrak{p}(\mho )=\mho$. □

Proposition 7.12

Let$(X,{\tau }_{p_{cel}})$be a$P{F}_{cel}$space and$\mathfrak{p}:(X,{\tau }_{p_{cel}})\to (X,{\tau }_{p_{cel}})$be a mapping. If ℧ is a $P{F}_{cel}{\mathcal{O}}^{⁎}$ open set under the mapping $\mathfrak{p}$, then $\mathfrak{p}(\mho )=\mho$.

Proof

The proof follows from the above remark. From the remark we have $\mathfrak{p}(\mho )=\mho$ this implies ${\mathfrak{p}}^2(\mho )=\mathfrak{p}(\mathfrak{p}(\mho ))=\mho$,${\mathfrak{p}}^3(\mho )=\mathfrak{p}({\mathfrak{p}}^2(\mho ))=\mho$...Hence $\mathfrak{p}(\mho )=\mho$. □

Proposition 7.13

Let $(X,{\tau }_{p_{cel}})$ be a $P{F}_{cel}$ space and $\mathfrak{p}:(X,{\tau }_{p_{cel}})\to (X,{\tau }_{p_{cel}})$ be a mapping. If ${\mho }_1,{\mho }_2$ is a $P{F}_{cel}{\mathcal{O}}^{⁎}$ open sets under the mapping $\mathfrak{p}$ , then $\mathfrak{p}({\mho }_1\cap {\mho }_2)=\mathfrak{p}({\mho }_1)\cap \mathfrak{p}({\mho }_2)$ .

Proof

There are three cases here,

Case 1 Suppose $\mathfrak{p}$ is bijective. $\mathfrak{p}(x_i)=x_j,x_i,x_j\in Xi\ne j$ for $i,j\in I$. Let ${\mho }_1$ and ${\mho }_2$ be $P{F}_{cel}{\mathcal{O}}^{⁎}$ open sets under the mapping p. Then there exists ${\varrho }_1,{\varrho }_2\in PF(X)$ defined as

$${\varrho }_1=(m_i,n_i),m_i,n_i\in PF(X),i\in I,{\varrho }_2=(s_i,r_i),s_i,r_i\in PF(X),i\in I,$$

such that $P{F}_{cel}{\mathcal{O}}^{⁎}({\varrho }_1)={\mho }_1$ $P{F}_{cel}{\mathcal{O}}^{⁎}({\varrho }_2)={\mho }_2$. From Proposition 7.8,

$$P{F}_{cel}{\mathcal{O}}^{⁎}({\varrho }_1)=min(m_i),max(n_i)={\mho }_1,P{F}_{cel}{\mathcal{O}}^{⁎}({\varrho }_2)=min(s_i),max(r_i)={\mho }_2$$

Thus $P{F}_{cel}{\mathcal{O}}^{⁎}({\varrho }_1)\cap P{F}_{cel}{\mathcal{O}}^{⁎}({\varrho }_2)=\{min(m_i,s_i),max(n_i,s_i)\}$. Hence $\mathfrak{p}({\varrho }_1\cap {\varrho }_2)={\mho }_1\cap {\mho }_2$, ${\mathfrak{p}}^2({\varrho }_1\cap {\varrho }_2)={\mho }_1\cap {\mho }_2$ ${\mathfrak{p}}^3({\varrho }_1\cap {\varrho }_2)={\mho }_1\cap {\mho }_2$....

Therefore by definition, $P{F}_{cel}{\mathcal{O}}^{⁎}({\mho }_1\cap {\mho }_2)$ and Proposition 7.8. $\mathfrak{p}({\mho }_1\cap {\mho }_2)=\mathfrak{p}({\mho }_1)\cap \mathfrak{p}({\mho }_2)$.

Case 2 Suppose $\mathfrak{p}$ is bijective. $\mathfrak{p}(x_i)=x_j,x_i,x_j\in Xi=j$ for $i,j\in I$. Let ${\mho }_{1}and{\mho }_2$ be $P{F}_{cel}{\mathcal{O}}^{⁎}$ open sets under the mapping p. Then there exists ${\varrho }_1,{\varrho }_2\in PF(X)$ defined as

$${\varrho }_1=(m_i,n_i),m_i,n_i\in PF(X),i\in I.{\varrho }_2=(s_i,r_i),s_i,r_i\in PF(X),i\in I.$$

From Proposition 7.8 $P{F}_{cel}{\mathcal{O}}^{⁎}({\varrho }_1)={\mho }_1$, $P{F}_{cel}{\mathcal{O}}^{⁎}({\varrho }_2)={\mho }_2$

$$\begin{gathered} P{F}_{cel}{\mathcal{O}}^{⁎}({\varrho }_1)=\{(m_i,n_i)fori=jmin(m_i),max(n_i)fori\ne j\}={\mho }_1 \\ P{F}_{cel}{\mathcal{O}}^{⁎}({\varrho }_2)=\{(s_i,r_i)fori=jmin(s_i),max(r_i)fori\ne j\}={\mho }_2 \end{gathered}$$

Thus $P{F}_{cel}{\mathcal{O}}^{⁎}({\varrho }_1)\cap {\mho }_2=\{min(m_i,s_i),max(n_i,r_i)\}$.

Hence $\mathfrak{p}({\mho }_1\cap {\mho }_2)=\mathfrak{p}({\mho }_1)\cap \mathfrak{p}({\mho }_2)$.

Case 3

If the mapping $\mathfrak{p}$ is constant mapping. The proof is obvious. □

Proposition 7.14

Let $(X,{\tau }_{p_{cel}})$ be a $P{F}_{cel}$ space and $\mathfrak{p}:(X,{\tau }_{p_{cel}})\to (X,{\tau }_{p_{cel}})$ be a mapping. If ${\mho }_1,{\mho }_2$ is a $P{F}_{cel}{\mathcal{O}}^{⁎}$ open sets under the mapping $\mathfrak{p}$ , then $\mathfrak{p}({\mho }_1\cup {\mho }_2)=\mathfrak{p}({\mho }_1)\cup \mathfrak{p}({\mho }_2)$ .

Proof

The proof is simple. □

Definition 7.15

Let $(X,{\tau }_{p_{cel}},T_D)$ be a $P{F}_{cel}TDS$ and let ϱ be a $P{F}_{cel}$ of X is a $P{F}_{cel}$ invariant set if $\mathfrak{p}(\varrho )=\varrho$ and every $P{F}_{cel}$ invariant set is $P{F}_{cel}P$.

Definition 7.16

A PFS in $(X,{\tau }_{p_{cel}})$ is called $P{F}_{cel}$ point if it takes the value zero for all $y\in X$ except one, say $x\in X$. If its value at x is $\lambda (0\le \lambda \le 1)$. The $P{F}_{cel}$ point is denoted by $x_{\lambda }$, where the point x is called its support.

Definition 7.17

Let $(X,{\tau }_{p_{cel}},T_D)$ be a $P{F}_{cel}$TDS and $P{F}_{cel}$point ${\varrho }_{\lambda }\in PF(X)$ is said to be a recurrent point if p for any $P{F}_{cel}$ orbit neighbourhood $U=U({\varrho }_{\lambda })$ of ${\varrho }_{\lambda }$ then there exists positive integer N such that $p^n{\varrho }_{\lambda }\in U({\varrho }_{\lambda })$. The collection of recurrent $P{F}_{cel}$ points is denoted by $P{F}_{cel}Rec(p)$.

Definition 7.18

Let ${\varrho }_{\lambda }\in PF(X)$ be any $P{F}_{cel}$ point is said to be recurrent $P{F}_{cel}$point of the $P{F}_{cel}$TDS $(X,{\tau }_{p_{cel}},T_D)$ whenever for any $P{F}_{cel}$ open neighbourhood, $\omega =\omega ({\varrho }_{\lambda })$ of ${\varrho }_{\lambda }$ then there exists an infinite set of positive integer n such that $p^n({\varrho }_{\lambda })\in \omega ({\varrho }_{\lambda })$. The collection of recurrent $P{F}_{cel}$ points is denoted by $P{F}_{cel}Rec(p)$.

8. Pythagorean fuzzy cellular topological transitivity

This section examines the fundamental attribute of a dynamical system called topological transitivity. Pythagorean fuzzy cellular topological dynamical systems investigates this fundamental topological transitivity property in this section.

Definition 8.1

Let ℧ be a PFS in $(X,{\tau }_{p_{cel}})$ then

i) ℧ is said to be $P{F}_{cel}$ dense if there exists no $P{F}_{cel}$ closed set $\beta \ne 0_X\in PF(X)$ such that $\mho \subset \beta \subset 1_X$. That is $P{F}_{cel}cl(\mho )=1_X$.

ii) ℧ is said to be $P{F}_{cel}$ no where dense if there exists no $P{F}_{cel}$ open set $\beta \ne 0_X\in PF(X)$ such that $\beta \subset P{F}_{cel}cl(\mho )$. That is $P{F}_{cel}int(P{F}_{cel}cl(\mho ))=0_X$.

Definition 8.2

Let $(X,{\tau }_{p_{cel}})$ be a $P{F}_{cel}$ compact. A homeomorphism $\mathfrak{p}:(X,{\tau }_{p_{cel}})\to (X,{\tau }_{p_{cel}})$ is called $P{F}_{cel}$ topological transitive, if there is some $\mho \in PF(X)$ such that $P{F}_{cel}{\mathcal{O}}^{⁎}S$ is $P{F}_{cel}$ dense in $(X,{\tau }_{p_{cel}})$ that is $P{F}_{cel}cl(P{F}_{cel}\mathcal{O}(\mho ))=1_X$

Definition 8.3

Let $\mathfrak{p}:(X,{\tau }_{p_{cel}})\to (X,{\tau }_{p_{cel}})$ is continuous, then $P{F}_{cel}$ topological transitivity is defined as an image of $P{F}_{cel}{\mathcal{O}}^{⁎}S$ ℧. If $P{F}_{cel}{{\mathcal{O}}^{⁎}}^{+}(\mho )$ is $P{F}_{cel}$ dense in $(X,{\tau }_{p_{cel}})$ for some $\mho \in PF(X)$ then it is called $P{F}_{cel}$ positive topological transitivity.

Definition 8.4

Let $(X,{\tau }_{p_{cel}},T_D)$ be a $P{F}_{cel}$TDS. Then the $P{F}_{cel}$ ϱ of $(X,{\tau }_{p_{cel}})$ is said to be

i)$P{F}_{cel}$ invariant if $\mathfrak{p}(\varrho )<\varrho$.

ii) $P{F}_{cel}$ inversely invariant if ${\mathfrak{p}}^{-1}(\varrho )<\varrho$.

Definition 8.5

Let $(X,{\tau }_{p_{cel}},T_D)$ be the $P{F}_{cel}$TDS and let $(X,{\tau }_{p_{cel}})$ be a $P{F}_{cel}$ compact topological space. For any pair $\varrho \ne 0_X$ and $\gamma \ne 0_X\in PF(X)$ is said to be $P{F}_{cel}$ topological transitivity $(P{F}_{cel}TT)$.

i) ${(P{F}_{cel}TT)}_N$, if there exists a positive integer n, such that ${\mathfrak{p}}^n(\varrho )\cap \gamma \ne 0_X$.

ii) ${(P{F}_{cel}TT)}_{N_0}$, if there exists a non-negative integer n, such that ${\mathfrak{p}}^n(\varrho )\cap \gamma \ne 0_X$.

Remark 8.6

Let $(X,{\tau }_{p_{cel}},T_D)$ be the $P{F}_{cel}$TDS. Then this system satisfies the property:

i) Property ${(P{F}_{cel}TT)}_N$ is called the $P{F}_{cel}$ topological N transitivity.

ii) property ${(P{F}_{cel}TT)}_{N_0}$ is called the $P{F}_{cel}$ topological $N_0$ transitivity.

iii)If both i) and ii) are true, then $P{F}_{cel}$ topological Z transitivity.

That is for any pair of $P{F}_{cel}$ open sets ϱ and $\gamma \in PF(X)$ and integer n such that ${\mathfrak{p}}^n(\varrho )\cap \gamma \ne 0_X$.

Definition 8.7

The $P{F}_{cel}{\mathcal{O}}^{⁎}S$ $\mho \in PF(X)$ of $(X,{\tau }_{p_{cel}})$ is defined as the set $P{F}_{cel}^{+}(\mho )=\bigcup _{n\in N_0}{\mathfrak{p}}^n(\mho )$. The collection of all pre-images of $\mho \in PF(X)$ of $(X,{\tau }_{p_{cel}})$ under the function is denoted by $P{F}_{cel}^{-}(\mho )=\bigcup _{n\in Z_0^{-}}{\mathfrak{p}}^n(\mho )$ and their union $P{F}_{cel}^{-}(\mho )\cup P{F}_{cel}^{+}(\mho )$ is denoted by $P{F}_{cel}^{\pm }(\mho )=\bigcup _{n\in Z}{\mathfrak{p}}^n(\mho )$.

Proposition 8.8

Let $(X,{\tau }_{p_{cel}})$ be a $P{F}_{cel}$ compact topological space and for a $P{F}_{cel}$ homeomorphism $\mathfrak{p}:(X,{\tau }_{p_{cel}})\to (X,{\tau }_{p_{cel}})$ the following are equivalent:

i) $P{F}_{cel}$ is topologically transitive.

ii) If β is $P{F}_{cel}$ closed sets (resp., open sets) of $(X,{\tau }_{p_{cel}})$ and $\mathfrak{p}(\beta )=\beta$ then either $\beta =1_X$ (resp., $0_X$ ) or β is nowhere $P{F}_{cel}$ dense (resp., $P{F}_{cel}$ dense).

iii) If $\beta \ne 0_X$ , $\varrho \ne 0_X\in PF(X)$ are Pythagorean fuzzy open sets, then there exists $n\in Z$ with ${\mathfrak{p}}^n(\beta )\cap \varrho \ne 0_X$ .

iv) The collection $\{\varrho \in PF(X):P{F}_{cel}cl(P{F}_{cel}{\mathcal{O}}^{⁎}S))=1_X\}$ is $P{F}_{cel}$ dense.

Proof

i)⇒ ii) Assume that $\mathfrak{p}$ is $P{F}_{cel}$ topological transitive. By definition, if there exists some $\varrho \in PF(X)$ such that $P{F}_{cel}{\mathcal{O}}^{⁎}(\varrho )$ is $P{F}_{cel}$ dense in $(X,{\tau }_{p_{cel}})$. $P{F}_{cel}cl(P{F}_{cel}\mathcal{O}(\varrho ))=1_X$.

Let β be a $P{F}_{cel}$ closed set of $(X,{\tau }_{p_{cel}})$ and $\mathfrak{p}(\beta )=\beta$. To prove that $\beta =1_X$ or β is nowhere $P{F}_{cel}$ dense. If there is $P{F}_{cel}$ open set $0_X\ne \xi <\beta$ then there exists $n\in Z$ such that ${\mathfrak{p}}^n(\xi )<\varrho <\beta$. So, $P{F}_{cel}$ orbit set of ϱ, $P{F}_{cel}{\mathcal{O}}^{⁎}(\varrho )<\beta$. Hence $1_X<\beta$. We get $\beta =1_X$, otherwise there is no $P{F}_{cel}$ open sets $\xi <\beta$ such that $\xi <P{F}_{cel}cl(\beta )=\beta$. That is $P{F}_{cel}int(P{F}_{cel}cl(\beta ))=0_X$. Thus β is $P{F}_{cel}$ nowhere dense.

ii) ⇒ iii) Suppose $\xi \ne 0_X$, $\varrho \ne 0_X\in PF(X)$ are $P{F}_{cel}$ open sets. Then $\bigcup _{n=-\infty }^{\infty }{\mathfrak{p}}^n(\xi )$ is a $P{F}_{cel}$ variant open set. So it is $P{F}_{cel}$ dense. By ii) $\bigcup _{n=-\infty }^{\infty }{\mathfrak{p}}^n(\xi )\cap \varrho \ne 0_X$.

iii) ⇒ iv) This proof is simple.

To prove iv) Let ${\xi }_1,{\xi }_2,...,{\xi }_n$ be a $P{F}_{cel}$ countable base for $(X,{\tau }_{p_{cel}})$, then its images of ${\beta }_n$ is also $P{F}_{cel}$ dense.

iv) ⇒ i)To show that $\mathfrak{p}$ is $P{F}_{cel}$ topological transitive. Since the collection $\{\varrho \in PF(X):P{F}_{cel}cl(P{F}_{cel}{\mathcal{O}}^{⁎}))=1_X\}$ is $P{F}_{cel}$ dense. It belongs to this collection. □

Remark 8.9

Let $(X,{\tau }_{p_{cel}})$ be a $P{F}_{cel}$compact topological space and for a $P{F}_{cel}$ homeomorphism $\mathfrak{p}:(X,{\tau }_{p_{cel}})\to (X,{\tau }_{p_{cel}})$ the following are equivalent:

i) $P{F}_{cel}$ is positive topologically transitive.

ii) If β is $P{F}_{cel}$ closed sets (resp., open sets) of $(X,{\tau }_{p_{cel}})$ and $\mathfrak{p}(\beta )=\beta$ then either $\beta =1_X$ (resp., $0_X$) or β is nowhere $P{F}_{cel}$ dense (resp., $P{F}_{cel}$ dense).

iii) If $\beta \ne 0_X$, $\varrho \ne 0_X\in PF(X)$ are Pythagorean fuzzy open sets, then there exists $n\in Z$ with ${\mathfrak{p}}^n(\beta )\cap \varrho \ne 0_X$.

iv) The collection $\{\varrho \in PF(X):P{F}_{cel}cl(P{F}_{cel}{\mathcal{O}}^{⁎}))=1_X\}$ is $P{F}_{cel}$ dense.

Proposition 8.10

Let $(X,{\tau }_{p_{cel}},T_D)$ be a $P{F}_{cel}$ TDS and $\mathfrak{p}:(X,{\tau }_{p_{cel}})\to (X,{\tau }_{p_{cel}}$ be a $P{F}_{cel}$ continuous. Then the following are equivalent:

i) $\mathfrak{p}$ is a $P{F}_{cel}$ topologically transitive.

ii) for any pair of $P{F}_{cel}$ open ϱ and $\gamma \in PF(X)$ there exists a positive $n\in Z-\{0\}$ such that ${\mathfrak{p}}^n\cap \gamma \ne 0_X$ .

iii) for any $P{F}_{cel}$ open set $\varrho \ne 0_X\in PF(X)\overline{\bigcup _{n=0}^{\infty }{\mathfrak{p}}^n(\varrho )}=1_X$ .

iv) for any $P{F}_{cel}$ open set $\varrho \ne 0_X\in PF(X)\overline{\bigcup _{n=0}^{\infty }{\mathfrak{p}}^n(\varrho )}=1_X$ .

v) for any pair of $P{F}_{cel}$ open sets $\gamma \ne 0_X$ and $\varrho \ne 0_X\in PF(X)$ , there exists a positive $n\in Z_0$ such that ${\mathfrak{p}}^n(\varrho )\cap \gamma \ne 0_X$ .

vi) for any $P{F}_{cel}$ open set $\varrho \ne 0_X\in PF(X)\overline{\bigcup _{n=1}^{\infty }{\mathfrak{p}}^{-n}(\varrho )}=1_X$ .

vii) for any $P{F}_{cel}$ open set $\varrho \ne 0_X\in PF(X)\overline{\bigcup _{n=1}^{\infty }{\mathfrak{p}}^{-n}(\varrho )}=1_X$

ix) If ϱ is $P{F}_{cel}$ open set of $(X,{\tau }_{p_{cel}})$ and ${\mathfrak{p}}^{-1}(\varrho )<\varrho$ then either $\varrho =0_X$ or ϱ is nowhere $P{F}_{cel}$ dense.

Proof

i)-vii) by the Definitions 7.4($P{F}_{cel}$ orbit*) and 7.15 (inversely variant)

viii), ix) by the above Proposition 8.8. □

Proposition 8.11

For the $P{F}_{cel}$ orbit set for the following properties hold:

a) i) $\mathfrak{p}(P{F}_{ce{l}^{+}}(\varrho ))$ = $P{F}_{cel}^{+}\mathfrak{p}(\varrho )<P{F}_{cel}^{+}(\varrho )<{\mathfrak{p}}^{-1}P{F}_{cel}^{+}\mathfrak{p}(\varrho )$ .

ii) $\mathfrak{p}(P{F}_{cel}^{-}(\varrho ))$ = ${\mathfrak{p}}^{-1}P{F}_{cel}^{-}(\varrho )<{\mathfrak{p}}^{-1}P{F}_{cel}^{-}(\varrho )$ .

iii) $\mathfrak{p}(P{F}_{cel}^{\pm }(\varrho ))$ = $P{F}_{cel}^{\pm }(\varrho )<{\mathfrak{p}}^{-1}P{F}_{cel}^{\pm }(\varrho )$ .

b)i) $P{F}_{cel}^{-}(\varrho )=x\in X:P{F}_{cel}^{+}(1_X)\cap \varrho \ne 0_X$ .

ii) $P{F}_{cel}^{+}(\varrho )=x\in X:P{F}_{cel}^{-}(1_X)\cap \varrho \ne 0_X$ .

iii) $P{F}_{cel}^{\pm }(\varrho )=x\in X:P{F}_{cel}^{\pm }(1_X)\cap \varrho \ne 0_X$ .

c) If $\beta \in PF(X)$ of $(X,{\tau }_{p_{cel}})$ , then

i) $P{F}_{cel}^{+}(\varrho \cup \beta )$ = $P{F}_{cel}^{+}(\varrho )\cup P{F}_{cel}^{+}(\beta )$

ii) $P{F}_{cel}^{-}(\varrho \cup \beta )$ = $P{F}_{cel}^{-}(\varrho )\cup P{F}_{cel}^{-}(\beta )$

iii) $P{F}_{cel}^{p}m(\varrho \cup \beta )$ = $P{F}_{cel}^{\pm }(\varrho )\cup P{F}_{cel}^{\pm }(\beta )$

d)i) $P{F}_{cel}^{+}(\varrho )={\mathfrak{p}}^{n^{+}}((\bigcup _{j=0}^{n-1}{\mathfrak{p}}^j(\varrho )))$

ii) $P{F}_{cel}^{-}(\varrho )={\mathfrak{p}}^{n^{-}}((\bigcup _{j=0}^{n-1}{\mathfrak{p}}^j(\varrho )))$

Proof

By the Proposition 8.10 and by Definition 8.7.

To prove a)i) $\mathfrak{p}(P{F}_{cel}^{+}(\varrho ))=\mathfrak{p}(\bigcup _{n\in N_0}{\mathfrak{p}}^n(\varrho ))$ = $(\bigcup _{n\in N_0}{\mathfrak{p}}^n\mathfrak{p}(\varrho ))=P{F}_{cel}^{+}(\varrho )<{\mathfrak{p}}^{-1}(P{F}_{cel}^{+}(\varrho ))$.

ii) $\mathfrak{p}(P{F}_{cel}^{-}(\varrho ))=\mathfrak{p}(\bigcup _{n\in N_0}{\mathfrak{p}}^n(\varrho ))$ = $(\bigcup _{n\in N_0}{\mathfrak{p}}^n{\mathfrak{p}}^{-1}(\varrho ))=P{F}_{cel}^{-}(\varrho )<{\mathfrak{p}}^{-1}(P{F}_{cel}^{-}(\varrho ))$.

iii) $\mathfrak{p}(P{F}_{cel}^{\pm }(\varrho ))=\mathfrak{p}(\bigcup _{n\in N_0}{\mathfrak{p}}^n(\varrho ))$ = $(\bigcup _{n\in N_0}{\mathfrak{p}}^n{\mathfrak{p}}^{-1}(\varrho ))=P{F}_{cel}^{\pm }(\varrho )<{\mathfrak{p}}^{-1}(P{F}_{cel}^{\pm }(\varrho ))$.

b)i) Assume i) $P{F}_{cel}^{-}(\varrho )=x\in X:P{F}_{cel}^{+}(1_X)\cap \varrho \ne 0_X$. Similarly the other two cases are easy.

c)i) Let $\beta \in PF(X)$, to prove this by using Definition 8.7

$P{F}_{cel}^{+}(\varrho \cup \beta )$ = $\bigcup _{n\in N_0}{\mathfrak{p}}^n(\varrho \cup \beta )=(\bigcup _{n\in N_0}{\mathfrak{p}}^n(\varrho ))\cup (\bigcup _{n\in N_0}{\mathfrak{p}}^n(\beta ))=P{F}_{cel}^{+}(\varrho )\cup P{F}_{cel}^{+}(\beta )$. Hence true for ii) and iii).

d)i) $P{F}_{cel}^{+}(\varrho )=\bigcup _{m\in N_0}(\bigcup _{j=0}^{n-1}{\mathfrak{p}}^{j+mn}(\varrho ))={\mathfrak{p}}^{n^{+}}(\bigcup _{j=0}^{n-1}{\mathfrak{p}}^j(\varrho ))$

ii) $P{F}_{cel}^{-}(\varrho )=\bigcup _{m\in Z_0^{-}}(\bigcup _{j=0}^{n-1}{\mathfrak{p}}^{-j-mn}(\varrho ))={\mathfrak{p}}^{n^{-}}(\bigcup _{j=0}^{n-1}{\mathfrak{p}}^{-j}(\varrho ))$. □

Proposition 8.12

The following statements are true for $P{F}_{cel}$ invariant sets:

i) If $\mathfrak{p}(\varrho )<\varrho$ , then $P{F}_{cel}^{+}(\varrho )=\varrho$ and ${\mathfrak{p}}^{-1}(\varrho )<\varrho$ then $P{F}_{cel}^{-}(\varrho )=\varrho$

ii) $\mathfrak{p}(\varrho )<\varrho$ if and only if ${\mathfrak{p}}^{-1}(1_X\cap {\varrho }^c)<(1_X\cap {\varrho }^c)$ .

iii) The intersection of any number of $P{F}_{cel}$ invariants sets ( $(P{F}_{cel}$ inversely invariant) is $P{F}_{cel}$ invariant sets ( $(P{F}_{cel}$ inversely invariant).

iv) If ${\mathfrak{p}}^n(\varrho )=\varrho$ , then $\mathfrak{p}(\bigcup _{j=0}^{n-1}{\mathfrak{p}}^j(\varrho ))\subset \bigcup _{j=0}^{n-1}{\mathfrak{p}}^j(\varrho )$ and $P{F}_{cel}^{+}(\varrho )=\bigcup _{j=0}^{n-1}{\mathfrak{p}}^j(\varrho )$ .

v) If ${\mathfrak{p}}^{-n}(\varrho )=\varrho$ , then ${\mathfrak{p}}^{-\mathfrak{1}}(\bigcup _{j=0}^{n-1}{\mathfrak{p}}^{-j}(\varrho ))\subset \bigcup _{j=0}^{n-1}{\mathfrak{p}}^{-j}(\varrho )$ and $P{F}_{cel}^{-}(\varrho )=\bigcup _{j=0}^{n-1}{\mathfrak{p}}^{-j}(\varrho )$ .

Proof

i) and iii) are by the Definition 8.4

ii) Assume $P{F}_{cel}(\varrho )=\varrho$, ${\mathfrak{p}}^{-1}(1_X\cap {\varrho }^c)={\mathfrak{p}}^{-1}(1_X)\cap {\mathfrak{p}}^{-1}(\varrho )=(1_X)\cap {\varrho }^c$ by the Definition 8.4. Conversely, assume that ${\mathfrak{p}}^{-1}(1_X)\cap {\varrho }^c)<1_X\cap {\varrho }^c$ by the Definition 8.4 $\mathfrak{p}(\varrho )<\varrho$.

iv) $\mathfrak{p}(\bigcup _{j=0}^{n-1}{\mathfrak{p}}^j(\varrho ))=\bigcup _{j=0}^{n-1}{\mathfrak{p}}^{j+1}(\varrho )$ = ${\mathfrak{p}}^n(\varrho )\cup \bigcup _{j=1}^{n-1}{\mathfrak{p}}^j(\varrho )\subset \bigcup _{j=0}^{n-1}{\mathfrak{p}}^{j+1}(\varrho )$ By Proposition 8.10 $P{F}_{cel}^{+}(\varrho )={\mathfrak{p}}^{n^{+}}(\bigcup _{j=0}^{n-1}{\mathfrak{p}}^j(\varrho ))\subset \bigcup _{j=0}^{n-1}{\mathfrak{p}}^j(\varrho )$.

v) is similar to the proof of iv). □

9. Conclusion and recommendation of the study

In this study, Pythagorean fuzzy cellular space is defined. The Pythagorean fuzzy cellular space is used to study continuity, compactness, normality, and homeomorphism. The $P{F}_{cel}$TDS is introduced and implemented using the Pythagorean fuzzy cellular space. The $P{F}_{cel}$TDS under the Pythagorean fuzzy cellular continuous map employs an iterative approach. Furthermore, the iterative technique generates a Pythagorean fuzzy orbit* set. This study depicts that $P{F}_{cel}$TDS is topologically transitive, which is a key feature of all fuzzy topological dynamical systems. Chaos and topological atoms can be defined and extended to $P{F}_{cel}TDS$. Pythagorean fuzzy cellular continuity, compactness, and normality may be extended in an assortment of ways. If the dynamical system is vague, this novel $P{F}_{cel}$TDS can be employed to improve comprehension. There are several real-time applications in image processing, pattern recognition, and control theory. For example, in control theory, two types of control system are there namely, open-loop and closed-loop control system. Iterative process is employed to each set in the system [21] under the continuous map. This kind of application may be considered for future study with the $P{F}_{cel}$TDS.

CRediT authorship contribution statement

Gnanachristy N B: Writing – original draft. Revathi G K: Writing – review & editing.

Declaration of Competing Interest

All authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability

No data was used for the research described in this article.