# Dynamics on Binary Relations over Topological Spaces

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

- (i)
- x is a universal element for the sequence ${\left({\rho}_{n}\right)}_{n\in \mathbb{N}}$ if $x\in {\bigcap}_{n\in \mathbb{N}}D\left({\rho}_{n}\right)$ and for each $n\in {\mathbb{N}}_{0}$ there exists an element ${y}_{n}\in {\rho}_{n}\left(x\right)$ such that the set $\{{y}_{n}:n\in \mathbb{N}\}$ is dense in Y. As a particular case, if ${\rho}_{n}:={\rho}^{n}$, then we say x ishypercyclicfor ρ.
- (ii)
- ρ is topologically transitive if, for every pair of non-empty open sets $U,V\subset X$, there is some $n\in \mathbb{N}$ such that $U\cap {\rho}^{-n}\left(V\right)\ne \varnothing $. If there is some ${n}_{0}\in \mathbb{N}$ such that this last condition holds for all $n\ge {n}_{0}$, we say that ρ is topologically mixing.

**Remark**

**1.**

- (i)
- Let $G=(X,\rho )$ be a graph with X equipped with the discrete topology. It can be simply proved that the graph G is connected if and only if ρ is topologically transitive or hypercyclic.
- (ii)
- In the Definition 1 (ii), it is irrelevant whether we write ${\rho}^{-n}\left(V\right)$ or ${\rho}^{n}\left(V\right)$. It is also worth noting that ρ does not need to be topologically mixing whenever ρ is topologically transitive: Let us consider a graph $G=(X,\rho )$ that is isomorphic to a square, that is $X=\{{x}_{1},{x}_{2},{x}_{3},{x}_{4}\}$ and $\rho =\{({x}_{1},{x}_{2}),({x}_{2},{x}_{3}),({x}_{3},{x}_{4}),({x}_{4},{x}_{1})\}$, see Figure 1a. Clearly, ρ is topologically transitive, but it does not hold the topologically mixing property since there is no odd number $n\in \mathbb{N}$ such that ${x}_{3}\in {\rho}^{n}\left({x}_{1}\right)$.
- (iii)
- Unlike the linear setting, in our framework, the notion of hypercyclicity cannot be connected to that of topological transitivity in any reasonable way. It is well known that these notions are equivalent for continuous linear operators on Fréchet spaces by Baire’s category theorem (see, for instance, [21,22]). However, there exist examples of continuous linear operators on non-metrizable locally convex spaces that are topologically transitive and not hypercyclic [23]. Moreover, for any non-trivial Banach space X there exists a multivalued linear operator $\mathcal{A}=\left\{0\right\}\times X$ that is hypercyclic and not topologically transitive (cf. [24]). It is very simple to construct an example of a hypercyclic relation on a finite set that is not topologically transitive, as well: Set the digraph $G=(X,\rho )$ with $X:=\{{x}_{1},{x}_{2},{x}_{3}\}$ and $\rho :=\{({x}_{1},{x}_{2}),({x}_{2},{x}_{1}),({x}_{1},{x}_{3})\}$, endowing X with the discrete topology, see Figure 1b. Then ${x}_{1}$ is a hypercyclic element for ρ, but ρ is not topologically transitive, since $\left\{{x}_{2}\right\}\cap {\rho}^{n}\left(\left\{{x}_{3}\right\}\right)=\varnothing $ for all $n\in \mathbb{N}$.

**Example**

**1.**

## 3. Hypercyclic and Chaotic Digraphs

**Definition**

**2.**

- 1.
- x is a periodic point of ρ if $x\in {D}_{\infty}\left(\rho \right)$ and there exists $n\in \mathbb{N}$ such that $x\in {\rho}^{n}\left(x\right)$.
- 2.
- ρ is Devaney-chaotic if it is topologically transitive and it has a dense set of periodic points.
- 3.
- ρ is chaotic if it is hypercyclic and it has a dense set of periodic points.

**Example**

**2.**

- (i)
- Let $X:=\{{x}_{1},{x}_{2},{x}_{3},{x}_{4}\}$ be equipped with discrete topology, and let $\rho :=\{({x}_{1},{x}_{2}),({x}_{2},{x}_{1}),({x}_{1},{x}_{3}),({x}_{3},{x}_{4}),({x}_{4},{x}_{3})\}$, see Figure 2a. Then ${x}_{3}\in \rho \left({x}_{1}\right),$${x}_{4}\in {\rho}^{2}\left({x}_{1}\right),$${x}_{1}\in {\rho}^{3}\left({x}_{1}\right),$${x}_{2}\in {\rho}^{2n+1}\left({x}_{1}\right)$, and ${x}_{1}\in {\rho}^{2n}\left({x}_{1}\right)$ ($n\ge 2$), which simply yields that ${x}_{1}\in HC\left(\rho \right)$. It is also clear that any element of X is periodic for ρ, such that ρ is chaotic. Since there is no path connecting ${x}_{3}$ and ${x}_{1}$ in $G,$G is not strongly connected and ρ is neither topologically transitive nor Devaney-chaotic. Therefore, the first implication in Equation (2) is strict.
- (ii)
- Let $C={x}_{1}\dots {x}_{n+1}$ be an oriented closed cycle of length n (with ${x}_{1}={x}_{n+1})$, and let $\rho :=C\cup \left\{({x}_{1},{x}_{n+2})\right\},$ where ${x}_{n+2}\ne {x}_{j}$ for $1\le j\le n+1$, see Figure 2b. Then it can be easily seen that $G=(X,\rho )$, endowed with the discrete topology, is hypercyclic, since any element lying on the cycle C is hypercyclic for ρ and that G is not chaotic, because the point ${x}_{n+2}$ cannot be a periodic element for $\rho .$
- (iii)
- Let $X:=\{{x}_{1},{x}_{2},{x}_{3}\}$ and $\rho :=\{({x}_{1},{x}_{2}),({x}_{3},{x}_{2})\}$, see Figure 2c. Then $G=(X,\rho )$ is weakly connected but, equipped with the discrete topology, it is not hypercyclic.

**Example**

**3.**

## 4. Dynamics on Tournaments

**Definition**

**3.**

**Example**

**4.**

**Theorem**

**1.**

**Proof.**

**Corollary**

**1.**

**Proof.**

## 5. Disjointness on Binary Relations, Graphs, and Digraphs

**Definition**

**4.**

**Definition**

**5.**

- 1.
- The sequences ${\left({\rho}_{1,n}\right)}_{n\in \mathbb{N}},\dots ,{\left({\rho}_{N,n}\right)}_{n\in \mathbb{N}}$ are d-topologically transitive if, for every non-empty open subsets $U\subset X$ and ${V}_{1},\dots ,{V}_{N}\subset Y$, there exists $n\in \mathbb{N}$ such that$$U\cap {\rho}_{1,n}^{-1}\left({V}_{1}\right)\cap \dots \cap {\rho}_{N,n}^{-1}\left({V}_{N}\right)\ne \varnothing .$$
- 2.
- The sequences ${\left({\rho}_{1,n}\right)}_{n\in \mathbb{N}},\dots ,{\left({\rho}_{N,n}\right)}_{n\in \mathbb{N}}$ are d-topologically mixing if, for every non-empty open subsets $U\subset X$ and ${V}_{1},\dots ,{V}_{N}\subset Y$, there exists ${n}_{0}\in \mathbb{N}$ such that, for every $n\ge {n}_{0},$ we have that Equation (3) holds.
- 3.
- The binary relations ${\rho}_{1},\xb7\xb7\xb7,{\rho}_{N}$ are d-topologically transitive (d-topologically mixing) if the sequences ${\left({\rho}_{1}^{n}\right)}_{n\in \mathbb{N}},\dots ,{\left({\rho}_{N}^{n}\right)}_{n\in \mathbb{N}}$ are d-topologically transitive (d-topologically mixing).

**Definition**

**6.**

**Theorem**

**2.**

- i.
- The graphs are d-Devaney-chaotic.
- ii.
- The graphs are d-hypercyclic.
- iii.
- Each graph contains $n\ge 3$ nodes.

**Proof.**

**Theorem**

**3.**

**Theorem**

**4.**

**Example**

**5.**

- (i)
- Let G be the graph appearing in Example 2 (i), and let ${K}_{4}$ be the complete graph of these 4 nodes, and both are equipped with discrete topologies, see Figure 4a. It can be easily seen that G and H are d-chaotic but not d-Devaney-chaotic.
- (ii)
- Let $X:=\{{x}_{1},{x}_{2},{x}_{3},{x}_{4}\},$ and let $\rho :={\bigcup}_{1\le i,j\le 3}({x}_{i},{x}_{j})\cup \{({x}_{1},{x}_{4}),({x}_{2},{x}_{4})\}$, see Figure 4b. If $G=(X,\rho )$ is equipped with the discrete topology, then the pair $G,G$ is d-hypercyclic, since, for any even number $k\in \mathbb{N}$ and for every $x\in X$, we have $x\in {\rho}^{k}\left({x}_{1}\right)$, but it is not d-chaotic since ${x}_{4}$ is not periodic in G.

**Theorem**

**5.**

**Proof.**

**Theorem**

**6.**

- 1.
- ${T}_{1},\dots ,\phantom{\rule{4pt}{0ex}}{T}_{N}$ are d-Devaney-chaotic.
- 2.
- ${T}_{1},\dots ,\phantom{\rule{4pt}{0ex}}{T}_{N}$ are d-topologically transitive.
- 3.
- ${T}_{l}$ is strongly connected for all $l\in {\mathbb{N}}_{N}$.
- 4.
- ${T}_{l}$ is a Hamiltonian tournament for all $l\in {\mathbb{N}}_{N}$.

**Proof.**

**Example**

**6.**

**Example**

**7.**

## Author Contributions

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) In Remark 1 (ii), we show a graph that, endowed with the discrete topology, it is topologically transitive but not topologically mixing. (

**b**) In Remark 1 (iii), we have a graph that, endowed with the discrete topology, it is hypercyclic but not topologically transitive. (

**c**) In Example 1, different topologies—${\tau}_{1}=\{\varnothing ,\left\{{x}_{1}\right\},\left\{{x}_{2}\right\},\left\{{x}_{3}\right\}$, $\{{x}_{1},{x}_{2}\}$, $\{{x}_{1},{x}_{3}\}$, $\{{x}_{2},{x}_{3}\},\{{x}_{1},{x}_{2},{x}_{3}\},\{{x}_{1},{x}_{2},{x}_{3},{x}_{4}\}\}$ and ${\tau}_{2}=\{\varnothing ,\{{x}_{1},{x}_{2}\},\{{x}_{1},{x}_{2},{x}_{3},{x}_{4}\}\}$—can be defined such that the graph is hypercyclic for ${\tau}_{1}$ but not for ${\tau}_{2}$.

**Figure 2.**The following graphs are endowing with the discrete topology. (

**a**) In Example 2 (i), we show that all in the elements in a graph can be periodic, but this does not imply strong connectivity, Devaney chaos, nor topological transitivity. (

**b**) In Example 2 (ii), we show that hypercyclicity does not imply density of periodic points. (

**c**) In Example 2 (iii), we show that weak connectivity does not imply hypercyclicity.

**Figure 3.**(

**a**) In Example 3, we show a Devaney-chaotic digraph, whose strongly connected components are not dense. (

**b**) In Example 4, we show an example of a tournament that, endowed with the discrete topology, it is hypercyclic but not topologically transitive.

**Figure 4.**The following graphs are endowed with the discrete topology. (

**a**) In Example 5 (i), we show that the graph of Example 2 (i) and ${K}_{4}$ are d-chaotic but not d-Devaney-chaotic. (

**b**) In Example 5 (ii), these two copies of the same graph are d-hypercyclic but not d-chaotic.

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**MDPI and ACS Style**

Chen, C.-C.; Conejero, J.A.; Kostić, M.; Murillo-Arcila, M.
Dynamics on Binary Relations over Topological Spaces. *Symmetry* **2018**, *10*, 211.
https://doi.org/10.3390/sym10060211

**AMA Style**

Chen C-C, Conejero JA, Kostić M, Murillo-Arcila M.
Dynamics on Binary Relations over Topological Spaces. *Symmetry*. 2018; 10(6):211.
https://doi.org/10.3390/sym10060211

**Chicago/Turabian Style**

Chen, Chung-Chuan, J. Alberto Conejero, Marko Kostić, and Marina Murillo-Arcila.
2018. "Dynamics on Binary Relations over Topological Spaces" *Symmetry* 10, no. 6: 211.
https://doi.org/10.3390/sym10060211