# Countably Expansiveness for Continuous Dynamical Systems

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

**Definition**

**1**

**Definition**

**2**

**Definition**

**3**

**Remark**

**1**

## 2. Countably Expansiveness for Suspension Flows

**Remark**

**2.**

**Definition**

**4.**

**Example**

**1.**

**Example**

**2.**

**Definition**

**5**

**Example**

**3.**

**Proof.**

- (i)
- $d({\sigma}^{n}(s),{\sigma}^{n}(t))=0$ for $\forall n$ implies $s=t.$
- (ii)
- If $s\ne t$, then $0<d({\sigma}^{n}(s),{\sigma}^{n}(t))\le \delta $ for $\forall n$ implies there exists $j\in \mathbb{Z}$ such that ${s}_{n+j}\ne {t}_{n+j}.$ That is, the jth coordinate of ${\sigma}^{n}(s)={s}_{n}{s}_{n+1}{s}_{n+2}\cdots {s}_{n+j}\cdots $ and ${\sigma}^{n}(t)={t}_{n}{t}_{n+1}{t}_{n+2}\cdots {t}_{n+j}\cdots $ have at least one different coordinate components.

**Lemma**

**1.**

**Proof.**

**Example**

**4.**

**Lemma**

**2.**

- (a)
- f is countably expansive if and only ${f}^{k}$ is countably expansive, for some $k\in \mathbb{Z}\backslash \{0\}.$
- (b)
- If f is the identity map ${\mathtt{I}}_{\mathtt{d}}$, then f is not countably expansive.

**Proof of (a).**

**Proof of (b).**

**Remark**

**3.**

**Definition**

**6**

- $\varphi :X\times \mathbb{R}\u27f6X$ satisfying $\varphi (x,0)=x,$
- $\varphi (\varphi (x,s),t)=\varphi (x,s+t)$ for $x\in X$ and $s,\phantom{\rule{0.277778em}{0ex}}t\in \mathbb{R}.$

**Definition**

**7.**

**Theorem**

**1.**

**Proof.**

**Example**

**5.**

**Example**

**6.**

**Remark**

**4.**

**Example**

**7.**

**Proof.**

## 3. ${\mathit{C}}^{\mathbf{1}}$ Stably Countably Expansive Vector Fields

**Note:**To study of dynamical systems, the properties of orbits (or points) are important: singular, periodic, non-wandering, etc. If a flow has a periodic orbit or singularity, then it causes a chaos phenomenon (for example, Geometric Lorenz attractor). This means that we cannot control the system. Especially, the countably expansive flow which we present in this paper does not have a singularity. Thus, we could investigate the stability of countably expansive flows.

**Definition**

**8.**

**Lemma**

**3.**

**Proof.**

**Lemma**

**4.**

**Proof.**

**Definition**

**9.**

**Theorem**

**2.**

**Lemma**

**5.**

**Proof.**

**Proof of Theorem 2.**

**Lemma**

**6.**

**Proof.**

**Theorem**

**3.**

**Proof.**

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Lee, M.; Oh, J. Countably Expansiveness for Continuous Dynamical Systems. *Mathematics* **2019**, *7*, 1228.
https://doi.org/10.3390/math7121228

**AMA Style**

Lee M, Oh J. Countably Expansiveness for Continuous Dynamical Systems. *Mathematics*. 2019; 7(12):1228.
https://doi.org/10.3390/math7121228

**Chicago/Turabian Style**

Lee, Manseob, and Jumi Oh. 2019. "Countably Expansiveness for Continuous Dynamical Systems" *Mathematics* 7, no. 12: 1228.
https://doi.org/10.3390/math7121228