# Some Chaos Notions on Dendrites

^{*}

## Abstract

**:**

## 1. Introduction

- transitive (for any non-empty open sets U and V, $\exists m\in \mathbb{N}$ s.t ${f}^{m}\left(U\right)\cap V\ne \varnothing $),
- contains a dense set of periodic points,
- sensitive dependence on initial conditions ($SDIC$) ($\exists \alpha >0$, s.t $\forall x\in X$ and $\forall \beta >0$, $\exists y\in X$ and $\exists m\in \mathbb{N}$ s.t $d(x,y)<\beta $ but $d({f}^{m}\left(x\right),{f}^{m}\left(y\right))\ge \alpha $).

## 2. Results

**Definition**

**1.**

**Theorem**

**1.**

**Proof.**

**Proposition**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3**

**.**Let f be a map on a continuum space X. f is mixing if and only if $\forall \u03f5>0$ and for all non-empty open set U, $\exists M\in \mathbb{N}$ such that

**Theorem**

**4.**

**Proof.**

**Theorem**

**5.**

**Proof.**

**Theorem**

**6**

**.**Let f be a map on a space X. f is weakly mixing if and only if, for any non-empty open sets U and V, $\exists m\in \mathbb{N}$ s.t ${f}^{m}\left(V\right)\cap U\ne \varnothing $ and ${f}^{m}\left(V\right)\cap V\ne \varnothing .$

**Theorem**

**7.**

**Proof.**

**Theorem**

**8.**

**Example**

**1.**

**Remark**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Lemma**

**1.**

**Proof.**

**Theorem**

**9**

**.**Let f be a weakly mixing map on a space X; then,

**Theorem**

**10.**

**Proof.**

**Remark**

**2.**

**Theorem**

**11.**

**Proof.**

**Theorem**

**12.**

**Proof.**

**Theorem**

**13**

**Proposition**

**2.**

**Proof.**

**Theorem**

**14**

**.**Let f be a map on a compact metric space with no isolated points. Then, Devaney chaos implies strong dense periodicity.

**Theorem**

**15.**

**Proof.**

**Proposition**

**3.**

**Proof.**

## 3. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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Fadel, A.; Dzul-Kifli, S.C.
Some Chaos Notions on Dendrites. *Symmetry* **2019**, *11*, 1309.
https://doi.org/10.3390/sym11101309

**AMA Style**

Fadel A, Dzul-Kifli SC.
Some Chaos Notions on Dendrites. *Symmetry*. 2019; 11(10):1309.
https://doi.org/10.3390/sym11101309

**Chicago/Turabian Style**

Fadel, Asmaa, and Syahida Che Dzul-Kifli.
2019. "Some Chaos Notions on Dendrites" *Symmetry* 11, no. 10: 1309.
https://doi.org/10.3390/sym11101309