# On the Boundedness and Symmetry Properties of the Fractal Sets Generated from Alternated Complex Map

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^{†}

## Abstract

**:**

## 1. Introduction

## 2. The Fractal Sets Generated from Alternated Complex Map

**Definition 1.**

**Theorem 1.**

**Theorem 2.**

- The Julia set is connected if and only if all the critical orbits are bounded.
- The Julia set is totally disconnected, a cantor set, if (but not only if) all the critical orbits are unbounded.
- For a polynomial with at least one critical orbit unbounded, the Julia set is totally disconnected if and only if all the bounded critical orbits are aperiodic.

**Definition 2.**

**Figure 1.**(

**a**) The 3-D slice of Mandelbrot-efficacy set: ${M}_{e}{\left({P}_{{c}_{1},{c}_{2}}\right)}_{{q}_{0}=0}$; (

**b**) The 2-D slice of ${M}_{e}{\left({P}_{{c}_{1},{c}_{2}}\right)}_{{q}_{0}=0}$ with ${p}_{0}=0$; (

**c**,

**d**) The partial enlarged views of ${M}_{e}{\left({P}_{{c}_{1},{c}_{2}}\right)}_{{p}_{0}=0,{q}_{0}=0}$.

## 3. Boundedness of the Fractal Sets Generated from Alternated Complex Map

**Theorem 3.**

**Proof.**

**Theorem 4.**

**Proof.**

**Remark 1.**

- Then Theorem 3 becomes: $J\left({P}_{c}\right)\subset \left\{z\phantom{\rule{3.33333pt}{0ex}}\right|\phantom{\rule{3.33333pt}{0ex}}\left|z\right|\phantom{\rule{3.33333pt}{0ex}}\le \phantom{\rule{3.33333pt}{0ex}}max\left\{\right|c|,2\}\}.$
- Theorem 4 becomes: $M\left({P}_{c}\right)\subset \left\{c\phantom{\rule{3.33333pt}{0ex}}\right|\phantom{\rule{3.33333pt}{0ex}}\left|c\right|\le 2\}.$

## 4. Symmetry of the Fractal Sets Generated from Alternated Complex Map

**Theorem 5.**

**Proof.**

**Figure 3.**(

**a**) ${M}_{e}{\left({P}_{{c}_{1},{c}_{2}}\right)}_{{q}_{0}=0,{s}_{0}=-0.9}$ and ${M}_{e}{\left({P}_{{c}_{1},{c}_{2}}\right)}_{{q}_{0}=0,{s}_{0}=0.9};$ (

**b**) ${M}_{e}{\left({P}_{{c}_{1},{c}_{2}}\right)}_{{q}_{0}=0,{s}_{0}=-0.75}$ and ${M}_{e}{\left({P}_{{c}_{1},{c}_{2}}\right)}_{{q}_{0}=0,{s}_{0}=0.75};$ (

**c**) ${M}_{e}{\left({P}_{{c}_{1},{c}_{2}}\right)}_{{q}_{0}=0,{s}_{0}=-0.3}$ and ${M}_{e}{\left({P}_{{c}_{1},{c}_{2}}\right)}_{{q}_{0}=0,{s}_{0}=0.3.}$

**Theorem 6.**

**Proof.**

**Figure 4.**(

**a**–

**c**) The connectivity loci of system (2) and its partial enlarged views; (

**d**) Connected $K\left({P}_{{c}_{1},{c}_{2}}\right)$ with ${c}_{1}=-$0.09, ${c}_{2}=0.021-0.75i$ (D point in (

**c**)); (

**e**) Disconnected $K\left({P}_{{c}_{1},{c}_{2}}\right)$ with ${c}_{1}=\phantom{\rule{3.33333pt}{0ex}}-$0.096, ${c}_{2}=0.028-0.75i$ (E point); (

**f**) Totally disconnected $K\left({P}_{{c}_{1},{c}_{2}}\right)$ with ${c}_{1}=\phantom{\rule{3.33333pt}{0ex}}-$0.098, ${c}_{2}=0.032-0.75i$ (F point).

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Wang, D.; Liu, S.
On the Boundedness and Symmetry Properties of the Fractal Sets Generated from Alternated Complex Map. *Symmetry* **2016**, *8*, 7.
https://doi.org/10.3390/sym8020007

**AMA Style**

Wang D, Liu S.
On the Boundedness and Symmetry Properties of the Fractal Sets Generated from Alternated Complex Map. *Symmetry*. 2016; 8(2):7.
https://doi.org/10.3390/sym8020007

**Chicago/Turabian Style**

Wang, Da, and ShuTang Liu.
2016. "On the Boundedness and Symmetry Properties of the Fractal Sets Generated from Alternated Complex Map" *Symmetry* 8, no. 2: 7.
https://doi.org/10.3390/sym8020007