﻿ 类帐篷映射上的链回归点集研究 Research on Chain Regression Point Set on Tent-Like Mapping

Pure Mathematics
Vol. 09  No. 03 ( 2019 ), Article ID: 30408 , 7 pages
10.12677/PM.2019.93059

Research on Chain Regression Point Set on Tent-Like Mapping

Zhanfu Huo, Ziqing Fu

School of Mathematics and Information Science, Guangxi University, Nanning Guangxi

Received: Apr. 29th, 2019; accepted: May 9th, 2019; published: May 24th, 2019

ABSTRACT

The study of chain regression points of continuous self-mapping on metric space has always been an important part of topological dynamical system. This paper mainly studies the dynamic properties of continuous mapping on compact metric, focusing on the characteristics of chain regression points of tent-like mapping.

Keywords:Compact Metric, Continuous Mapping, Tent-Like Mapping, Chain Regression Point

1. 引言与预备知识

${g}_{t,\lambda }\left(x\right)=\left\{\begin{array}{ll}\frac{1-\lambda }{t}x+\lambda \hfill & 0\le x\le t\hfill \\ \frac{x}{t-1}+\frac{1}{1-t}\hfill & t\le x\le 1\hfill \end{array}$

$t=0$ 时， ${g}_{t,\lambda }\left(x\right)=-x+1$，当 $t=1$ 时， ${g}_{t,x}\left(x\right)=\left(1-\lambda \right)x+\lambda$。记 ${a}_{t}$${g}_{t,\lambda }\left(x\right)$ 的不动点，记 ${k}_{1}$${g}_{t,\lambda }\left(x\right)$$\left[0,t\right]$ 上的斜率， ${k}_{2}=-\frac{1}{1-t}$${g}_{t,\lambda }\left(x\right)$$\left[t,1\right]$ 上的斜率。 ${U}_{\delta }^{-}=\left[0,\delta \right]$。本节主要讨论 $0 时的情形，当 $0\le \lambda \le t$ 时，记 ${a}_{t}=\frac{1}{2-t}$${{a}^{\prime }}_{t}=\frac{t+\lambda {t}^{2}-2\lambda t}{\left(1-\lambda \right)\left(2-t\right)}$，则 ${g}_{t,\lambda }\left({a}_{t}\right)={g}_{t,\lambda }\left({{a}^{\prime }}_{t}\right)={a}_{t}$，且 $0\le {{a}^{\prime }}_{t}\le t\le {a}_{t}\le 1$

2. 相关引理

，因为，所以，即有

，则

，则由引理2.3，

，所以，因为，即，所以

，则，且由情形1~3知

.

.

；②；③；④； ⑤

3. 主要定理的证明

Research on Chain Regression Point Set on Tent-Like Mapping[J]. 理论数学, 2019, 09(03): 441-447. https://doi.org/10.12677/PM.2019.93059

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