關(guān)于非自治動(dòng)力系統(tǒng)中的h-極小覆蓋

趙佳琪,王延庚

(西北大學(xué)數(shù)學(xué)系,陜西西安 710127)

關(guān)于非自治動(dòng)力系統(tǒng)中的h-極小覆蓋

趙佳琪,王延庚

(西北大學(xué)數(shù)學(xué)系,陜西西安 710127)

設(shè)(X,d1,f1,∞)與(Y,d2,g1,∞)為兩個(gè)非自治動(dòng)力系統(tǒng),h是從(X,d1,f1,∞) 到(Y,d2,g1,∞)的拓?fù)浒牍曹?通過對(duì)自治動(dòng)力系統(tǒng)中的h-極小覆蓋的研究,本文得到了以下結(jié)論:1)對(duì)于任意的y∈Y及x∈h-1(y),orb(x,f1,∞)被h映射為orb(y,g1,∞),ω(x,f1,∞)被h映射為ω(y,g1,∞);2)在(X,d1,f1,∞)中引入關(guān)于拓?fù)浒牍曹椀膆-極小覆蓋的定義,證明了h-極小覆蓋的存在性;3)對(duì)于任意的x∈X 和y∈Y,在(ω(x,f1,∞),f1,∞|ω(x,f1,∞))與(ω(y,g1,∞),g1,∞|ω(y,g1,∞))均構(gòu)成原系統(tǒng)的子系統(tǒng)的前提下,R(f1,∞)被h映射為R(g1,∞).這些結(jié)論豐富了非自治動(dòng)力系統(tǒng)的內(nèi)容.

自治動(dòng)力系統(tǒng);非自治動(dòng)力系統(tǒng);h-極小覆蓋;拓?fù)涔曹?拓?fù)浒牍曹?/p>

1 引言

設(shè)(X,d)為緊致度量空間,f:X→X為連續(xù)自映射,用(X,f)表示由緊致度量空間X上的連續(xù)自映射f生成的離散拓?fù)浒雱?dòng)力系統(tǒng)[1],簡(jiǎn)稱緊致系統(tǒng),也被稱作自治動(dòng)力系統(tǒng).

對(duì)于兩個(gè)拓?fù)涔曹椀木o致系統(tǒng)(X,f)與(Y,g),系統(tǒng)(X,f)和(Y,g)有著完全相同的動(dòng)力性狀.但對(duì)于拓?fù)浒牍曹椀膬蓚€(gè)緊致系統(tǒng),擴(kuò)充和因子的動(dòng)力性狀卻可以相去甚遠(yuǎn).1995年,文獻(xiàn)[2]中對(duì)自治動(dòng)力系統(tǒng)引進(jìn)了關(guān)于拓?fù)浒牍曹椀臉O小覆蓋的概念.從而在擴(kuò)充中找到了某個(gè)子系統(tǒng),使得它與因子在性質(zhì)上更為相近.

1996年,文獻(xiàn)[3-4]中提出了非自治動(dòng)力系統(tǒng)的定義.2004年,文獻(xiàn)[5]中提出了非自治動(dòng)力系統(tǒng)的子系統(tǒng)的定義.2006年,文獻(xiàn)[6]中提出了非自治動(dòng)力系統(tǒng)中的拓?fù)涔曹椗c拓?fù)浒牍曹椀亩x.這些都為非自治動(dòng)力系統(tǒng)中h-極小覆蓋定義的提出提供了條件.

本文通過對(duì)自治動(dòng)力系統(tǒng)中h-極小覆蓋的研究,在非自治離散拓?fù)鋭?dòng)力系統(tǒng)中給出了關(guān)于拓?fù)浒牍曹椀膆-極小覆蓋的概念,并且證明了h-極小覆蓋的存在性.

2 基本定義

3 非自治動(dòng)力系統(tǒng)中的h-極小覆蓋及其存在性

[1]周作領(lǐng).符號(hào)動(dòng)力系統(tǒng)[M].上海:上?？萍冀逃霭嫔?1997.

[2]周作領(lǐng),何偉弘.軌道結(jié)構(gòu)的層次與拓?fù)浒牍曹梉J].中國(guó)科學(xué):A輯,1995,25(5):457-464.

[3]Kolyada S,Snoha L.Topological entropy of non-autonomous dynamical systems[J].Random Comput. Dynam,1996,4(2/3):205-233.

[4]Kolyada S,Misiurewicz M,Snoha L.Topological entropy of non-autonomous piecewise monotone dynamical systems on the interval[J].Fund.Math.,1999,160:161-181.

[5]Kolyada S,Snoha L,Trofimchuk S.On minimality of non-autonomous dynamical systems[J].Nonlinear Oscil.,2004,7(1):83-89.

[6]Tian Chuanjun,Chen Guanrong.Chaos of a sequence of maps in a metric space[J].Chaos,Solitons and Fractals,2006,28:1067-1075.

[7]Liu Lei,Chen Bin.Onω-limit sets and attraction of non-autonomous discrete dynamical systems[J].J. Korean Math.Soc.,2012,49(4):703-713.

On h-minimal covering of non-autonomous dynamical systems

Zhao Jiaqi,Wang Yangeng
(Department of Mathematics,Northwest University,Xi′an710127,China)

Let(X,d1,f1,∞)and(Y,d2,g1,∞)are non-autonomous discrete dynamical systems,(Y,d2,g1,∞) is quasiconjugate to(X,d1,f1,∞)viah:X→Y.By using the h-minimal covering of autonomous discrete dynamical systems,we can obtain the following resluts:1)For any pointy∈Y,x∈h-1(y),there are h(orb(x,f1,∞))=orb(y,g1,∞)andh(ω(x,f1,∞))=ω(y,g1,∞);2)We define the h-minimal covering of nonautonomous discrete dynamical systems(X,d1,f1,∞).In addition,the existence of the h-minimal covering is studied;3)For any point x∈X,y∈Y,while(ω(x,f1,∞),f1,∞|ω(x,f1,∞))and(ω(y,g1,∞),g1,∞|ω(y,g1,∞))are subsystems of the original systems,we have h(R(f1,∞))=R(g1,∞).These conclusions enriched the contents of non-autonomous discrete dynamical systems.

autonomous discrete dynamical system,non-autonomous discrete dynamical system, the h-minimal covering,topological conjugation,topological quasiconjugation

O178

A

1008-5513(2013)02-0179-06

10.3969/j.issn.1008-5513.2013.02.011

2012-09-10.

陜西省自然科學(xué)基金(2012JM1016).

趙佳琪(1987-),碩士生,研究方向：拓?fù)鋭?dòng)力系統(tǒng).

2010 MSC:54B20