Chain transitivity and uniform persistence

Chain transitivity and uniform persistence

Chaos, Solitons and Fractals 14 (2002) 1071–1076 www.elsevier.com/locate/chaos Chain transitivity and uniform persistence Zheng Zuo-Huan a a,*,1 , ...

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Chaos, Solitons and Fractals 14 (2002) 1071–1076 www.elsevier.com/locate/chaos

Chain transitivity and uniform persistence Zheng Zuo-Huan a

a,*,1

, Huang Tu-Sen

b,2

Institute of Applied Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Science, Beijing 100080, China b Department of Mathematics, Ningbo University, Ningbo City, Zhejiang Province 315211, China Accepted 5 February 2002

Abstract Some properties of attractivity, strong repellors and uniform persistence for continuous maps in metric spaces are presented. A result in [J. Dynam. Differential Equations 13 (2001) 107] on uniform persistence is improved. Ó 2002 Elsevier Science Ltd. All rights reserved.

1. Introduction Pseudo trajectory and chain transitivity have been introduced by Conley [1] and Bowen [2] for flows and diffeomorphisms, respectively. Easton [4] gave a criterion of Lipschitz ergodicity by using strong chain transitivity. Zheng [12] obtained a necessary and sufficient condition of Lipschitz ergodicity by using E-chain transitivity. Some results about chain transitivity and chain recurrence can be found in [5,11,13–16]. In this paper, we discuss some properties of chain transitivity for a discrete semiflow generated by a continuous map f on a metric space X, and then investigate the uniform persistence by these properties. Hirsch et al. [5] gave a sufficient condition that any compact internally chain transitive set is a fixed point when the fixed points are all isolated invariant sets (see [5, Theorem 3.2]). We prove that this condition is also necessary (see Theorem 2.2). Hirsch et al. [5] showed that if a compact internally chain transitive set had a nonempty intersection with the stable set of an attractor then the compact internally chain transitive set would be contained in the attractor. We prove that the compact internally chain transitive set is, in fact, contained in a Morse set of any Morse decomposition of this attractor (see Theorem 2.1). Hirsch et al. [5] proposed some sufficient conditions of stability of uniform persistence (see [5, Theorem 4.4]). We give a necessary and sufficient condition of stability of uniform persistence (see Theorem 3.2). We also generalize a result about strong repellors (see Theorem 3.1).

2. Chain transitivity Let Z be the set of integers and Zþ the set of nonnegative integers. Let X be a metric space with metric d and f : X ! X be a continuous map. A subset A  X is said to be an attractor for f if A is nonempty, compact and invariant ðf ðAÞ ¼ AÞ, and there exists some open neighbourhood U of A in X such that limn!1 supx2U fdðf n ðxÞ; AÞg ¼ 0. A global attractor for f is an attractor which attracts every point in X. For a nonempty invariant set M, the set

*

Corresponding author. Tel.: +86-10-6265-2362; fax: +86-10-6254-1689. E-mail address: [email protected] (Z. Zuo-Huan). 1 This work was supported by National Science Foundation of China (Grant No. 19901034), 10171099. 2 Research of H.T.S. funded by Academy of Mathematics and System Sciences, Chinese Academy of Sciences. 0960-0779/02/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 0 - 0 7 7 9 ( 0 2 ) 0 0 0 4 7 - 4

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W s ðMÞ ¼ fx 2 X j limn!1 dðf n ðxÞ; MÞ ¼ 0g is called the stable set of M. The omega limit set of x is defined in the usual way as xðxÞ ¼ fy 2 X j f nk ðxÞ ! y for some nk ! 1g. A negative orbit through x ¼ x0 is a sequence c ðxÞ ¼ fxk g0k¼ 1 such that f ðxk 1 Þ ¼ xk for integers k 6 0. There may be no negative orbit through x, and even if there is one, it may not be unique. Of course, a point of an invariant set always has at least one negative orbit contained in the invariant set. For a given negative orbit c ðxÞ we define its alpha limit set as aðc Þ ¼ fy 2 X j xnk ! y for some nk ! 1g. Definition 2.1 [1,4,5]. A point x 2 X is said to be chain recurrent if for any  > 0, there exists a finite sequence of points x1 ; x2 ; . . . ; xm in X ðm > 1Þ with x1 ¼ x ¼ xm such that dðf ðxi Þ; xiþ1 Þ <  for all 1 6 i 6 m 1. The set of all chain recurrent points for f : X ! X is denoted RðX ; f Þ. Let A  X be a nonempty invariant set. We call A internally chain recurrent if RðA; f Þ ¼ A, and internally chain transitive if the following stronger condition holds: for any a; b 2 A and  > 0, there exists a finite sequence x1 ; x2 ; . . . ; xm in A with x1 ¼ a; xm ¼ b such that dðf ðxi Þ; xiþ1 Þ <  for all 1 6 i 6 m 1. The sequence fx1 ; x2 ; . . . ; xm g is called an -chain in A connecting a and b. Bowen [2] proved that omega limit sets of any precompact positive orbits of continuous invertible maps are internally chain transitive. Robinson [3] proved that omega limit sets of any precompact positive orbits of continuous maps are internally chain recurrent. Hirsch et al. [5] proved that omega limit sets of any precompact positive orbits of continuous maps are internally chain transitive. Let fSn : X ! X gn P 0 be a sequence of continuous maps. The discrete dynamical process (or process for short) generated by fSn g is the sequence fTn : X ! X gn P 0 defined by T0 ¼ I ¼ the identity map of X and Tn ¼ Sn 1 Sn 2    S1 S0 ;

n P 1:

The orbit of x 2 X under this process is the set cþ ðxÞ ¼ fTn ðxÞ j n P 0g, and its omega limit set is   xðxÞ ¼ y 2 X j ð9nk ! 1Þ lim Tnk ðxÞ ¼ y : k!1

The process fTn : X ! X g is asymptotically autonomous, if there exists a continuous map S : X ! X such that nj ! 1;

xj ! x ) lim Snj ðxj Þ ¼ SðxÞ: j!1

We also say that Tn is asymptotic to S. Zhao [10] proved that omega limit set x of any precompact positive orbit of an asymptotically autonomous process fTn : X ! X gn P 0 with limit S : X ! X is nonempty, compact, invariant, and internally chain recurrent for S. Hirsch et al. [5] proved that x is internally chain transitive for S. Let A and B be two nonempty compact subsets of X. The Hausdorff distance between A and B is defined by dH ðA; BÞ ¼ maxðsupfdðx; BÞ j x 2 Ag; supfdðx; AÞ j x 2 BgÞ:

ð2:1Þ

Definition 2.2 [5,10,11]. Let S : X ! X be a continuous map. A sequence fxn g in X is an asymptotic pseudo orbit of S if lim dðSðxn Þ; xnþ1 Þ ¼ 0:

n!1

The omega limit set of fxn g is the set of limits of subsequences. Lemma 2.1 (Hirsch–Smith–Zhao Lemma). Let S; Sn : X ! X for n P 1 be continuous. Let fDn g be a sequence of nonempty compact subsets of X with limn!1 dH ðDn ; DÞ ¼ 0 for some compact subset S D of X. Assume that for each n P 1; Dn is invariant and internally chain transitive for Sn . If Sn ! S uniformly on D [ ð n P 1 Dn Þ, then D is invariant and internally chain transitive for S. Let f : X ! X be a continuous map. A nonempty invariant subset M of X is said to be isolated for f if it is the maximal invariant set in some neighbourhood of itself. Lemma 2.2 (Strong attractivity). Let A be an attractor and L be a compact internally chain transitive set for f : X ! X . If L \ W s ðAÞ 6¼ ;, then L  A. Let S be a compact metric space and f : S ! S be a continuous map with f ðSÞ ¼ S. An ordered collection fM1 ; . . . ; Mk g of pairwise disjoint, compact and invariant subsets of S is called a Morse decomposition of S if for each

Z. Zuo-Huan, H. Tu-Sen / Chaos, Solitons and Fractals 14 (2002) 1071–1076

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Sk

x 2 S n i¼1 Mi , there exists an i with xðxÞ  Mi and for any negative orbit c through x there exists a j > i with aðc Þ  Mj . Theorem 2.1. Let f : X ! X be a continuous map. Let A be an attractor for f and fM1 ; . . . ; Mk g be a Morse decomposition of A. Let L be a compact internally chain transitive set for f. If L \ W s ðAÞ 6¼ ;, then there exists an i such that L  Mi . Proof. By Lemma 2.2, L \ W s ðAÞ 6¼ ; implies that L  A. It is easy to see that L is internally S chain recurrent as L is internally chain transitive. It then follows that [11, Lemma 4.3], implies that L  RðA; f Þ  ki¼1 Mi . Then the invariant connectedness of L implies that L  Mi for some i.  Let A and B be two isolated invariant sets (not necessarily distinct). A is said to be chained to B, written A ! B, if there exists a full orbit through some x 62 A [ B such that xðxÞ  B and aðxÞ  A. A finite sequence fM1 ; . . . ; Mk g of invariant sets is called a chain if M1 ! M2 !    ! Mk . The chain is called a cycle if Mk ¼ M1 . When each fixed point of f is an isolated invariant set, Hirsch et al. [5] proved: if there is no cycle of fixed points and that every precompact orbit converges to some fixed point of f, then any compact internally chain transitive set is a fixed point of f (see [5, Theorem 3.2]). We shall show that the converse is also true. Theorem 2.2. Assume that each fixed point of f is an isolated invariant set. Then any compact internally chain transitive set is a fixed point of f if and only if that there is no cycle of fixed points and that every precompact orbit converges to some fixed point of f. Proof. The sufficiency follows from [5, Theorem 3.2]. To prove the necessity, we can suppose that positive orbit cþ ðxÞ through x 2 X is precompact. It is easy to see that xðxÞ is internally chain transitive (see [5, Lemma 2.1]), hence xðxÞ is a fixed point of f by the assumption, and limn!1 dðf n ðxÞ; xðxÞÞ ¼ 0. Suppose there exists a finite sequence fq1 ; q2 ; . . . ; qm gm P 1 , where qi ði ¼ 1; . . . ; mÞ is a fixed point of f, such that q1 ! q2 !    ! qm ! q1 . For i ¼ 1; . . . ; m 1, choose a full orbit cðxi Þ through some xi such that xðxi Þ ¼ fqiþ1 g and aðxi Þ ¼ fqi g. ForSi ¼ m, choose a full orbit cðxm Þ through some xm such that xðxm Þ ¼ fq1 g and aðxm Þ ¼ fqm g. Set L ¼ fq1 ; q2 ; . . . ; qm g [ ð mi¼1 cðxi ÞÞ. Then L is a compact internally chain transitive set and L is not a fixed point of f. But this contradicts the assumption.  3. Uniform persistence Throughout this section, X is a metric space with metric d; f : X ! X is a continuous map. Let X0  X be an open set with f ðX0 Þ  X0 . Define oX0 ¼ X n X0 , and Mo ¼ fx 2 oX0 j f n ðxÞ 2 oX0 ; nSP 0g, which may be empty. We assume hereafter that every positive orbit of f is precompact. If S  X , define XðSÞ ¼ x2S xðxÞ. For any subset A of X, we shall  to denote its closure. use A Theorem 3.1. Assume that f has a global attractor A and the maximal compact invariant set Ao ¼ A \ Mo of f in oX0 , possibly empty, admits a Morse decomposition fM1 ; . . . ; Mk g. Let B ¼ fi 2 f1; . . . ; kg j Mi \ X0 6¼ ;g. If for each j 2 B, there hold (i) Mj is isolated in X, (ii) W s ðMj Þ \ X0 ¼ ;. Then there exists d > 0 such that for any compact internally chain transitive set L with L 6 Mi for all 1 6 i 6 k, there holds inf x2L dðx; oX0 Þ > d. Proof. We first prove the following claim.



Claim. There is an  > 0 such that if L is a compact internally chain transitive set not contained in any Mi , then supx2L dðx; oX0 Þ > . Proof of the Claim. Assume that, by contradiction, there exists a sequence of compact internally chain transitive sets fDn j n P 1g with Dn  6 Mi ; 1 6 i 6 k, such that lim sup dðx; oX0 Þ ¼ 0:

n!1 x2Dn

ð3:1Þ

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Since W s ðAÞ ¼ X , by Lemma 2.2, we have Dn  A for all n P 1. Suppose that there is an n 2 f1; 2; . . .g such that S Dn  oX0 . By the invariance of Dn , we have Dn  Ao and [11, Lemma 4.3], implies that Dn  RðAo ; f Þ  ki¼1 Mi . Then the invariant connectedness of Dn implies that Dn  Mi for some i, contradicting to our assumption. Hence Dn \ X0 6¼ ;

ð3:2Þ

for all n P 1. In the compact metric space of compact nonempty subsets of A with Hausdorff distance dH , the sequence fDn j n P 1g has a convergent subsequence. Without loss of generality, we assume that for some nonempty compact set D  A; limn!1 dH ðDn ; DÞ ¼ 0. Then for any x 2 D, there exists xn 2 Dn such that limn!1 xn ¼ x. By (3.1), there exists yn 2 oX0 such that limn!1 dðxn ; yn Þ ¼ 0. Hence limn!1 yn ¼ x and x 2 oX0 ¼ oX0 . Therefore D  oX0 . Lemma 2.1 implies that D is invariant S and internally chain transitive for f. It then follows that D  Ao and [11, Lemma 4.3] imply that D  RðAo ; f Þ  ki¼1 Mi . Then the invariant connectedness of D implies that D  Mi for some i. Eq. (3.2) and the fact that limn!1 dH ðDn ; DÞ ¼ 0 imply that D \ X0 6¼ ;, hence i 2 B. Because Mi is isolated in X, it follows that there exists an integer N P 0 such that Dn  Mi for all n P N , contradicting to our assumption again. This proves the claim.  Proof of Theorem 3.1 (continued). We now prove the theorem by contradiction. Assume that there exists a sequence of compact internally chain transitive sets fLn j n P 1g with Ln 6 Mi ; 1 6 i 6 k; n P 1, such that lim inf dðx; oX0 Þ ¼ 0:

ð3:3Þ

n!1 x2Ln

As in the proof of the claim, we can assume that Ln  A; n P 1, Ln \ X0 6¼ ;;

n ¼ 1; 2 . . . ;

ð3:4Þ

and limn!1 dH ðLn ; LÞ ¼ 0, where L is a compact internally chain transitive set for f : X ! X ; L \ X0 6¼ ;; L \ oX0 6¼ ; and L 6 Mi for each 1 6 i 6 k. By the above claim, we can choose a 2 L \ oX0 and b 2 L with dðb; oX0 Þ > . Let fxn j n P 0g be the asymptotic pseudo orbit determined by a and b in L. Then there exist two subsequences fxmj g and fxrj g such that xmj ¼ a and xrj ¼ b for all j P 1. Note that dðxsj þ1 ; f ðxÞÞ 6 dðxsj þ1 ; f ðxj ÞÞ þ dðf ðxsj Þ; f ðxÞÞ: By induction, it then follows that for any convergent subsequence xsj ! x 2 X ; j ! 1, there holds limj!1 xsj þn ¼ f n ðxÞ for any integer n P 0. We can further choose two sequences flj g and fnj g with lj < mj < nj and limj!1 lj ¼ 1 such that dðxlj ; oX0 Þ > ; dðxk ; oX0 Þ 6 

dðxnj ; oX0 Þ > ;

j P 1;

for any integer k 2 ðlj ; nj Þ; j P 1:

ð3:5Þ ð3:6Þ

Since L is compact, we can assume that, after taking a convergent subsequence, limj!1 xlj ¼ x 2 L. Then (3.5) implies that dðx; oX0 Þ P  and hence x 2 X0 . So xðxÞ \ X 0 6¼ ;:

ð3:7Þ

We further claim that the sequence fnj lj g is unbounded. Assume that, by contradiction, fnj lj g is bounded. Then fmj lj g is also bounded and hence we can assume that, after choosing a subsequence, mj lj ¼ m, where m is an integer. Since f ðX0 Þ  X0 , we have a ¼ limj!1 xmj ¼ f m ðxÞ 2 X0 , which contradicts a 2 oX0 . Thus we can assume that, by taking a subsequence, limj!1 ðnj lj Þ ¼ 1. Then for any integer n P 1, there exists an integer J ðnÞ P 1 such that nj lj > n for all j P J ðnÞ. Eq. (3.6) implies that dðxlj þn ; oX0 Þ 6  for all j P J ðnÞ. Since f n ðxÞ ¼ lim xlj þn , it folj!1 lows that dðf n ðxÞ; oX0 Þ 6 ;

n P 1:

ð3:8Þ

By the compactness of L, it follows that xðxÞ is a nonempty, compact, internally chain transitive set for f : X ! X . Eq. (3.8) implies that sup dðy; oX0 Þ 6 :

ð3:9Þ

y2xðxÞ

Appealing again to the claim, we conclude that xðxÞ  Mi for some 1 6 i 6 k, and hence x 2 W s ðMi Þ \ X0 . Now (3.7) implies i 2 B. But these contradict the assumption (ii) and the theorem is proved.  Remark 3.1. Hirsch et al. [5] proved the similar result (see [5, Theorem 4.3]). But, in [5, Theorem 4.3], the conditions (i) and (ii) must hold for all 1 6 i 6 k.

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Proposition 3.1. Assume that f has a nonempty compact invariant susbset A in X, Ao ¼ A \ Mo , possibly empty, admits a Morse decomposition fM1 ; . . . ; Mk g. Let B ¼ fi 2 f1; . . . ; kg j Mi \ X0 6¼ ;g. If for any j 2 B, there hold (i) Mj is isolated in A, (ii) W s ðMj Þ \ X0 \ A ¼ ;. Then there exists d > 0 such that for any compact internally chain transitive set L with L 6 Mi for all 1 6 i 6 k, there holds inf x2L dðx; oX0 Þ > d. Proof. This proof is similar to that of Theorem 3.2 and is omitted.



Remark 3.2. By Lemma 2.2, A is a global attractor for f : X ! X implies that L  A for any compact internally chain transitive set L for f. Hence Proposition 3.1 generalizes Theorem 3.1 and [5, Theorem 4.3]. Definition 3.1 [5–9]. f : X ! X is said to be uniformly persistent with respect to ðX0 ; oX0 Þ if there exists g > 0 such that lim inf n!1 dðf n ðxÞ; oX0 Þ P g for all x 2 X0 . If ‘‘inf’’ in this inequality is replaced with ‘‘sup’’, f is said to be weakly uniformly persistent with respect to ðX0 ; oX0 Þ. Let Sm : X ! X ; m P 0, be a sequence of continuous maps such that every positive orbit for Sm has compact closure, and Sm ðX0 Þ  X0 . Assume that Sm satisfies the conditions of Theorem 3.1. Precisely, for any fixed integer m P 0; Sm has a global attractor Am , the maximal compact invariant set Amo ¼ Am \ Mom of Sm in oX0 , possibly empty, admits a Morse decomposition m g; fM1m ; . . . ; Mlm

Bm ¼ fi 2 f1; . . . ; lm g j Mim \ X0 6¼ ;g;

such that Mjm is isolated in X and W s ðMj Þ \ X0 ¼ ; for any j 2 Bm . By Theorem 3.1, there exists dm > 0 such that for any compact internally chain transitive set L for Sm with L 6 Mim for all 1 6 iS 6 lm , there holds inf x2L dðx; oX0 Þ > dm . Let xm ðxÞ denote the omega limit set of x for discrete semiflow Sm , and set W ¼ m P 0;x2X xm ðxÞ. Because the omega limit set of any precompact positive orbit is compact internally chain transitive, it follows that for any precompact positive orbit m cþ m ðxÞ with xm ðxÞ 6 Mi ; 1 6 i 6 lm , there holds inf dðy; oX0 Þ P dm :

y2xm ðxÞ

ðÞ

Let d0m denote the supremum of dm satisfying (*). Theorem 3.2. Assume that W is compact and Sm ! S0 uniformly on W. In addition, assume that Sm satisfies (i) and (ii) of Theorem 3.1 for all m P 0. Then there exists g > 0 and a positive integer N such that lim infn!1 dðSmn x; oX0 Þ P g for m P N and x 2 X0 if and only if there exists g0 > 0 such that inffd0m j m > 0g P g0 . Proof. Sufficiency. By the assumption, there exists g0 > 0 such that inffd0m j m > 0g P g0 . Assume that, by contradiction, there exist a point sequence fxk g  X0 and a positive integer sequence fmk g with limk!1 mk ¼ 1 such that lim lim inf dðSmn k ðxk Þ; oX0 Þ ¼ 0:

k!1

n!1

ð3:10Þ

[5, Lemma 2.1], implies that xmk ðxk Þ is compact internally chain transitive with respect to Smk . In the compact metric space of compact nonempty subsets of W with Hausdorff distance dH , the sequence xmk ðxk Þ has a convergent subsequence. Without loss of generality, we assume that for some nonempty compact L  W , limk!1 dH ðxmk ðxk Þ; LÞ ¼ 0. By Lemma 2.1, L is internally chain transitive for S0 . Clearly, there exists yk 2 xmk ðxk Þ such that lim dðyk ; oX0 Þ ¼ 0, and k!1 hence L \ oX0 6¼ ;. Theorem 3.1, applied to S0 , implies that L  Mi0 for some 1 6 i 6 l0 . Because fxk g  X0 and 0 Sm ðX0 Þ  X0 ; xmk ðxk Þ  X0 , hence i 2 B . Choose k sufficiently large such that n g o xmk ðxk Þ  x 2 X j dðx; Mi0 Þ < 0 : ð3:11Þ 2 Now Mi0  oX0 implies that inf x2xmk ðxk Þ dðx; oX0 Þ < g20 < d0mk . Therefore, there exists 1 6 j 6 lmk such that xmk ðxk Þ  Mjmk , and hence xk 2 W s ðMjmk Þ. Since xmk ðxÞ  X0 , we have j 2 Bmk , which is a contradiction to that W s ðMjmk Þ \ X0 ¼ ; for all j 2 Bmk .

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Necessity. Assume that there exist g > 0 and a positive integer N such that lim dðSmn ðxÞ; oX0 Þ P g

n!1

for all m P N and x 2 X0 . Let g0 ¼

1 inffg; d01 ; d02 ; . . . ; d0N 1 g: 2

ð3:12Þ

Let m be any fixed positive integer. If m 6 N 1, then d0m > g0 . If m P N, then for any x 2 X with xm ðxÞ 6 Mim ; 1 6 i 6 lm , there holds inf dðy; oX0 Þ P d0m :

ð3:13Þ

y2xm ðxÞ

Therefore, there exists a positive integer n1 such that Smn1 ðxÞ 2 X0 , and hence lim inf dðSmn ðSmn1 ðxÞÞ; oX0 Þ P g:

ð3:14Þ

n!1

Then inf dðy; oX0 Þ ¼

y2xm ðxÞ

inf

y2xm ðSmn1 ðxÞÞ

dðy; oX0 Þ P g:

Thus d0m P g P g0 and inffd0m j m > 0g P g0 .



Remark 3.3. Theorem 3.2 holds if the corresponding conditions are replaced by (C1) and (C20 ) in [5].

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