THE SET OF RECURRENT POINTS OF A CONTINUOUS SELF-MAP ON AN INTERVAL AND STRONG CHAOS

Transcription

1 J. Appl. Math. & Computing Vol. 4(2004), No. - 2, pp THE SET OF RECURRENT POINTS OF A CONTINUOUS SELF-MAP ON AN INTERVAL AND STRONG CHAOS LIDONG WANG, GONGFU LIAO, ZHENYAN CHU AND XIAODONG DUAN Abstract. In this paper, we discuss a continuous self-map of an interval and the existence of an uncountable strongly chaotic set. It is proved that if a continuous self-map of an interval has positive topological entropy, then it has an uncountable strongly chaotic set in which each point is recurrent, but is not almost periodic. AMS Mathematics Subject Classification : 28D20, 54H20, 58F, 58F3. Key word and phrases : Strong chaos, topological entropy, recurrence, regular shift invariant.. Introduction Throughout this paper, X will denote a compact metric space with metric d, I is the closed interval [0, ]. f : X X is continuous, f n will denote the n-fold iterates of f. Let (X, f) be a compact system. If α is an open cover of X let N(α) denote the number of sets in a finite subcover of α with smallest cardinality. If α and β are two open covers of X let α β = {A B : A α, B β} and f (α) ={f (A) :A α}. For an open α let n i=0 f i (α) =α f (α) f (n ) (α) and ( n ) h(f,α) = lim n n log N f i (α). i=0 The topological entropy of f is defined by h(f) =suph(f,α). α Received March 5, Revised August 26, c 2004 Korean Society for Computational & Applied Mathematics and Korean SIGCAM. 277

2 278 L. Wang, G. Liao, Z. Chu and X. Duan Let x X. Then y X is said to be an ω limit point of x if the sequence f(x),f 2 (x), has a subsequence converging to y. The set of all ω limit points of x is called ω limit set of x, denoted by ω(x, f). The set of all ω limit points of f is denoted by ω(f). x X is called a recurrent point of f if the sequence f(x),f 2 (x), has a subsequence converging to x. The set of all recurrent points of f is denoted by R(f). x X is called an almost periodic point of f if for any ε>0, one can find N>0such that for any integer q, there is an integer r with q r<n+ q satisfying d(f r (x),x) <ε. The set of all almost periodic points of f is denoted by A(f). For x, y in X, any real number t, and positive integer n, let ξ n (f,x,y,t)=#{ i d(f i (x),f i (y)) <t, i n }, where we use #( ) to denote the cardinal number of a set. Let and let F (f,x,y,t) = lim inf n F (f,x,y,t) = lim sup n n ξ n(f,x,y,t), n ξ n(f,x,y,t). Definition. Call x, y X a pair of points displaying strong chaos if () F (f,x,y,t) = 0 for some t>0, and (2) F (f,x,y,t) = for any t>0. Definition 2. f is said to display strong chaos if there exists an uncountable set D X such that any different two points in D display strong chaos. For a continuous map f : I I, Schweizer and Smítal [] have proved: (C ) If f has zero topological entropy, then no pair of points can form a strongly chaotic set. (C 2 ) If f has positive entropy, then there exists an uncountable strongly chaotic set in which each member is an ω-limit point of f. One may pose the following questions: (Q ) Is (C ) still true for a continuous map of any compact metric space X? (Q 2 ) Is there an uncountable strongly chaotic set in which each member is a recurrent point of f under the hypothesis of (C 2 )? A negative answer to (Q ) has been given in [2], where a minimal strongly chaotic sub-shift having zero topological entropy was constructed. In this paper, a positive answer to (Q 2 ) is given. In fact, we will prove The Main Theorem. Let f : I I be continuous. If f has positive topological entropy, then it has an uncountable strongly chaotic set in which each point is recurrent, but is not almost periodic.

3 The set of recurrent points of a continuous self-map 279 Corollary. Let f : I I be continuous. Then the following are equivalent : () f has positive topological entropy. (2) f has an uncountable strongly chaotic set in which each point is recurrent. (3) f has an uncountable strongly chaotic set in which each point is an ω- limit one. (4) f has an uncountable strongly chaotic set containing two points. Remark. In the Corollary above, () (3) and (4) () are the results in [3]. We can obtain (2) (3), because each recurrent point must be an ω-limit one. (3) (4) is obvious. However, () (2) is the Main Theorem. Many authors are interested in the chaotic set in the sense of Li and Yorke (see [4]). Since any strongly chaotic set must be chaotic, some results in [4], [5], [6] and [7] may be also deduced directly from the Main Theorem of this paper. 2. Basic definitions and preparations Let S = {0, }, Σ={x = x x 2 x i S, i =, 2, } and define ρ : Σ Σ R as follows : for any x, y Σ, if x = x x 2 and y = y y 2, 0 if x = y, ρ(x, y) = 2 k if x y, and k = min{n x n y n }. It is not difficult to check that ρ is a metric on Σ. The space (Σ,ρ) is compact and called the one-sided symbolic space with two symbols. Define σ :Σ Σ by σ(x x 2 )=x 2 x 3 for any x = x x 2 Σ. Then σ is continuous and called the shift on Σ. Call E a tuple ( over S = {0, }) if it is a finite arrangement of elements in S. IfE = e e 2 e m where e i S, i m, the length of E is said to be m, denoted by E = m. For an arbitrary tuple B = b b 2 b n, [B] ={x = x x 2 Σ x i = b i, i n} is called a cylinder generated by B. For any n, let ß n = { [b b 2 b n ] b i =0 or, i n }. Then the collection ß n is a subalgebra which generates the σ-algebra of Borel n= subsets of Σ. Let h : X Σ be a continuous map. We use I [B] to denote h [B] for any [B] ß n. Definition 3. Let f : I I be continuous. A compact set Λ I is said to be a regular shift invariant set of f if :

4 280 L. Wang, G. Liao, Z. Chu and X. Duan () f(λ) Λ, (2) There exists a continuous surjection h : Λ Σ satisfying (a) h f Λ =σ h, (b) For any n and any different [B], [C] ß n, I [B] I [C] =. Definition 4. ß(X) is an σ-algebra of Borel subsets of X (see [8]). A probability measure µ is an invariant measure of f on (X, ß(X)), if µ(f (B)) = µ(b) for any B ß(X). We denote the set of all invariant measures of f by M(X, f). µ M(X, f) is ergodic (f can be regarded ergodic too) if the only members B of ß(X) with f (B) =B satisfy µ(b) =0orµ(B) =. If µ is the unique member of M(X, f), it must be ergodic [8] and we say that f is uniquely ergodic. Lemma. Let f : X X, g : Y Y be continuous, where X and Y are both compact metric spaces. If there exists a continuous surjection h : X Y such that g h = h f, then () h(a(f)) = A(g), (2) h(r(f)) = R(g). For a proof see [9]. Lemma 2. There exists an uncountable set T on the one-sided symbolic space, satisfying () T R(σ) A(σ), (2) σ T is strongly chaotic, (3) σ T is uniquely ergodic. For a proof see [0]. Lemma 3. If f has positive topological entropy, then there exists N > 0 such that f N has a regular shift invariant set. For a proof see []. Lemma 4. Let σ :Σ Σ be continuous. If µ is the only invariant probability measure of σ R(σ) A(σ), then µ({x}) equals to zero for any x R(σ) A(σ). Proof. Let x R(σ) A(σ). We first claim that {x}, {σ (x)}, {σ 2 (x)}, are pairwise disjoint. Assume the claim to be false. Then {σ m (x)} {σ n (x)} for some m and n with m>n 0. Take arbitrarily y {σ m (x)} {σ n (x)}. Then we have σ m (y) =σ n (y) =x. Furthermore, σ m n (x) =σ m n (σ n (y)) = σ m (y) =x,

5 The set of recurrent points of a continuous self-map 28 i.e., x is a periodic point, which contradicts with x R(σ) A(σ). Since µ is an invariant probability measure of σ R(σ) A(σ) and the set of simple point on (Σ,ρ) is closed, {x} ß(Σ) and µ({x}) =µ(σ ({x})) = µ(σ 2 ({x})) = = µ(σ n ({x})). By the countable additivity of µ, we get µ({x}) =0. Lemma 5. Suppose T = R(σ) A(σ). If µ is the only invariant probability measure of σ T, then the sequence { µ ([b 2 b 2 b n ])} of real numbers converges to zero uniformly in b i {0, }, i n, asn. Proof. We use diam( ) to denote the diameter of a set. For any ε>0and any x T, by Lemma 4, there is an open neighborhood V x of x such that µ(v x ) <ε. And by the definition of [ b b 2 b n ], there exists N>0such that diam([ b b 2 b n ]) <εholds uniformly in b i {0, }, i n, when n N. Thus for any x [ b b 2 b n ] T, there exists N > 0 such that x must be contained in some V x when n N. So µ ([b b 2 b n ])=µ([b b 2 b n ] T ) <ε. Lemma 6. Let f : X X be continuous. Then f is strongly chaotic if and only if f n is strongly chaotic for any positive integer n. To prove Lemma 6, we first prove two propositions. Proposition. Let f : X X be continuous. then the following results hold for x, y X, N>0 and t>0. (i) If F (f,x,y,t)=0, then F (f N,x,y,t)=0, (ii) If F (f,x,y,t)=, then F (f N,x,y,t)=. Proof. (i) If F (f,x,y,t) = 0, then there is a sequence such that ξ nk (f,x,y,t) 0 (k ). () Put where [ nk ] N m k = [ nk ], N denotes the integral part of N. Then for each k, ξ mk ( f N,x,y,t ) ξ nk (f,x,y,t).

6 282 L. Wang, G. Liao, Z. Chu and X. Duan It follows from () that for k, ξ mk (f N,x,y,t) 0, moreover, N ξ mk (f N,x,y,t) 0. This gives ξ mk (f N,x,y,t) 0 (k ). m k So F (f N,x,y,t)=0. (ii) If F (f,x,y,t) =, then there exists a sequence such that ξ nk (f,x,y,t) (k ). (2) Let δ nk (f,x,y,t)=#{ i d(f i (x),f i (y)) t,0 i< }. (3) Then by (2), since for all > 0, δ nk (f,x,y,t) 0 (k ), ξ nk (f,x,y,t)+ δ nk (f,x,y,t)=. (4) Put [ nk ] m k =. N To be similar to the argument given above, we get that for k, δ mk (f N,x,y,t) 0, m k and furthermore, ( ξ mk f N,x,y,t ) = ( δ mk f N,x,y,t ). m k m k This proves F ( f N,x,y,t ) =. Proposition 2. Let f : X X be continuous. then the following results hold for x, y X, N>0. (i) If F ( f N,x,y,s ) = 0 for s > 0, then there exists t > 0 such that F (f,x,y,t)=0, (ii) If F ( f N,x,y,s ) =for all s>0, then F (f,x,y,t)=for all t>0.

7 The set of recurrent points of a continuous self-map 283 Proof. (i) If F (f N,x,y,s) = 0 for s>0, then there exists a sequence such that ξ nk (f N,x,y,s) 0 (k ). (5) Since X is compact, f i is uniformly continuous for each i =, 2,,N. Therefore, for fixed s>0, there exists t>0such that for all p, q X and each i =, 2,,N, d(f i (p),f i (q)) t provided d(f N (p),f N (q)) s. So we have N(δ nk (f N,x,y,s) ) δ nk N(f,x,y,t), (6) where δ nk ( ) and δ nk N( ) are as in (3). Put m k = N. According to (6), we obtain ξ mk (f,x,y,t)= δ mk (f,x,y,t) m k m k = N m k Nδ nk (f N,x,y,s)+ N m k m k δ nk (f N,x,y,s)+ N m k = ξ nk (f N,x,y,s)+. Noting that 0ask, by (5) and (7), we have ξ mk (f,x,y,t) 0 (k ). m k This shows F (f,x,y,t)=0. (ii) Suppose F (f N,x,y,s) = for all s>0. For any given t>0, since f i is uniformly continuous for each i =, 2,,N, there exists s>0 such that for all p, q X and each i =, 2,,N, d(f i (p),f i (q)) <tprovided d (p, q) <s. For such s, F (f N,x,y,s) = by the hypothesis. So there exists a sequence, such that for k, ξ nk (f N,x,y,s). Put m k = N. It is easy to see that ( Nξ nk f N,x,y,s ) ξ mk (f,x,y,t). Dividing both sides by m k gives ξ nk ( f N,x,y,s ) m k ξ mk (f,x,y,t) for each k. (7)

8 284 L. Wang, G. Liao, Z. Chu and X. Duan Therefore, by we have So ξ nk ( f N,x,y,s ) m k ξ mk (f,x,y,t) F (f,x,y,t)=. (k ), (k ). Proof of Lemma 6. The necessity holds by Proposition and the sufficiency holds by Proposition 2. Lemma 7. Let f : X X be continuous. Then R(f) =R(f n ) and A(f) = A(f n ) for any positive integer n. For a proof see [2]. Lemma 8. Period 2 n if and only if topological entropy is positive. For a proof see [5]. 3. Proof of the main theorem Proof. By the hypothesis, f has positive entropy, and by Lemma 3, there exists N>0such that f N has a regular shift invariant set, denoted by Λ. Thus there is a continuous surjection h :Λ Σ such that for any x Λ, h f N (x) =σ h(x). According to Lemma 2, there is an uncountable set T R(σ) A(σ) such that T is a strongly chaotic set and σ T has the only ergodic measure µ. Denote, for simplicity, g = f N Λ. For any y T, by Lemma, there exists x R(g) A(g) such that h(x) = y. Let D = { x x R(g) A(g),h(x) =y and y T }. Then D Λ and D is an uncountable set. To complete the theorem, it suffices to show that D is a strongly chaotic set of f. For any distinct x,x 2 D, there exists y,y 2 T such that h(x i )=y i, i =, 2. Since y and y 2 are in a strongly chaotic set of σ, there exists s>0 and a sequence such that ξ nk (σ, y,y 2,s) 0 (k ). (8)

9 The set of recurrent points of a continuous self-map 285 Choose an N>0such that diam([b]) <s for any [B] ß N. Let t = min { d (I [B],I [C] ) [B], [C] ß N and [B] [C] }, where d (I [B],I [C] ) = inf{ d ( p, q) p I [B], q I [C] }. By the properties of g, d (I [B],I [C] ) > 0 for any distinct [B], [C] ß N and so t>0. It is easily seen that for any i 0, ρ(σ i (y ),σ i (y 2 )) s σ i (y ) [B],σ i (y 2 ) [C] for some distinct [B], [C] ß N (since diam([b]) <s) g i (x ) I [B],g i (x 2 ) I [C] and d(i [B],I [C] ) t d(g i (x ),g i (x 2 )) t, therefore, we have for each k ξ nk (g, x,x 2,t) ξ nk (σ, y,y 2,s). By (8), we get ξ nk (g, x,x 2,t) 0 (k ), and hence F (g, x,x 2,t)=0. (9) We now prove F (g, x,x 2,t) = for any t>0. For any given t>0 and ε>0, choose an integer k>0, such that tk >. And by Lemma 5, we may also choose an N large enough, such that for any [B] ß N T, µ([b]) < ε, i.e., for any 2k y T, lim n n #{ i σi (y) [B], 0 i n } < ε 2k. (0) Put s = 2. N Since F (σ, y,y 2,s) =, there exists a sequence n j such that ξ nj (σ, y,y 2,s) n j (j ). () Denote, for simplicity, θ nj = #{ i g i (x ),g i (x 2 ) I n [B], 0 i<n j }. j Note that [B] ß N T ρ(σ i (y ),σ i (y 2 )) <s σ i (y ),σ i (y 2 ) [B] for some [B] ß N T g i (x ),g i (x 2 ) I [B] for some [B] ß N T. (2)

10 286 L. Wang, G. Liao, Z. Chu and X. Duan According to (), we have θ nj (j ). (3) Thus we can from (0), (2) and (3) choose N large enough, such that for n j >N, #{ i g i (x ),g i (x 2 ) I [B], 0 i<n j } < ε for any [B] ß N T, n j 2k and θ nj < ε 2. (4) On one hand, by the definition of θ nj, ε θ nj 2k = θ nj [B] ß N T diam(i [B] )<t [B] ß N T diam(i [B] ) t [B] ß N T diam(i [B] ) t n j #{ i g i (x ),g i (x 2 ) I [B], 0 i<n j } (5) n j #{ i g i (x ),g i (x 2 ) I [B], 0 i<n j } n j ξ nj (g, x,x 2,t). On the other hand, because of the choice of k, there exist at most k different [B] with diam(i [B] ) t in ß N T. In fact, since tk >, if there exists k >k such that k different [B] satisfy diam(i [B] ) t, the diam(i [B] ) tk >. However, diam(i [B] ) k diam(i [B] ), [B] ß N T this is contradictious. By (5), we have θ nj ε 2 = θ ε n j k 2k ξ nj (g, x,x 2,t). n j Combining this with (4), we see that for n j >N, 0 n j ξ nj (g, x,x 2,t) θ nj + ε 2 < ε 2 + ε 2 = ε, which gives F (g, x,x 2,t)=. (6) By (9), (6) and the arbitrariness of x and x 2, we know that D is an uncountable strongly chaotic set of g. Thus, we have proved that f N has an uncountable strongly chaotic set D in R(f N ) A(f N ). By Lemma 6, D is also a strongly chaotic set of f. And according to Lemma 7, D R(f) A(f).

11 The set of recurrent points of a continuous self-map 287 Example. Let x + f(x) = 2, if 0 x 2 2x +2, if <x. 2 It is easy to see that f C 0 (I,I) and zero is a point of period 3 of f. By Lemma 8, the topological entropy of f is positive. And by the Main Theorem, f is strongly chaotic on the set R(f) A(f). References. B. Schweizer and J. Smítal, Measures of chaos and a spectral decomposition of dynamical systems on the interval, Trans. Amer. Math. Soc. 344 (994), P. Erdő and A. H. Stone, Some remarks on almost periodic transformations, Bull. Amer. Math. Soc. 5 (945), T. Y. Li, M. Misiurewicz, G. Pianigiam and J. Yorke, No division implies chaos, Trans. Amer. Math. Soc. 273 (982), R. S. Yang, Pseudo-shift-invariant sets and chaos, Chinese Ann. Math. Ser. A 3 (992), (in chinese) 5. W. H. Gottschalk, Orbit-closure decompositions and almost periodic properties, ibid. 50 (944), T. Y. Li, M. Misiurewicz, G. Pianigiam and J. Yorke, Odd chaos, Phys. Lett. A 87 (982), G. F. Liao, Chian recurrent orbits of mapping of the interval, Northeast Math. J. 2 (986), P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, New York Heidelberg Berlin, Z. L. Zhou and W. H. He, Administrative levels of orbit structure and topological semiconjugate, Science of China (A), 25(5) (995), L. D. Wang, Z. Y. Chu and G. F. Liao, Recurrence, strong chaos and unique ergodicity, Journal of Jilin Normal University ( Natural Science Edition), 24() (2003), -5. (in Chinese). L. S. Block and W. A. Coppel, Dynamics in One Dimension, Springer-Verlag, New York Heidelberg Berlin, (992), J. C. Xiong, Set of almost periodic points of a continuous self-map of an interval, Acta. Math. Sci., New Series, 2 (986), B. S. Du, Every chaotic interval map has a scrambled set in the recurrent set, Bull. Austral. Math. Soc. 39 (989), T. Y. Li and J. A. Yorke, Period 3 implies chaos, Amer. Math. Monthly, 82 (975), Y. Oono, Period = 2 n implies chaos, Prog. Theor. Phys. F 59 (978), Eleonor Ciurea, An algorithm for minimal dynamic flow, J. Appl. Math. & Computing(old KJCAM) 7(2) (2000), Lidong Wang is a professor and doctor in the Research Institute of Nonlinear Information & Technology, Dalian Nationalities University. His research interests are major in chaos, topological dynamics and ergodic theory.

12 288 L. Wang, G. Liao, Z. Chu and X. Duan The Research Institute of Nonlinear Information & Technology, Dalian Nationalities University, Dalian 6600, P. R. China Gongfu Liao is a professor of the Reaearch Institute of Mathematics in Jilin University. His research interests are major in chaos, topological dynamics and ergodic theory. The Research Institute of Mathematics, Jilin University, Changchun 30023, P. R. China liaogf@public.cc.jl.cn Zhenyan Chu is a graduate student of Jilin Normal University and a teacher in the Research Institute of Nonlinear Information & Technology, Dalian Nationalities University. Her research interests are major in chaos, topological dynamics and ergodic theory. The Research Institute of Nonlinear Information & Technology, Dalian Nationalities University, Dalian 6600, P. R. China chuzhenyan8@sohu.com Xiaodong Duan is a professor and doctor in the Research Institute of Nonlinear Information & Technology, Dalian Nationalities University. His research interests are major in chaos and fractal. The Research Institute of Nonlinear Information & Technology, Dalian Nationalities University, Dalian 6600, P. R. China