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Chaos in a topologically transitive system XIONG Jincheng South China Normal University, Guangzhou 510631, China (email:
[email protected]) Received June 2, 2004; revised November 22, 2004
Abstract The chaotic phenomena have been studied in a topologically transitive system and it has been shown that the erratic time dependence of orbits in such a topologically transitive system is more complicated than what described by the well-known technology “Li-Yorke chaos”. The concept “sensitive dependency on initial conditions” has been generalized, and the chaotic phenomena has been discussed for transitive systems with the generalized sensitive dependency property. Keywords: chaos, topological transitive, topologically dynamical systems, sensitive dependence. DOI: 10.1360/04ys0120
1
Introduction
Ruelle and Takens[1] considered a chaotic system a transitive system with sensitive dependency on initial conditions. Li and Yorke[2] though that a system is chaotic if there is an uncountable scrambled set in its domain. The chaotic system defined by Devaney[3] is a system chaotic in the sense of Ruelle and Takens with a dense set of periodic points. However, many authors (for example, ref. [4]) found that a transitive system with a dense set of periodic points has to be of sensitive dependence on initial conditions. Huang and Ye[5] studied the transitive system with a fixed point, and showed that such a system is chaotic in the sense of Li and Yorke. Their work led an open problem (i.e. whether or not Devaney’s chaos implies Li-Yorke’s chaos) closed. The approach used by Huang and Ye is quite compact. Mai[6] constructed a scrambled set crueller than that in Li-Yorke original sense when the transitive system is of sensitive dependency on initial conditions. On the other hand, Xiong and Yang[7] , Xiong and Chen[8] described the chaotic phenomena caused by topologically mixing map, topologically weak mixing map or measurable mixing map using the words different from Li and Yorke’s. In the present paper we follow the approaches in refs. [7,8] to discover the chaotic phenomena in a transitive system. From Main Theorem in Section 6 of the present paper, one may find that to describe the chaotic phenomena it is not enough only by the technology of Li and Yorke. Two generic properties of the power systems of a transitive system are given in Section 3, in which the main idea for the relative results comes from Huang and Copyright by Science in China Press 2005
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Ye[5] . In Section 4,we generalize the concept of sensitive dependency on initial conditions. In Section 5, we show another generic property of the power systems which is relative to the generalized sensitive dependency on the initial conditions. The Main Theorem of the present paper is put in Section 6. 2
Preliminaries
Throughout the present paper, (X, d) denotes a complete metric space, f : X −→ X denotes a continuous map. We say that (X, f ) is a topological dynamical system or a dynamical system, or a system which means that X is a complete metric space and f is a continuous self-map on the space X. A set of a metric space is called a Gδ set if it is a countable intersection of open subsets. A set of a metric space is called residual if it contains a dense Gδ subset. The complement of a residual set is said to be of the first category. We will use the notation N to denote the set of positive integers, R the set of real numbers. Notation [0, +∞] denotes the set [0, +∞) of non-negative real numbers with an infinite point +∞, endowed with the topology “one point compactification”. (Therefore, as a topological space, [0, +∞] is homeomorphic to the unit closed interval with the usual topology.) In addition, for any a, b ∈ [0, +∞], a b, the notations (a, b), (a, b], [a, b) are defined as usual. Recall some well-known definitions and results as follows. A real-valued function g : X −→ [0, +∞] is upper semi-continuous if for any a ∈ [0, +∞], f −1 ([a, +∞]) is a closed set in X ; a map g is lower semi-continuous if for any a ∈ [0, +∞], f −1 ([0, a]) is a closed set in X. Lemma 2.1. [0, +∞].
Suppose G is a family of real-valued functions from X to
(1) Define a real-valued function g : X −→ [0, +∞] such that for any x ∈ X g(x) = inf{h(x) | h ∈ G}. If every h ∈ G is upper semi-continuous, then g is upper semi-continuous. (2) Define a real-valued function g : X −→ [0, +∞]such that for any x ∈ X g(x) = sup{h(x) | h ∈ G}. If every h ∈ G is lower semi-continuous, then g is lower semi-continuous. Proof.
(1) is true because for any a ∈ [0, +∞] we have g −1 ([a, +∞]) =
h∈G
The proof of the statement (2) is similar.
h−1 ([a, +∞]). 2
Lemma 2.2. (1) Suppose hi : X −→ [0, +∞], i ∈ N, are upper semicontinuous. Define the function g : X −→ [0, +∞] such that for any x ∈ X g(x) = sup hi (x). i∈N
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Then g satisfies the condition for any a ∈ [0, +∞], if the set g −1 ([0, a]) is dense in X, then it is a dense Gδ set in X. (2) Suppose hi : X −→ [0, +∞], i ∈ N, are lower semi-continuous. Define the function g : X −→ [0, +∞] such that for any x ∈ X g(x) = inf hi (x). i∈N
Then g satisfies the condition for any a ∈ [0, +∞], if the set g −1 ([a, +∞]) is dense in X, then it is a dense Gδ set in X. Proof.
(1) is true because for any a ∈ [0, +∞] we have g
−1
([0, a]) =
∞ i=1
∞
n=1
h−1 i
1 [0, a + ) n
.
Therefore whenever g −1 ([0, a]) is dense, every set h−1 [0, a + n1 ) is a dense i open set. And the countable intersection of dense open sets is dense. The proof of the statement (2) is similar. 2 Corollary 2.3. (1) Suppose hi : X −→ [0, +∞], i ∈ N, are upper semicontinuous. Define the function g : X −→ [0, +∞] such that for any x ∈ X g(x) = lim inf hi (x). i→∞
Then the function g satisfies the condition for any a ∈ [0, +∞], if the set g −1 ([0, a]) is dense in X, then it is a dense Gδ set in X. (2) Suppose hi : X −→ [0, +∞], i ∈ N, are lower semi-continuous. Define the function g : X −→ [0, +∞] such that for any x ∈ X g(x) = lim sup hi (x). Then the function g satisfies the condition for any a ∈ [0, +∞] if the set g −1 ([a, +∞]) is dense in X, then it is a dense Gδ set in X. Proof.
(1) Note that ∞
g(x) = lim inf hi (x) = sup inf hk (x). i→∞
i∈N k=i
By Lemma 2.1, for every i ∈ N, the function inf ki hk (x) is upper semicontinuous. By Lemma 2.2 the function g satisfies the condition of this lemma. The proof of the statement (2) is similar. 2 We need a corollary (Corollary 2.4) of the following Mycielski Theorem. Theorem[9] . Suppose X is a complete metric space dense in itself. Suppose for every number N ∈ N, RN is a set of the first category in the product space X rN , and {Gj }j∈N is a sequence of non-empty open sets in X. Then for each j there is a non-empty compact perfect set Cj ⊂ Gj such that the set K = ∪∞ j=1 Cj satisfies the condition that for every N ∈ N, and for any pairwise different rN points x1 , x2 , · · · , xrN in K, we have (x1 , x2 , · · · , xrN ) ∈ / RN . We will call a countable union of non-empty compact perfect sets Mycielski set. www.scichina.com
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Corollary 2.4. Suppose X is a seperable complete metric space dense in itself. Suppose for every N ∈ N, the set PN is residual in the product space X N . Then there is a Mycielski set K in X such that for any non-empty open set U the intersection U ∩ K contains non-empty compact perfect set and for any N ∈ N and any pairwise different N points x1 , x2 , · · · , xrN in K, we have (x1 , x2 , · · · , xN ) ∈ PN . Proof. Apply Mycielski Theorem to RN = X N − PN , rN = N and {Gi , i ∈ N} which is the family of all non-empty sets in a countable base of X. 2 3
Two generic properties of the power systems of a transitive system For each N ∈ N, denote f (N ) = f × f × · · · × f : X N −→ X N .
N times
The system (X N , f (N ) ) is called the N -power system of the system (X, f ). A point x ∈ X is said to be a transitive point of the system (X, f ) if the orbit O+ (x, f ) of x is dense in X, i.e. O+ (x, f ) = X, where O+ (x, f ) = {f n (x) : n ∈ N}. The system (X, f ) is said to be a topologically transitive system or a transitive system if there is a transitive point in X. A point x ∈ X is called a recurrent point of the system (X, f ) if there is an increase sequence {ni }i∈N of positive integers such that limi→∞ f ni (x) = x. The set, denoted by Rec(f ), consisting of all recurrent points of the system (X, f ) is called the set of recurrent points. Lemma 3.1. If (X, f ) is a transitive system, then the set of transitive points of f is a dense Gδ set. Proof. We prove a statement first: if w ∈ X is a transitive point of the system (X, f ), then every point in the positive orbit of the point w is transitive. It is sufficient to prove that the point f (w) is transitive. Otherwise, if f (w) is / U not transitive, then there is a nonempty open subset U such that f k (w) ∈ for any k 2. Since w is transitive, we have f (w) ∈ U . Then there is a neighborhood V of w such that f (V ) ⊂ U and f (w) ∈ V for some 1. Thus f +1 (w) ∈ U , a contradiction. Now give the proof of this lemma. Since (X, f ) is a transitive system, X is separable. Let {Un : n ∈ N} be the family of all nonempty sets of a countable base of the space X. The set of transitive points of f will be ∞ ∞
n=1
f −k (Un ) .
k=1
Because the system (X, f ) has a transitive point, from the statement at the beginning of the proof the set of transitive points is dense, i.e. ∞ ∞
n=1
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k=1
f −k (Un ) = X.
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−k Therefore, the set ∞ (Un ) is a dense open set of X for any n ∈ N. Thus, k=1 f the set of transitive points of the system (X, f ) is a dense Gδ set. 2
Lemma 3.2. Suppose the recurrent set Rec(f ) of the system (X, f ) is dense in X, where X is a complete metric space. Then the recurrent set Rec(f ) of f is a dense Gδ set. Proof.
Define a real-valued function F : X −→ [0, +∞] such that F (x) = lim inf d(f n (x), x) n→∞
for any x ∈ X. It is obvious that the point x ∈ X is a recurrent point of the system (X, f ) if and only if F (x) = 0. By Corollary 2.3, since the set F −1 (0) is dense, the set F −1 (0) is a dense Gδ set in X. 2 Proposition 3.3. Suppose the system (X, f ) is transitive, where X is a complete metric space. Then for every N ∈ N the set Rec(f (N ) ) of recurrent points of the N -power system (X N , f (N ) ) is a dense Gδ set in X N . Proof. Suppose w ∈ X is a transitive point of the system (X, f ). Then for every N ∈ N the set (O+ (w, f ))N is dense in X N , and every point of (O+ (w, f ))N is a recurrent point of the power system (X N , f (N ) ). Therefore, the set Rec(f (N ) ) of recurrent points of the power system (X N , f (N ) ) is dense in X N . By Lemma 3.2 the set Rec(f (N ) ) is a dense Gδ set in X N . 2 Remark 3.4. If the point (x1 , x2 , · · · , xN ) ∈ X N is a recurrent point of the power system (X N , f (N ) ) then there is an increasing sequence {ni } of positive integers such that limi→∞ f ni (xj ) = xj for every j ∈ {1, 2, · · · , N }. In this case if xi = xj , whenever i = j, then we have lim sup min{d(f n (xi ), f n (xj )) : i, j ∈ {1, 2, · · · , N }; i = j} > 0. n→∞
Proposition 3.5. Suppose the system (X, f ) is transitive, where X is a complete metric space. Let r0 0 be the infimum of the diameters of all invariant sets of the system (X, f ). Then for every N ∈ N there is a dense Gδ set LN ⊂ X N satisfying the condition for any point (x1 , x2 , · · · , xN ) in LN , we have lim inf max {d(f n (xi ), f n (xj )) : i, j ∈ {1, 2, · · · , N }} r0 . n→∞
Proof. Given an integer N 2. Define a real-valued function Flinf : X N −→ R such that for any (x1 , x2 , · · · , xN ) ∈ X N , Flinf (x1 , x2 , · · · , xN ) = lim inf max {d(f n (xi ), f n (xj )) : i, j ∈ {1, 2, · · · , N }} . n→∞
Suppose S ⊂ X is an invariant subset of the system (X, f ) with |S| = rS r0 , where |S| denotes the diameter of the set S. Suppose w ∈ X is a transitive point of the system (X, f ). We now prove that for every point z = (z1 , z2 , · · · , zN ) ∈ (O+ (x, f ))N we have Flinf (z) rS as follows. Suppose zi = f ki (w) for some ki ∈ N, i ∈ {1, 2, · · · , N }. Choose www.scichina.com
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arbitrarily a point y ∈ S. There is an increasing sequence {nj }j∈N of positive integers such that limj→∞ f nj (w) = y. Then for every i = 1, 2, · · · , N , lim f nj (zi ) = lim f nj (f ki (w)) = f ki (y) ∈ S.
j→∞
j→∞
Obviously, there are i0 , i1 ∈ {1, 2, · · · , N } and a subsequence {mj }j∈N of the sequence {nj }j∈N such that d(f mj (zi0 ), f mj (zi1 )) = max {d(f mj (zi ), f mj (zj )) : i, j ∈ {1, 2, · · · , N }} . Therefore, Flinf (z) = lim inf max {d(f n (zi ), f n (zj )) : i, j ∈ {1, 2, · · · , N }} n→∞
lim max {d(f mj (zi ), f mj (zj )) : i, j ∈ {1, 2, · · · , N }} j→∞
= lim d(f mj (zi0 ), f mj (zi1 )) j→∞
= d(f ki0 (y), f ki1 (y)) rS . −1 Since (O+ (x, f ))N is dense in X N the set Flinf ([0, rS ]) is dense as well. By −1 Corollary 2.3 Flinf ([0, rS ]) is a dense Gδ set in X N .
If in X there is an invariant subset S0 with |S0 | = rS0 = r0 , then the set LN = is a dense Gδ set in X, which is required. If in X any invariant subset S has |S| = rS > r0 , choose a sequence of invariant subsets {Si }i∈N in X −1 such that limi→∞ rSi = r0 . In this case the set LN = ∩∞ i=1 Flinf ([0, rSi ]) is also a dense Gδ set, as an intersection of many countable dense Gδ sets. It is easily 2 seen that the set LN is required. −1 ([0, rS0 ]) Flinf
4
Generalized sensitive dependency on initial conditions
In topological dynamical systems there is a remarkable concept “sensitive dependency on initial conditions”, which can be generalized as follows. Given an integer N 2. If there is a real number λ 0 such that for every non-empty set U open in X there are N points y1 , y2 , · · · , yN such that min {d(f n (xi ), f n (xj )) : i, j ∈ {1, 2, · · · , N }; i = j} λ for some n 0, then the number λ will be called an N -sensitive coefficient of the system (X, f ). The supremum of all N -sensitive coefficients of the system (X, f ), denoted by λN , is called the N -critically sensitive coefficient of the system (X, f ). It is easily seen that a system with a positive 2-critically sensitive coefficient is a system of sensitive dependence on initial conditions in common parlance. Example 1. Suppose Σ2 is the symbolic space on two symbols {0, 1}, i.e. Σ2 = Π∞ i=1 {0, 1}. The metric ρ on the space Σ2 is defined as follows: for any x = x1 , x2 , · · · , y = y1 , y2 , · · · ,
ρ(x, y) =
0 2−k
: :
if x = y, if x = y and k = min{n 1 : xn = yn } − 1.
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The shift σ : ΣN → ΣN on the symbolic space ΣN is defined as follows: for any x = x1 , x2 , · · · ∈ ΣN , σ(x) = x2 , x3 , · · · , i.e. for any n 1, the n-th coordinate of σ(x) is exactly the n + 1-th coordinate of x. For the dynamical system (Σ2 , σ), we have 1 if 2M −1 < N 2M . λN = M −1 2 Obviously, every N + 1-sensitive coefficient of a system is also an N -sensitive coefficient. Now we give. Proposition 4.1. then for every N 2
For a system (X, f ) if the space X is locally connected λN
In other words,
λ2 N −1
λ2 . N −1
is an N -sensitive coefficient of the system (X, f ).
Proof. Let α be a base of the locally connected space X, where every member is connected. For every U ∈ α there are x, y ∈ U such that d(f n (x), f n (y)) λ2 for some n 0. Define a function h : U −→ R for every N 2 such that h(z) = d(f n (z), f n (x)) for every z ∈ U . Since U is connected there are N points x1 = x, x2 , · · · , xN = y ∈ U such that i−1 d(f n (x), f n (y)). h(xi ) = N −1 For any i, j = 1, 2, · · · , N , i < j, we have d(f n (x1 ), f n (xj )) d(f n (x1 ), f n (xi )) + d(f n (xi ), f n (xj )). Hence, (j − i)d(f n (x), f n (y)) N −1 d(f n (x), f n (y)) λ2 . N −1 N −1 This shows that λ2 /(N − 1) is an N -sensitive coefficient of the system (X, f ), and λN λ2 /(N − 1). The proof is complete. 2 d(f n (xi ), f n (xj ))
5 Another generic property of the power systems of a transitive system For every N 2 and every i ∈ {1, 2, · · · , N }, define a real-valued function φN,i and φN : X N −→ [0, +∞] such that φN,i (x1 , x2 , · · · , xN ) = sup min{d(f n (xi ), f n (xj )) : j = 1, 2, · · · , N ; j = i}, n∈N
φN (x1 , x2 , · · · , xN ) = min φN,i (x1 , x2 , · · · , xN ) 1iN
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for any (x1 , x2 , · · · , xN ) ∈ X N . Since the functions “d” “min” are continuous, the real-valued functions φN,i and φN are lower semi-continuous. Lemma 5.1. Suppose the system (X, f ) is transitive and w ∈ X is a transitive point of the system. Suppose N 2 and i ∈ {1, 2, · · · , N }. If λ 0 is an N -sensitive coefficient of the system (X, f ). Then for any k1 , k2 , · · · , kN ∈ N in any neighborhood W of the point (f k1 (w), f k2 (w), · · · , f kN (w)) ∈ X N , there is a point (x1 , x2 , · · · , xN ) ∈ W such that φN,i (x1 , x2 , · · · , xN ) λ/2. Proof. Suppose W is a neighborhood of the point (f k1 (w), f k2 (w), · · · , f kN (w)) in the product space X N . Choose a neighborhood U of the transitive point w ∈ X such that |U |, |f (U )|, · · · , |f m (U )| < λ whenever λ > 0, where m = max{ki : i = 1, 2, · · · , N }, and f k1 (U ) × f k2 (U ) × · · · × f kN (U ) ⊂ W. Choose y1 , y2 , · · · , yN ∈ U and n ∈ N such that min{d(f n (yi ), f n (yj )) : i, j ∈ {1, 2, · · · , N }; i = j} λ. Clearly, we have n m. For every i ∈ {1, 2, · · · , N } denote Bi = {f ki (y ) : = 1, 2, · · · N }. Given arbitrarily i, j ∈ {1, 2, · · · , N }, i = j. We show that for every point f kj (y ) in Bj there is at most one point f ki (y0 ) in Bi such that d(f n−ki (f ki (y0 )), f n−ki (f kj (y ))) < λ/2. The reason is that if another point f ki (y1 ) satisfies the following formula d(f n−ki (f ki (y1 ), f n−ki (f kj (y ))) < λ/2, it would follow that d(f n (y0 ), f n (y1 )) < λ. This is a contradiction. Now we define a map Fji : Bj −→ P(Bi ) such that for every z ∈ Bj Fji (z) = {y ∈ Bi : d(f n−ki (z), f n−ki (y)) λ/2}, where P(Bi ) is the power set of the set Bi . Note that for every x ∈ Bj we have cardFji (x) N − 1. Therefore,
j=1,2,···,N ;j=i
Fji (x) = ∅
x∈Bj
because the complementary set of the left set in the above formula (with respect to the set Bi ) contains at most N − 1 points. This shows that the points x1 ∈ B1 , x2 ∈ B2 , · · · , xN ∈ BN satisfy the condition d(f n−ki (xi ), f n−ki (xj )) λ/2 whenever j ∈ {1, 2, · · · , N }, j = i. Obviously, 2 (x1 , x2 , · · · , xN ) ∈ W and φN,i (x1 , x2 , · · · , xN ) λ/2. For every N 2 define a real-valued function ΦN : X N −→ [0, +∞] such that for every (x1 , x2 , · · · , xN ) ∈ X N , Copyright by Science in China Press 2005
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ΦN (x1 , x2 , · · · , xN ) = min lim sup min{d(f n (xi ), f n (xj )) : j ∈ {1, 2, · · · , N }; j = i}. 1iN
n→∞
Proposition 5.2. Suppose (X, f ) is a transitive system, where X is a complete metric space. Suppose λ 0 is an N -sensitive coefficient of the system (X, f ), where N 2. Then the set consisting of the points (x1 , x2 , · · · , xN ) with ΦN (x1 , x2 , · · · , xN )
λ 2
is a residual set in the product space X N . Proof. Suppose N ∈ N. Since for every i ∈ {1, 2, · · · , N } the real-valued function φN,i is lower semi-continuous the set φ−1 N,i ([λ/2, +∞]) is a dense set. Hence, it is a dense Gδ set. Therefore, the set φ−1 N ([λ/2, +∞]) =
N i=1
φ−1 N,i ([λ/2, +∞])
is also a dense Gδ set as well. By Proposition 3.3 the set Rec(f (N ) ) of recurrent points of the power system (X N , f (N ) ) is a dense Gδ set in the space X N . Therefore, the set φ−1 N ([λ/2, +∞]) Rec(f (N ) ) is a dense Gδ set in the space X N . Obviously, if the point x ∈ X N is a recurrent point of the power system (X N , f (N ) ) then φN (x) = ΦN (x). It follows that (N ) ) ⊂ Φ−1 φ−1 N ([λ/2, +∞]) ∩ Rec(f N ([λ/2, +∞]). −1 Hence Φ ([λ/2, +∞]) is a residual set. The proof is complete.
6
2
Main theorem
Theorem (Main Theorem). Suppose (X, f ) is a transitive topological system, where X is a complete metric space dense in itself, with a metric d. Then there is a subset K of the space X, of which the intersection with any non-empty open set in X contains a non-empty compact perfect set. For the set K the following conditions are satisfied: (1) Every point of K is transitive; (2) For every integer N 2 and for any pairwise different N points x1 , x2 , · · · , xN in K, lim sup min {d(f n (xi ), f n (xj )) : i, j ∈ {1, 2, · · · , N }; i = j} > 0; n→∞
(3) For every integer N 2 and for any pairwise different N points x1 , x2 , · · · , xN in K, lim inf max {d(f n (xi ), f n (xj )) : i, j ∈ {1, 2, · · · , N }} d0 , n→∞
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where d0 is the infimum of the diameters of the invariant sets of the system (X, f ); (Note that if there is a fixed point of the system (X, f ), then d0 = 0.) (4) For every integer N 2 and for any pairwise different N points x1 , x2 , · · · , xN in K, λN , min lim sup min {d(f n (xi ), f n (xj )) : j ∈ {1, 2, · · · , N }; j = i} 1iN n→∞ 2 where λN is the N -critically sensitive coefficent of the system (X, f ) and if the space X is locally connected then for every integer N 2 and for any pairwise different N points x1 , x2 , · · · , xN in K, min lim sup min {d(f n (xi ), f n (xj )) : j ∈ {1, 2, · · · , N }; j = i}
1iN
n→∞
λ2 . 2(N − 1) Proof. For N = 1 let P1 be the set consisting of all transitive points of the system (X, f ). For every N 2 let
PN = Rec(f (N ) ) ∩ LN ∩
∞
k=1
and if X is locally connected let
PN = Rec(f
(N )
) ∩ LN ∩
∞
k=1
Φ−1 N
Φ−1 N
1 λN (1 − ) , +∞ k 2
1 1− k
λ2 , +∞ 2(N − 1)
,
where LN is that as in Proposition 3.5 and the definition of the real-valued function ΦN is in Section 5. By Lemma 3.1, Proposition 3.3, Proposition 3.5, Proposition 5.2, and Proposition 4.1, for every N 1, in both cases whether the space X is locally connected or not, the set PN is a residual set in X N . Since the system (X, f ) is transitive, the space is separable. Applying Corollary 2.4 of the Mycielsky Theorem, the space X has a dense Mycielski set K satisfying the condition: for any nonempty open subset U of X the intersection U ∩K contains a nonempty compact perfect set and for any N 1 any pairwise different N point x1 , x2 , · · · , xN in K, we have that (x1 , x2 , · · · , xN ) ∈ PN . Then, by the definitions of the sets −1 λ2 1 λN 1 Rec(f (N ) ), LN , Φ−1 N ([(1 − k ) 2 , +∞]) and ΦN ([(1 − k ) 2(N −1) , +∞]), we know immediately that the set K is required. 2 Acknowledgements The author would like to send his thanks to two referees of this paper and Dr. Wang Huoyun, Ms. Tan Feng, Ms. Liu Xiaoling and Mr. Fu Heman for their useful suggestions and their careful reading of the manuscript. This work was supported by the National Natural Science Foundation of China (Grant No. 10171034).
References 1. Ruelle, D., Takens, F., On the natural of turbulence, Comm. Math. Phys, 1971, 20: 167–192. 2. Li, T. Y., Yorke, J., Period three implies chaos, Amer. Math. Monthly, 1975, 82: 985–992. 3. Devaney, R., An Introduction to Chaotic Dynamical Systems, Reading MA: Addison-Wesley, 1989. Copyright by Science in China Press 2005
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