3-Adic System and Chaos

Let (Z(3),τ) be a 3-adic system. we prove in (Z(3),τ) the existence of uncountable distributional chaotic set of A(τ), which is an almost periodic points set, and further come to a conclusion that τ is chaotic in the sense of Devaney and Wiggins.


Introduction
In 1975, Li and Yorke introduced in 1 a new definition of chaos for interval maps.The central point in their definition is the existence of a scrambled set.Later, it was observed that positive topological entropy of interval map implies the existence of a scrambled set 2 .Many sharpened results come into being in succession see 3-11 .One can find in 3, 4, 12 equivalent conditions for f to be chaotic and in 13 or 14 a chaotic map with topological entropy zero, which showed that positive topological entropy and Li-Yorke chaos are not equivalent.
By the result, it became clear that the positive topological entropy is a much stronger notion than the definition of chaos in the sense of Li and Yorke.To remove this disadvantage, Zhou 15 introduced the notion of measure center and showed importantly dynamical properties of system on its measure center.To decide the concept of measure center, he defined weakly almost periodic point, too, showing that the closure of a set of weakly almost periodic points equals to its measure center and the set of weakly almost periodic points is a set of absolutely ergodic measure 1.These show that it is more significant to discuss problems on a set of weakly almost periodic points.On the other hand, one important extensions of Li-Yorke definition were developed by Schweizer and Smítal in 16 ; this paper introduced the definition of distributional chaos and prove that this notion is equivalent to positive topological entropy for interval maps.And many scholars such as Liao, Du, and Zhou, Wang proved that the positive topological entropy of interval map is equivalent to the uncountable Li-Yorke chaotic set and the uncountable distributional chaotic set for A f , W f , and R f .Meanwhile Liao showed that the equivalent characterization is no longer valid when f acts on more general compact metric spaces.
In this paper, we discuss the existence of uncountable distributional chaotic set of A f in 3-adic system.
The main results are stated as follows.
Main Theorem.Let Z 3 , τ be a 3-adic system.Then 1 A τ contains an uncountable distributional chaotic set of τ; 2 τ is chaotic in the sense of Devaney; 3 τ is chaotic in the sense of Wiggins.

Basic Definitions and Preparations
Throughout this paper, X will denote a compact metric space with metric d, I is the closed interval 0, 1 .For a continuous map f: X → X, we denote the set of almost periodic points of f by A f and denote the topological entropy of f by ent f , whose definitions are as usual; f n will denote the n-fold iteration of f.For x, y in X, any real number t and positive integer n, let where we use # • to denote the cardinality of a set.Let then f and g are said to be topologically conjugate.
The notion of adic system is defined as follows.
Definition 2.6.Put We use the sequence a a

2.4
It is not difficult to check that ρ is a metric on Z 3 and Z 3 , ρ is a compact abelian group.Define τ : Z 3 → Z 3 by τ a a 1 for a a 1 a 2 • • • ∈ Z 3 ; τ or Z 3 , τ is called the 3-adic system.see 17 Call an invariant closed set A ⊂ I 3-adic, if the restriction f| A is topologically conjugate to the 3-adic system.
Consider the following functional equation: where λ ∈ 0,1 is to be determined, x ∈ 0, 1 and f 3 is the 3-fold iteration of f.By F we denote the set of continuous solutions of 2.5 such that any f ∈ F satisfies: p 1 there exists α ∈ λ, 1 such that f α 0; the restrictions f| λ,α and f| α,1 are both once continuously differentiable, and The following Lemma can be concluded by in 18, Theorem 2.1 .Lemma 2.7.Let 0 < λ < 1, α ∈ λ, 1 .Let f 0 : λ, 1 → 0, 1 be C 1 on each of the interval λ, α and α, 1 , and satisfy Then there exists a unique f ∈ F with f| λ,1 f 0 .Conversely, if f 0 is the restriction on λ, 1 of some f ∈ F, then it must satisfy ( 1)-( 4).

2.6
It is not difficult to check that f 0 satisfies the condition 1 -4 in Lemma 2.7.So F / ∅.We will be concerned in the notions of Hausdorff metric and Hausdorff dimension, whose definitions can be found in 19 .

Lemma
Moreover, suppose that for each i, there exists r i , such that

Proof of Main Theorem
In the sequel, we always suppose that f ∈ F and f take the minimum at α ∈ λ, 1 .Let I 0, 1 , f f| α,1 .For i 0, 1, 2, define φ i : Then φ i is a contraction for every i 0, 1, 2. Let φ x 2 i 0 φ i x .By Lemma 2.9, there exists a unique nonempty compact set E with φ E E.

3.1
For simplicity, we write Step 1. Prove that for any x ∈ I, f Proof.Letting f act on both sides of the equality φ 0 x f −1 φ 1 x , we get immediately the first equality.A similar argument yields the second equality.To show the third equality, we write 2.5 as f f f φ 2 x λf x .Since φ 2 x ∈ 0, λ , it follows from Lemma 2.7 that f • φ 2 x ∈ α, 1 and f 2 • φ 2 x ∈ α, 1 .By this and definitions of φ 0 and φ 1 , we get Step 2. Prove that for any subsets Using this repeatedly, we get for any k > 0 If for each r 1, 2, . . ., k, we all have i r j r , then from 3.3 , Journal of Applied Mathematics nothing that f I I .Thus the lemma holds for this special case.Assume that there exists some r, 1 ≤ r ≤ k, such that i q j q for q < r, but i r ≤ j r .Then by using 3.3 repeatedly, we know that where l q j q for q 1, . . ., r. Continuing this procedure, we must get some n, such that In, Steps 3, 5, and 6, we always suppose that the notation E is as in 3.1 .
Step 3. Prove that 3.5 Proof.Since φ I ⊂ I, we have Step 4. Prove that for any k > 0, Thus by f I ⊂ I, we have f Step 5. Prove that the restriction f| E is topologically conjugate to τ, where τ is the 3-adic system as defined in Section 1.
Proof.By the definition of φ, we have φ I 2 i 0 φ i I with this union disjoint.Then transforming by again with a disjoint union.Thus the sets {φ i 1 •••i k I } with k arbitrary form a net in the sense that any pair of sets from the collection are either disjoint or such that one is included in the other.It follows from Step 3 that for any a a then φ a I ⊂ E is nonempty, and if x ∈ E, then there exists a unique a ∈ Z 3 with x ∈ φ a I .
We now define a map H of E onto Z 3 by setting H x a if x ∈ φ a I .Then H is well defined.It is easy to see that for each i 0, 1, 2, the contraction ratio of φ i ≤ λ, so the contraction ratio of Step 6. Prove that if f has an n-adic set and the n is not a power of 2, then ent f > 0.
Proof.Write n k • 2 m , where k ≥ 3 is odd and m ≥ 0 is an integer.Let A be the n-adic set of f and p min A. There exists a homeomorphism H : A → Z n such that for x ∈ A, τ • H x H • f x .We may assume without loss of generality that Then V ⊂ Z n is an open neighborhood of the sequence a.There exists an ε > 0, such that for any q ∈ A, if q − p < ε, there H q ∈ V .Note that for l → ∞, τ n l a → a and furthermore f n l p → p, we have that there exists an l ≥ 0 such that f n l p − p < ε.

3.12
Let g f 2 lm .Since we easily see that H f s p τ s H p ∈ V if and only if n divides s, it follows that H f 2 lm p / ∈ V , since n can not divide 2 lm .And so g p f 2 lm p ≥ p ε.In particular, g p > p.By the same argument, we also have g 2 p f 2 lm 1 p ≥ p ε.In particular, g 2 p > p.Since n l k • 2 m l k l • 2 lm , from 3.12 , g k l p − p f n l p − p < ε, that is, g k l p < p ε. Thus we have for the odd k l , g k l p < g 2 p .

3.13
Note that a is current and nonperiodic for τ 2 lm , and so is p for g.By Lemma 2.13 we get ent g > 0. Moreover ent f > 0. Finally, we prove that A τ contains an uncountable distributional chaotic set of τ.By Step 5, the restriction f| E is topologically conjugate to τ.Thus there is a homeomorphism According to Lemma 2.11, there is an uncountable set ∧ ⊂ A f , which is distributional chaotic.By Lemma 2.12 for any y ∈ ∧, there exists x ∈ A τ such that h x y.Let Then D is an uncountable set.
To complete the proof, it suffices to show that D is a distributional chaotic set for τ.
First of all, we prove that for any x 1 , x 2 ∈ D, if F f, h x 1 , h x 2 , t 0 for some t > 0, then F τ, x 1 , x 2 , s 0 for some s > 0. For given t > 0, by uniform continuity of h, there exists s > 0, such that for any p, q ∈ D, |h p − h q | < t, provided ρ p, q < s.Since we easily see that h • τ i f i • h, it follows that if ρ τ i x 1 , τ i x 2 < s, then 3.16 This implies ξ n τ, x 1 , x 2 , s ≤ ξ n f, h x 1 , h x 2 , t 3.17 for any n ≥ 0. Thus by the definition of F, we immediately have the following result: F τ, x 1 , x 2 , s 0.

3.18
Secondly, we prove that if F * f, h x 1 , h x 2 , s 1 for all s > 0, then F * τ, x 1 , x 2 , t 1 for all t > 0. Since h is homeomorphism, h −1 : E → Z n is a surjective continuous map.By the first proof, we have to zero uniformly for i r ∈ {0, 1, 2} as k → ∞ where diam denotes diameter .Thus φ a I is a single point for each a ∈ Z 3 .And so H is injective.Moreover the map H is continuous.Let δ k > 0 be the least distance between any two of the 3 k interval φ a 1 •••a k I .If x ∈ φ α I , y ∈ φ β I , and |x − y| < δ, then ρ α, β < 3 −k .Finally, since f φ a I φ τ a I by 3.7 , we have H • f x τ • H x for each x ∈ E.
2.9 see 19, Theorem 8.3 .Let φ 1 , φ 2 , . . ., φ m be contractions on R n .Then there exists a unique nonempty compact set E such that n .Furthermore, for any nonempty compact subset F of R n , the iterates φ k F converge to E in the Hausdorff metric as k → ∞.Lemma 2.10 see 19, Theorem 8.8 .Let {φ i } m 1 be contractions on R for which the open set condition holds; that is, there is an open interval V such that where dim • denotes the Hausdorff dimension and t is defined by Lemma 2.12 see 21 .Let f : X → X, g : Y → Y be continuous, where X, Y are compact metric spaces.If there exists a continuous surjection h :X → Y such that g • h h • f, then h A fA g .Let ent f 0 and x ∈ I be recurrent but not periodic such that f x > x.Then the inequality f m x < f n x holds for all even m and all odd n.Lemma 2.14 see 23, Theorem 6.1.4 .Let f : I → I be an interval map.Then ent f > 0 if and only if there exists a closed invariant subset ∧ ⊂ I such that f| ∧ is chaotic in the sense of Devaney.Lemma 2.15 see 23, Theorem 6.2.4 .Let f : I → I be an interval map.If ent f > 0, then f is chaotic in the sense of Wiggins.
x 1 , x 2 , t , By 3.18 , 3.20 , and the arbitrariness of x 1 and x 2 , we conclude that D is an uncountable distributional chaotic set of τ.The proofs of 2 and 3 of the Main Theorem are obvious.