3-Adic System and Chaos

Main Theorem 1. 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.

Definition 2.1. Call x, y ∈ X a pair of points displaying distributional chaos, if

• (1)

  F(f, x, y, t) = 0 for some t > 0;

• (2)

  F(f, x, y, t) = 1 for any t > 0.

Definition 2.2. f is said to display distributional chaos, if there exists an uncountable set D ⊂ X such that any two different points in D display distributional chaos.

Definition 2.3. Let X be a metric space and f : X → X be a continuous map. The dynamical system (X, f) is called chaotic in the sense of Devaney, if

• (1)

  (X, f) is transitive;

• (2)

  the periodic points are dense in X;

• (3)

  (X, f) is sensitive to initial conditions.

Definition 2.4. Let X be a metric space and f : X → X be a continuous map. The dynamical system (X, f) is called chaotic in the sense of Wiggins, if there exists a compact invariant subset Y ⊂ X such that

• (1)

  f∣_Y is sensitive to initial conditions;

• (2)

  f∣_Y is transitive.

Definition 2.5. Let (X, f) and (Y, g) be dynamical systems; if there exists a homeomorphism h : X → Y such that h∘f = g∘h, then f and g are said to be topologically conjugate.

Definition 2.6. Put

We use the sequence a = a₁a₂⋯ to denote simply the member $$ in Z(3). Define ρ : Z(3) × Z(3) → R as follows: for any a, b ∈ Z(3), if a = a₁a₂⋯, b = b₁b₂⋯, then 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 = a₁a₂ ⋯ ∈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 f³ 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: (p₁) 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 f^′(x) ≥ 1 on [α, 1], f^′(x)<−2 on [λ, α]; (p₂)f(λα) < f(λ).

Lemma 2.7. Let 0 <λ < 1, α ∈ (λ, 1). Let f₀ : [λ, 1]→[0,1] be C¹ on each of the interval [λ, α] and [α, 1], and satisfy

• (1)

  f₀(α) = 0;

• (2)

  $$ on [λ, α] and $$ on [α, 1];

• (3)

  there exists α₀ ∈ (α, 1) such that f₀(α₀) = α and α < f₀(1) < α₀ < f₀(λ) < 1;

• (4)

  $$.

Then there exists a unique f ∈ ℱ with f|_[λ,1] = f₀. Conversely, if f₀ is the restriction on [λ, 1] of some f ∈ ℱ, then it must satisfy (1)–(4).

Proposition 2.8. ℱ ≠ ∅.

Proof. Let λ = 2/9, α = 1/2. Define f₀ : [λ, 1]→[0,1] by

It is not difficult to check that f₀ 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 2.9 (see [19], Theorem 8.3.)Let ϕ₁, ϕ₂, …, ϕ_m be contractions on Rⁿ. Then there exists a unique nonempty compact set E such that

where is a transformation of subsets of Rⁿ. Furthermore, for any nonempty compact subset F of Rⁿ, the iterates ϕ^k(F) converge to E in the Hausdorff metric as k → ∞.

Lemma 2.10 (see [19], Theorem 8.8.)Let $$ be contractions on R for which the open set condition holds; that is, there is an open interval V such that

• (1)

  $$,

• (2)

  ϕ₁(V), ϕ₂(V), …, ϕ_m(V) are pairwise disjoint.

Moreover, suppose that for each i, there exists r_i, such that |ϕ_i(x) − ϕ_i(y)| ≤ r_i|x − y| for all $$. Then dimE ≤ t, where dim(·) denotes the Hausdorff dimension and t is defined by

Lemma 2.11 (see [20], Theorem 3.2, [21].)Let f : I → I be continuous. Then the followings are equivalent:

• (1)

  ent(f) > 0;

• (2)

  A(f) contains an uncountable distributional chaotic set of f.

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(f)) = A(g).

Lemma 2.13 (see [22].)Let ent(f) = 0 and x ∈ I be recurrent but not periodic such that f(x) > x. Then the inequality f^m(x) < fⁿ(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.

Step 1. Prove that for any x ∈ I, f∘ϕ₀(x) = ϕ₁(x), f∘ϕ₁(x) = ϕ₂(x), f∘ϕ₂(x) = ϕ₀∘f(x).

Proof. Letting f act on both sides of the equality $$, 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(ϕ₂(x)))) = λf(x). Since ϕ₂(x) ∈ [0, λ], it follows from Lemma 2.7 that f∘ϕ₂(x) ∈ [α, 1] and f²∘ϕ₂(x) ∈ [α, 1]. By this and definitions of ϕ₀ and ϕ₁, we get

Step 2. Prove that for any subsets $$ and $$, there is an n > 0 such that $$.

Proof. If x ∈ I, i = 0,1, 2, then f³∘ϕ_i(x) = ϕ_i∘f(x) by Step 1. 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), (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 $$ or $$ has the form $$, where l_q = j_q for q = 1, …, r. Continuing this procedure, we must get some n, such that $$.

In, Steps 3, 5, and 6, we always suppose that the notation E is as in (3.1).

Step 3. Prove that

Proof. Since ϕ(I) ⊂ I, we have ϕ^k+1(I) = ϕ^k∘ϕ(I) ⊂ ϕ^k(I) for any k > 0. So from Lemma 2.9 we get

Step 4. Prove that for any k > 0, $$ is an invariant set of f, that is, f(ϕ^k(I)) ⊂ ϕ^k(I).

Proof. Note that each $$ has the form ϕ_22⋯2 or $$ or $$. Then, by using Step 1 repeatedly, we have

Thus by f(I) ⊂ I, we have $$. Moreover,

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 $$ with this union disjoint. Then transforming by $$,

again with a disjoint union. Thus the sets $$ (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 = a₁a₂ ⋯ ∈Z(3), if let 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 $$. It follows that diam $$ converges 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 $$. 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.

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 H(p) = a = 0a₂a₃⋯. Put

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 $$ and furthermore $$, we have that there exists an l ≥ 0 such that Let $$. Since we easily see that H(f^s(p)) = τ^s(H(p)) ∈ V if and only if n divides s, it follows that $$, since n can not divide 2^lm. And so $$. In particular, g(p) > p. By the same argument, we also have $$. In particular, g²(p) > p. Since $$, from (3.12), $$, that is, $$. Thus we have for the odd k^l, Note that a is current and nonperiodic for $$, 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 h : Z(3) → E such that for any x ∈ Z(3),

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 x₁, x₂ ∈ D, if F(f, h(x₁), h(x₂), t) = 0 for some t > 0, then F(τ, x₁, x₂, 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∘τⁱ = fⁱ∘h, it follows that if ρ(τⁱ(x₁), τⁱ(x₂)) < s, then

This implies for any n ≥ 0. Thus by the definition of F, we immediately have the following result:

Secondly, we prove that if F^*(f, h(x₁), h(x₂), s) = 1 for all s > 0, then F^*(τ, x₁, x₂, 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

which gives By (3.18), (3.20), and the arbitrariness of x₁ and x₂, we conclude that D is an uncountable distributional chaotic set of τ.

The proofs of (2) and (3) of the Main Theorem are obvious.