The Retentivity of Several Kinds of Chaos under Uniformly Conjugation

This paper is concerned with chaotic of maps on general metric spaces. It is proved that uniformly conjugation preserves AuslanderYorke’s chaoticity, dense chaoticity, dense δ-chaoticity, distributional chaoticity, and distributional chaoticity in a sequence. Mathematics Subject Classification: 54H20, 37C15


Introduction
The complexity of a topological dynamical system is intensively discussed since the introduction of the term chaos in 1975 by Li and Yorke [12] .That is, if a dynamical system (X, f ) has an uncountable set S ⊂ X, and (x, y) is a Li-Yorke pair for ∀x, y ∈ S : x = y, then (X, f ) is said to be chaotic in the sense of Li-Yorke.While, the definition of chaos in the sense of Li-Yorke is inconvenient in engineering applications.In 1989, R. L. Devaney [11] stated a definition of chaos, known as Devaney chaos today.A map f is said to be chaotic in the sense of Devaney on X if f is transitive on X, the set of periodic points of f is dense in X and f has sensitive dependence on initial conditions.Then, in 1992, Banks [6] proved that if f : (X, d) → (X, d) is transitive and has dense periodic points then f has sensitive dependence on initial conditions (where (X, d) is a metric space which has no isolated point).This causes that Devaney's chaoticity is preserved under topological conjugation on generally infinite metric space.And in 2007, C. Tian and G. Chen [2] obtained that Li-Yorke's chaoticity is preserved under (topological) uniformly conjugation on metric spaces.
In this paper, the notation N denotes the set of natural numbers and R + denotes the positive real numbers.B(x, ε)(ε > 0) denotes the ε-neighborhood of x in a space X.The closure of a set A ⊂ X is denoted by A. Let f : X → X be a map on a metric space (X, d) and x 0 ∈ X.The (positive or forward) orbit of the point x 0 is denoted by Other definitions (e.g., metric space, continuous map, period, periodic point) are as usual.
Definition 1.1 Let h : X → Y be a map from a metric space (X, d) into a metric space (Y, d).The map h is uniformly continuous if for any ε > 0, there exists a δ > 0 such that for ∀x 1 , x 2 ∈ X : The map h is said to be a homeomorphism if it is one-toone and onto, and both h and h −1 are continuous.In particular, the map h is said to be an uniform homeomorphism if it is one-to-one and onto, and both h and h −1 are uniformly continuous.Definition 1.3 Let (X, d) and (Y, d) be two metric spaces, and f : X → X and g : Y → Y be two maps.Maps f and g are said to be (topologically) conjugate if there exists a homeomorphism h : where "•" denotes composition of two maps.In particular, if h is an uniform homeomorphism and f and g are h-conjugate, f and g are said to be uniformly conjugate.
If f and g are uniformly conjugate, obviously they are conjugate.

Auslander-Yorke chaos
Since sensitive dependence on initial conditions (sensitivity for short) is rather intuitively a chaotic property, Auslander and Yorke [5] introduced a definition of chaos by associating sensitivity and transitivity.Definition 2.1 Let (X, d) be a metric space, f : X → X be a map.f is said to be Auslander-Yorke chaotic if f is (topological) transitive and f has sensitive dependence on initial conditions.Where, f is (topological) transitive if for arbitrary two nonempty open sets U, V ⊂ X, there exists m ∈ Z such that f m (U) ∩ V = φ.f has sensitive dependence on initial conditions if there exists a sensitivity constant ε > 0 such that for any x ∈ X, any δ > 0, one can find y ∈ X with d(x, y) < δ and Proof Necessity.f is Auslander-Yorke chaotic in X, then f is transitive and f has sensitive dependence on initial conditions.
For any Hence, g is transitive on Y .On the other hand, f is sensitivity, i.e. there exists a sensitivity constant ε > 0 such that for any x ∈ X, any δ > 0, one can find y ∈ X : for the above y ∈ X : d(x, y) < δ, since h is uniformly continuous, i.e. y converges to x implies h(y) converges to h(x), one has d( x, y) < δ (where y = h(y)) and Hence, g is sensitivity.So g is Auslander-Yorke chaos in Y .By the similar argument, the sufficiency is follows immediately.

Dense chaos
Chaotic in the sense of Li-Yorke means that a system has an uncountable scrambled set S in which arbitrary (x, y) ∈ S × S : x = y is a Li-Yorke pair.The definition of dense chaos is based on Li-Yorke pairs too.While, different from Li-Yorke chaos, dense chaos described that whether the Li-Yorke pairs are "everywhere" in the space.
Denotes the set of Li-Yorke pairs of f by And denotes the set of Li-Yorke pairs with modulus δ by Definition 3.2 Let f : X → X be a map on a metric space (X, d).The map f is said to be densely chaotic if LY (f ) = X × X. Definition 3.3 Let f : X → X be a map on a metric space (X, d).The map f is said to be densely δ-chaotic if LY (f, δ) = X × X. Theorem 3.1 Let (X, d) and (Y, d) be two metric spaces, f : X → X and g : Y → Y be two maps, and h : X → Y is an uniform homeomorphism.If f and g are h-conjugate, then f is densely chaotic on X if and only if g is densely chaotic on Y .
Proof Necessity (Sufficiency is similar).For every y = (y For every ε > 0, because h and h −1 are continuous maps, there exists ))) = 0, then, the limit of an arbitrary convergent subsequence of sequence ) < σ 1 for every σ 1 > 0. So the limit of an arbitrary convergent subsequence of sequence Similarly, since h is uniformly continuous, one has lim inf Finally, g thus is densely chaotic on Y .
Similarly to the proof of Theorem 4.

Distributional chaos
Let (X, d) be a metric space, f : X → X be a map, x, y ∈ X, t ∈ R + , n ∈ N, the upper and lower (distance) distribution functions F * xy (t, f ) and F xy (t, f ) are defined as follows: where Definition 4.1 Dynamical system (X, f ) is said to be distributional chaotic if there exists an uncountable set S ⊂ X such that (1) ∀t > 0, ∀x, y ∈ S : We called S be a distributionally scrambled set of X.
Suppose {p k } k∈AE be a strictly increasing sequence of positive integers.x, y ∈ X, t ∈ R + , the upper and lower (distance) distribution functions F * xy (t, {p k } k∈AE , f) and F xy (t, {p k } k∈AE , f) are defined as follows: Definition 4.2 Dynamical system (X, f ) is said to be distributional chaotic in a sequence if there exists an uncountable set S ⊂ X such that (1) ∀t > 0, ∀x, y ∈ S : Proof Necessity.f be distributional chaotic, then there exists an uncountable set S ⊂ X such that (1) ∀t > 0, ∀x, y ∈ S : Since S is uncountable and h is one-to-one, then h(S) ⊂ Y and h(S) is uncountable.
For ∀t > 0, ∀h(x), h(y) ∈ h(S) : h(x) = h(y), one has x, y ∈ S, x = y, and Since h is uniformly continuous, then for the above t > 0, ∃δ(t) > 0, if some Combined with condition (1), one has lim sup And because On the other hand, by condition (2), there exists t 0 > 0 such that for any x, y ∈ S : x = y, This implies that, Remark 2. There are some problems for research.For example, topological conjugation preserves Li-Yorke's chaoticity (Auslander-Yorke chaoticity, dense chaoticity, dense δ-chaoticity, distributional chaoticity and distributional chaoticity in a sequence) or not?Can the chaos be extended to topological spaces?If they can be, topological conjugation (or uniformly conjugation) preserves them or not?

Theorem 4 . 1
Let (X, d) and (Y, d) be two metric spaces.f : X → X and g : Y → Y be two maps, and h : X → Y is an uniform homeomorphism.If f and g are h-conjugate, then dynamical system (X, f ) is distributional chaotic if and only if dynamical system (Y, g) is distributional chaotic.
Definition 3.1 Let (X, d) is a metric space.A pair of points {x, y} ⊂ X is said to be a Li-Yorke pair if one has simultaneously lim sup n→∞ d(f n (x), f n (y)) > 0 and lim inf n→∞ )) ( d(g i (h(x)), g i (h(y)))) = 0. Therefore, h(S) is an uncountable distributionally scrambled set of Y .We thus conclude that (Y, g) is distributional chaotic.Similarly to the discussion of Theorem 4.1, Theorem 4.2 is obtained.It is easy to check that chaos are preserved under uniformly conjugation if they are preserved under topological conjugation.
Theorem 4.2 Let (X, d) and (Y, d) be two metric spaces.f : X → X and g : Y → Y be two maps, and h : X → Y is an uniform homeomorphism.If f and g are h-conjugate, then dynamical system (X, f ) is distributional chaotic in a sequence if and only if dynamical system (Y, g) is distributional chaotic in a sequence.Remark 1.