2.1. Several Definitions and Lemmas
D
⊂
X
is said to be a chaotic set of f if for any pair (x,y)∈D×D, x≠y, we have
(1)limn→∞infd(fn(x),fn(y))=0,limn→∞supd(fn(x),fn(y))>0.

Definition 4.
f
is said to be chaotic in the sense of Li and Yorke (for short: Li-Yorke chaotic), if it has a chaotic set D which is uncountable.

Let {pi} be an increasing sequence of positive integers, x,y∈X,t>0. Let
(2)Fxy(t,{pi})=limn→∞inf1n∑k=1nχ[0,t)(d(fpk(x),fpk(y))),Fxy*(t,{pi})=limn→∞sup1n∑k=1nχ[0,t)(d(fpk(x),fpk(y))),
where χA(y) is 1 if y∈A and 0 otherwise. Obviously, Fxy and Fxy* are both nondecreasing functions. If for t≤0 we define Fxy(t)=Fxy*(t)=0, then Fxy and Fxy* are probability distributional functions.

Definition 5.
Let D⊂X, If ∀x,y∈D,x≠y, we have
(3)(1) ∃δ>0, Fxy=(δ,{pi})=0,(2) ∀t>0, Fxy*(t,{pi})=1,

then D is said to be a distributively chaotic set in a sequence. The two points are said to be distributively chaotic point pair in a sequence. f is said to be distributively chaotic in a sequence, if f has a distributively chaotic set in a sequence which is uncountable.

Definition 6.
Let S⊂X. If there exist two strictly increasing sequences of positive integers {pi} and {qi} such that for any x,y∈S, x≠y,
(4)limi→∞d(fpi(x),fpi(y))=0,limi→∞d(fqi(x),fqi(y))>0,
then S is said to be a strong scrambled set. f is said to be chaotic in the strong sense of Li-Yorke, if f has an uncountable strong scrambled set.

Definition 7.
Let {pi} be an increasing sequence of positive integers, then,
(5)PR(f,{pi})={(x,y)∈X×X∣∀ε>0, ∃i∈N such that d(fpi(x),fpi(y))<εX×X}
is called proximal relation with respect to {pi}.

Thus
(6)AR(f,{pi})={(x,y)∈X×X∣limi→∞d(fpi(x),fpi(y))=0}
is called asymptotic relation with respect to {pi}.

Thus
(7)DR(f,{pi})=X×X-PR(f,{pi})
is called distal relation with respect to {pi}.

So
(8)DCR(f,{pi}) ={(x,y)∈X×X∣(x,y) is a distributively chaotic point pair of f in piX×X}
is called distributively chaotic respect to {pi}.

Definition 8 (see [<xref ref-type="bibr" rid="B3">3</xref>]).
f
is (topologically) transitive if for any two nonempty open sets U,V⊂X there exists n>0 such that fn(U)∩V≠∅. f is (topologically) weakly mixing if for any three nonempty open sets U,V,W⊂X there exists n>0 such that fn(W)∩V≠∅ and fn(W)∩U≠∅.

Definition 9 (see [<xref ref-type="bibr" rid="B4">4</xref>–<xref ref-type="bibr" rid="B6">6</xref>]).
Let f be a continuous map from a compact metric space (X,d) into itself. The orbit of a point x∈X is said to be unstable if there exists r>0 such that for every ϵ>0 there are y∈X and n≥1 satisfying inequalities d(x,y)<ϵ and d(fn(x),fn(y))>r. The map f is said to be chaotic in the sense of Martelli if there exists x0∈X such that x0 has dense orbit which is unstable.

Definition 10 (see [<xref ref-type="bibr" rid="B7">7</xref>]).
Let f be a continuous map from a compact metric space (X,d) into itself. We say f has sensitive dependence on initial conditions if there exists r>0 such that for any x∈X and ϵ>0, there is some y∈X and a nonnegative integer n satisfying d(x,y)<ϵ and d(fn(x),fn(y))>r. f is said to be chaotic in the sense of Wiggins, if f is transitive and has sensitive dependence on initial conditions.

Definition 11.
Let (X,d) be a compact metric space, f:X→X be a continuous map, and D be an uncountable distributively scrambled set in a sequence.

We say that f exhibits dense distributional chaos in a sequence if the set D may be chosen to be dense. If D is not only dense but additionally consists of points with dense orbits, then we say that f exhibits transitive distributional chaos in a sequence.

Lemma 12.
Let Σ be an infinite sequence set of {0,1}. Then, there exists an uncountable subset E⊂Σ such that for any different points s=s1s2,…,t=t1t2,…,sm≠tm for infinitely many m and sn=tn for infinitely many n.

Proof.
For a proof, see [8].

Lemma 13.
If {pi} and {qi} are both infinite increasing subsequences of {mi} which is a sequence of positive integers, then there exists an infinite increasing subsequence {ti}⊂{mi} such that
(9)AR(f,{pi})∩DR(f,{qi})⊂DCR(f,{ti}).

Proof.
For a proof, see [9].

Lemma 14.
f
is weakly mixing if and only if for any m≥2,fm is transitive.

Proof.
For a proof, see [10].