# Furstenberg Families and Sensitivity

Authors: Huoyun Wang; Jincheng Xiong; Feng Tan
Journal: Discrete Dynamics in Nature and Society
 (2010)

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## Abstract

We introduce and study  some concepts of sensitivity via Furstenberg
families. A dynamical system(X,f) is ℱ-sensitive if there exists a positive ε such that for every x∈X and every open neighborhood U of x there exists y∈U such
that the pair (x,y) is not ℱ-ε-asymptotic; that is, the time set {n:d(fn(x),fn(y))>ε} belongs to ℱ, where ℱ is a Furstenberg family. A dynamical system
(X,f) is (ℱ1, ℱ2)-sensitive if there is a positive ε such that every x∈X is a limit of points y∈X such that the pair (x,y) is ℱ1-proximal but not ℱ2-ε-asymptotic; that is, the time set {n:d(fn(x),fn(y))<δ} belongs to ℱ1 for any positive δ but the time set {n:d(fn(x),fn(y))>ε} belongs to ℱ2, where
ℱ1  and ℱ2 are Furstenberg families.


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## Body

## 1. Introduction

Throughout this paper a topological dynamical system (TDS) is a pair(X,f), where X is a compact metric space with a metric d and f:X→X is a continuous surjective map. Let ℤ+ be the set of nonnegative  integers.The phrase—sensitive dependence on initial condition—was first used by Ruelle [1], to indicate some exponential rate of divergence of orbits of nearby points. Following the work by Guckenheimer [2], Auslander and Yorke [3], Devaney [4], a TDS (X,f) is called sensitive if there exists a positive ɛ such that for every x∈X and every open neighborhood U of x, there exist y∈U and n∈ℤ+ with d(fn(x),fn(y))>ɛ; that is, there exists a positive ɛ′ such that in any opene (= open and nonempty) set there are two distinct points whose trajectories are apart from ɛ′ (at least one moment).Recently, several authors studied the sensitive property (cf. Abraham et al. [5], Akin and Kolyada [6]). The following proposition holds according to [6].Proposition 1.1.
Let(X,f) be a TDS. The following conditions are equivalent.(1)
(X,f) is sensitive.(2)
There exists a positiveɛ such that for every x∈X and every open neighborhood U of x there exists y∈U with lim supn→∞d(fn(x),fn(y))>ɛ.(3)
There exists a positiveɛ such that in any opene set U⊂X there exist x,y∈U and n∈ℤ+ with d(fn(x),fn(y))>ɛ.(4)
There exists a positiveɛ such that in any opene set U⊂X there exist x,y∈U with lim supn→∞d(fn(x),fn(y))>ɛ.From Proposition1.1, we know that a TDS (X,f) is sensitive if and only if there exists a positive ɛ such that in any opene set there are two distinct points whose trajectories are infinitely many times apart at least of ɛ.Some authors introduced concepts which link the Li-Yorke versions of chaos with the sensitivity in the recent years. Blanchard et al. [7] introduced the concept of spatiotemporal chaos. A TDS (X,f) is called spatiotemporally chaotic if every x∈X is a limit of points y∈X such that the pair (x,y) is proximal but not asymptotic; that is, the pair (x,y) is a Li-Yorke scrambled pair [8]. That is(1.1)lim infn→∞d(fn(x),fn(y))=0,lim supn→∞d(fn(x),fn(y))>0.Akin and Kolyada [6] introduced the concept of Li-Yorke sensitivity. A TDS (X,f) is called Li-Yorke sensitive if there is a positive ɛ such that every x∈X is a limit of points y∈X such that the pair (x,y) is proximal but not ɛ-asymptotic. That is,(1.2)lim infn→∞d(fn(x),fn(y))=0,lim supn→∞d(fn(x),fn(y))>ɛ.We see that Li-Yorke sensitivity clearly implies spatiotemporal chaos, but the latter property is strictly weaker (see [6]).LetJ⊂ℤ+. The upper density of J is(1.3)μ¯(J)=lim supn→∞♯(J∩{0,1,…,n-1})n,where♯ denotes the cardinality of the set. The lower density of J is(1.4)μ̲(J)=lim infn→∞♯(J∩{0,1,…,n-1})n.A pair(x,y)∈X×X isdistributively scrambled pair [9] if there is positive ɛ such that μ̲({n∈ℤ+:d(fn(x),fn(y))<ɛ}=0, that is, μ¯({n∈ℤ+:d(fn(x),fn(y))≥ɛ}=1, and μ¯({n∈ℤ+:d(fn(x),fn(y))<δ})=1 for any positive δ.Letℤ+ be the set of nonnegative integers, and let 𝒫 be the collection of all subsets of ℤ+. A subset ℱ of 𝒫 is called a Furstenberg family [10] if it is hereditary upwards; that is, F1⊂F2 and F1∈ℱ imply F2∈ℱ.In the past few years, some authors [11–14] investigated proximity, mixing, and chaos via Furstenberg family. In [13], ℱ-scrambled pair was defined via a Furstenberg family ℱ. A pair (x,y) is called  ℱ-scrambled pair if there is positive ɛ such that {n∈ℤ+:d(fn(x),fn(y))≥ɛ}∈ℱ, and {n∈ℤ+:d(fn(x),fn(y))<δ}∈ℱ for any positive δ. In [14], (ℱ1,ℱ2)-scrambled pair was defined via Furstenberg families ℱ1 and ℱ2. A pair (x,y) is called  (ℱ1,ℱ2)-scrambled pair if there is positive ɛ such that {n∈ℤ+:d(fn(x),fn(y))<δ}∈ℱ1 for any positive δ, and {n∈ℤ+:d(fn(x),fn(y))≥ɛ}∈ℱ2.In this paper we investigate the sensitivity from the viewpoint of Furstenberg families.A dynamical system(X,f) is  ℱ-sensitive if there exists a positive ɛ such that for every x∈X and every open neighborhood U of x there exists y∈U such that {n∈ℤ+:d(fn(x),fn(y))>ɛ} belongs to ℱ, where ℱ is a Furstenberg family.A dynamical system(X,f) is  (ℱ1,ℱ2)-sensitive if there is a positive ɛ such that every x∈X is a limit of points y∈X such that {n∈ℤ+:d(fn(x),fn(y))<δ} belongs to ℱ1 for any positive δ but {n∈ℤ+:d(fn(x),fn(y))>ɛ} belongs to ℱ2, where ℱ1 and ℱ2 are Furstenberg families.In Section2, some basic notions related to Furstenberg families are introduced. In Section 3, we introduce and study the concept of ℱ-sensitivity. In Section 4, the notion of (ℱ1, ℱ2)-sensitivity is introduced and investigated, and the sensitivity of symbolic dynamics in the sense Furstenberg families is discussed finally.


## 2. Preliminary

In this section, we introduce some basic notions related Furstenberg families (for details see [10]). For a Furstenberg family ℱ, the dual family is(2.1)kℱ={F∈𝒫:F∩F′≠∅,∀F′∈ℱ}={F∈𝒫:ℤ+∖F∉ℱ}.Clearly, ifℱ is a Furstenberg family then so is kℱ. Let 𝒫 be the collection of all subsets of ℤ+. It is easy to see that k𝒫=∅,k∅=𝒫. Clearly, k(kℱ)=ℱ and ℱ1⊂ℱ2 implies kℱ2⊂kℱ1. Let ℬ be the family of all infinite subsets of ℤ+. It is easy to see that ℬ is a Furstenberg family and kℬ is the family of all cofinite subsets.A Furstenberg familyℱ is proper if it is a proper subset of 𝒫. It is easy to see that a Furstenberg family ℱ is proper if and only if ℤ+∈ℱ and ∅∉ℱ. Any subset 𝒜 of 𝒫 can generate a Furstenberg family [𝒜]={F∈𝒫:F⊃A for some A∈𝒜}.A Furstenberg familyℱ is countably generated [10, 13] if there exists a countable subset 𝒜 of 𝒫 such that [𝒜]=ℱ. Clearly, kℬ is a countably generated proper family.For Furstenberg familiesℱ1 and ℱ2, let ℱ1·ℱ2={F1∩F2:F1∈ℱ,F2∈ℱ2}. A Furstenberg family ℱ is full if it is proper and ℱ·kℱ⊂ℬ. It is easy to see that a Furstenberg family ℱ is full if and only if kℬ·ℱ⊂ℱ. Clearly, kℬ and ℬ are full. Clearly, if ℱ is full then kℬ⊂ℱ. A Furstenberg family ℱ is a filterdual if ℱ is proper and kℱ⊃kℱ·kℱ.For everys∈[0,1], let(2.2)M¯(s)={F∈ℬ:μ¯(F)≥s}.Clearly,M¯(0)=ℬ and every M¯(s) is a full Furstenberg family (see [13]).Let(X,f) be a TDS and U,V⊂X. We define the meeting time set(2.3)N(U,V)={n∈ℤ+:fn(U)∩V≠∅}.In particular we haveN(x,V)={n∈ℤ+:fn(x)∈V} for x∈X.LetA⊂X and x∈X. If N(x,A)∈ℱ, x is called an ℱ-attaching point of A. The set of all ℱ-attaching points of A is called the set of ℱ-attaching of A, denoted by ℱ(A). Clearly,(2.4)ℱ(A)=⋃F∈ℱ‍⋂n∈Ff-n(A)=⋂F∈kℱ‍⋃n∈Ff-n(A).Letℱ⊂ℬ be a Furstenberg family. Recall that a TDS (X,f) is ℱ-transitive if for each pair of opene subsets U and V of X, N(U,V)∈ℱ. (X,f) is ℱ-mixing if (X×X,f×f) is ℱ-transitive.Let(X,f) be a TDS. A Furstenberg family ℱ is compatible with the system (X,f) [13] if the set of ℱ-attaching of U is a Gδ set of X for each open set U of X.


## 3.ℱ-Sensitivity

In this section, we introduce and study the concept ofℱ-sensitivity. Let (X,f) be a TDS and ℱ a Furstenberg family. Suppose that A⊂X. Let [A]δ={x∈X:d(x,A)<δ}. A¯ denotes the closure of A. A subset B of X is called invariant for f if f(B)⊂B.We will use the following relations onX:(3.1)Δ={(x,x):x∈X},Vɛ={(x,y):d(x,y)<ɛ},V¯ɛ={(x,y):d(x,y)≤ɛ}.For any subsetR⊂X×X and any point x∈X, we write(3.2)R(x)={y:(x,y)∈R}.We define the sets ofℱ-asymptotic pairs(3.3)Asymɛ(ℱ)={(x,y):N((x,y),V¯ɛ)∈kℱ}=kℱ(V¯ɛ),Asymɛ(ℱ)(x)={y:(x,y)∈Asymɛ(ℱ)},Asym(ℱ)=⋂ɛ>0Asymɛ(ℱ),Asym(ℱ)(x)=⋂ɛ>0Asymɛ(ℱ)(x).We say that(X,f) is weaklyℱ-sensitive [10] if there is a positive ɛ—a weakly ℱ-sensitive constant—such that in every opene subset U of X there  exist x and y of U such that the pair (x,y) is not ℱ-ɛ-asymptotic. That is, {n∈ℤ+:d(fn(x),fn(y))>ɛ}∈ℱ, or N((x,y),X×X∖V¯ɛ)∈ℱ.We say that(X,f) is  ℱ-sensitive if there exists a positive ɛ—a ℱ-sensitive constant—such that for every x∈X and every open neighborhood U of x there exists y∈U such that the pair (x,y) is not ℱ-ɛ-asymptotic.Theorem 3.1.
Let(X,f) be a TDS. Let ℱ1 and ℱ2 be Furstenberg families. Suppose that kℱ1⊃kℱ2·kℱ2. If (X,f) is weakly ℱ1-sensitive, then (X,f)that is ℱ2-sensitive.Proof.
If(X,f) is not ℱ2-sensitive, then for each ɛ>0 there exists a x∈X and there exists an open neighborhood U of x such that N((x,y),X×X∖V¯ɛ)∉ℱ2 for each y∈U. Thus ℤ+∖N((x,y),V¯ɛ)∉ℱ2, this implies N((x,y),V¯ɛ)∈kℱ2. Since kℱ1⊃kℱ2·kℱ2, by the triangle inequality we have N((a,b),V¯2ɛ)∈kℱ1 for any a and b of U. Then ℤ+∖N((x,y),X×X∖V¯2ɛ)∈kℱ1. So N((x,y),X×X∖V¯2ɛ)∉ℱ1, this contradicts the (X,f) is weakly ℱ1-sensitive.Corollary 3.2.
Let(X,f) be a TDS and ℱ a filderdual. The system (X,f) is weakly ℱ-sensitive if and only if it is ℱ-sensitive.Lemma 3.3.
Let(X,f) be a TDS. A Furstenberg family ℱ is compatible with the system (X,f) if and only if the set of kℱ-attaching of V is an Fσ set of X for each closed subset V of X.Proof.
Suppose thatV is a closed subset of X, then
(3.4)x∈kℱ(V)⇔N(x,V)∈kℱ⇔ℤ+∖N(x,X∖V)∈kℱ⇔N(x,X∖V)∉ℱ⇔x∉ℱ(X∖V).
Hence,kℱ(V)=X∖ℱ(X∖V). Thus ℱ(X∖V) is a Gδ set of X, if and only if kℱ(V) is an Fσ set of X.Lemma 3.4.
Let(X,f) be a TDS and ℱ1 and ℱ2 Furstenberg families. Suppose that ℱ2 is compatible with the system (X×X,f×f), and kℱ1⊃kℱ2·kℱ2. If (X,f) is weakly ℱ1-sensitive, then there exists a positive ɛ such that for every x∈X, X∖Asymε(ℱ2)(x) is a dense Gδ set.Proof.
Sinceℱ2 is compatible with the system (X×X,f×f), then Asymɛ(ℱ2) is an Fσ set of X×X by Lemma 3.3. Suppose that Asymɛ(ℱ2)=⋃n=1∞Cn, where every Cn is a closed subset of X×X, then Asymɛ(ℱ2)(x)=⋃n=1∞Cn(x). Suppose that for each ɛ>0 there exists x∈X such that Asymɛ(ℱ2)(x) is not first category. By Baire theorem there exists an opene subset U of X for some n such that U⊂Cn(x). Hence for each y∈U, N((x,y),V¯ɛ)∈kℱ2. Since kℱ1⊃kℱ2·kℱ2, by the triangle inequality we have N((a,b),V¯2ɛ)∈kℱ1 for any a and b of U. Then ℤ+∖N((a,b),X×X∖V¯2ɛ)∈kℱ1. So N((a,b),X×X∖V¯2ɛ)∉ℱ1, this contradicts the (X,f) that is weakly ℱ1-sensitive.The following lemma is proved in [13]. We give another proof here for completeness.Lemma 3.5.
Let(X,f) be a TDS and ℱ a Furstenberg family. If kℱ is a countably generated proper family, or ℱ=M¯(t),t∈[0,1], then ℱ is compatible with the system (X,f).Proof.
(1) LetV be a closed subset of X. Suppose that kℱ is a proper family countably generated by 𝒜, where 𝒜 is countable set, then
(3.5)kℱ(V)=⋃F∈kℱ‍⋂n∈Ff-n(V)=⋃F∈𝒜‍⋂n∈Ff-n(V).
Hence,kℱ(V) is an Fσ set.
(2) Suppose thatℱ=M¯(t),t∈[0,1]. If t=0, then ℱ=ℬ. Since kℬ is a countably generated proper family, the result is true by (1).
Suppose thatt∈(0,1]. It is easy to see that kℱ={F∈ℬ:μ̲(F)>1-t}(3.6)kℱ(V)={x:lim infm→∞♯{j∈{1,2,…,m}:x∈f-j(V)}m>1-t}={x:∃n∈ℤ+,∀m>n,♯{j∈{1,2,…,m}:x∈f-j(V)}m>1-t}={x:∃n∈ℤ+,∀m>n,∃l∈{1,2,…,m}:lm>1-t,∃(r1,r2,…,rl):1≤r1≤⋯≤rl≤m∀i∈{1,2,…,l},fri(x)∈Vlm}=⋃n=1∞‍⋂m=n+1∞‍⋃l∈Θm‍⋃(r1,…,rl)∈Λl,m‍⋂i=1lf-ri(V),
where
(3.7)Θm={1≤l≤m:lm>1-t},Λl,m={(r1,…,rl):1≤r1≤⋯≤rl≤m}.
Hence kℱ(V) is an Fσ set.By Lemma3.3, 3.5 holds.Example 3.6.
Letℱ1={F∈ℬ:μ̲(F)>0.8} and ℱ2={F∈ℬ:μ¯(F)≥0.4}. If (X,f) is weakly ℱ1-sensitive, then (1)
(X,f) is ℱ2-sensitive,(2)
there exists a positiveɛ such that for every x∈X, X∖Asymɛ(ℱ2)(x) is a dense Gδ set.
Sinceℱ1={F∈ℬ:μ̲(F)>0.8}, then kℱ1={F∈ℬ:μ¯(F)≥0.2}. Since ℱ2={F∈ℬ:μ¯(F)≥0.4}, then kℱ2={F∈ℬ:μ̲(F)>0.6}. For any F1,F2∈kℱ2, then
(3.8)μ¯(F1∩F2)=1-μ̲(ℤ+∖(F1∩F2))=1-μ̲((ℤ+∖F1)∪(ℤ+∖F2))≥1-0.4-0.4=0.2.
Hencekℱ1⊃kℱ2·kℱ2. If (X,f) is weakly ℱ1-sensitive, then (X,f) is ℱ2-sensitive by Theorem 3.1. By Lemmas 3.5 and 3.4 if (X,f) is weakly ℱ1-sensitive, then there exists a positive ɛ such that for every x∈X, X∖Asymɛ(ℱ2)(x) is a dense Gδ set.The following theorem is based on arguments in Huang and Ye [15]. It is called Huang-Ye equivalences in [6]. We state it here via Furstenberg families.Theorem 3.7.
Let(X,f) be a TDS. If ℱ is a filterdual and is compatible with (X×X,f×f), then the following statements are equivalent.(1)
(X,f) is weakly ℱ-sensitive.(2)
There exists a positiveɛ such that Asymɛ(ℱ) is a first category subset of X×X.(3)
There exists a positiveɛ such that for every x∈X,  Asymɛ(ℱ)(x) is a first category subset of X.(4)
There exists a positiveɛ such that for every x∈X, x∈X∖Asymɛ(ℱ)(x)¯.(5)
There exists a positiveɛ such that
(3.9)ℱ(X×X∖V¯ɛ)
is dense in X×X.(6)
(X,f) is ℱ-sensitive.Proof.
(1)⇔ (6). By Corollary 3.2, it holds.
(2)⇒ (1). If the system is not weakly ℱ-sensitive then for every ɛ>0, there exists an opene subset U of X such that N((x,y),X×X∖V¯ɛ)∉ℱ for each (x,y)∈U×U, that is, ℤ+∖N((x,y),V¯ɛ)∉ℱ. Then N((x,y),V¯ɛ)∈kℱ, this implies (x,y)∈Asymɛ(ℱ). Hence, U×U⊂Asymɛ(ℱ). So Asymɛ(ℱ) is not of first category.
(3)⇒ (2). By Lemma 3.3, we know that Asymɛ(ℱ)=kℱ(Vɛ¯) is an Fσ set. Suppose that Asymɛ(ℱ)=⋃i=1∞Ci, where Ci is a closed subset of X×X. Then Asymɛ(ℱ)(x)=⋃i=1∞Ci(x). If Asymɛ(ℱ) is not first category then by the Baire category theorem some Ci has nonempty interior. If U×V⊂Ci and x∈U, then V⊂Ci(x). So Asymɛ(ℱ)(x) is not first category.
(1)⇒ (3). By Lemma 3.4, it holds.
Thus, we have proved that (1)–(3) are equivalent.
(4)⇒ (1). If (X,f) is not weakly ℱ-sensitive, then for any ɛ>0 there exists an opene subset U⊂X such that U×U⊂Asymɛ(ℱ). Let x∈U. Then x∈U⊂Asymɛ(ℱ)(x), this implies x∉X∖Asymɛ(ℱ)(x)¯.
(3)⇒ (4). If there exists a positive ɛ such that for every x∈X,Asymɛ(ℱ)(x) is a first category subset of X, then X∖Asymɛ(ℱ)(x) is a dense Gδ subset of X. Thus (4) is true.
(2)⇔ (5). At first, we note that
(3.10)ℱ(X×X∖V¯ɛ)=X×X∖kℱ(V¯ɛ)=X×X∖Asymɛ(ℱ).
By Baire theorem, (2) ⇔ (5).Theorem 3.8.
Let(X,f) be a TDS. Suppose that (X,f) have two nonempty invariant subsets A and B of X with d(A,B)=inf{d(a,b):a∈A,b∈B}>0 such that ⋃i=1∞f-i(A) and ⋃i=1∞f-i(B) are dense subsets of X, then there exists a positive ɛ such that kℬ(X×X∖V¯ɛ) is a dense subset of X×X, and if ℱ is a full Furstenberg family then (X,f) is weakly ℱ-sensitive.Proof.
Sinced(A,B)>0, there exist positive numbers δ and ɛ such that [A]δ×[B]δ⊂X×X∖V¯ɛ. Since ⋃i=1∞f-i(A) and ⋃i=1∞f-i(B) are dense subsets of X, it is easy to check that so are kℬ([A]δ) and kℬ([B]δ). Since kℬ(X×X∖V¯ɛ)⊃kℬ([A]δ×[B]δ)⊃kℬ([A]δ)×kℬ([B]δ), then kℬ(X×X∖V¯ɛ) is a dense subset of X×X. Since ℱ is full then kℬ⊂ℱ, this implies that ℱ(X×X∖V¯ɛ) is a dense subset of X×X. Hence, (X,f) is weakly ℱ-sensitive.A map is semiopen if the image of an opene subset contains an opene subset. A factor mapπ:(X,f)→(Y,g) between dynamical systems is a continuous surjective map π:X→Y such that g∘π=π∘f. The weakly ℱ-sensitivity can be lifted up by a semi-open factor map.Theorem 3.9.
Let(X,f) and (Y,g) be TDS and π:X→Y semi-open factor map. Let ℱ be a Furstenberg family. If (Y,g) is weakly ℱ-sensitive, so is (X,f).Proof.
Letɛ be a weakly ℱ-sensitive constant for (Y,g). Since π is continuous then there is δ>0 such that if d2(π(x),π(y))>ɛ then d1(x,y)>δ.
LetU be an opene subset of X. As π is semi-open, π(U) contains an opene subset V of Y. Since (Y,g) is weakly ℱ-sensitive, then there exist y1 and y2 of V such that {n∈ℤ+:d2(gn(y1),gn(y2))>ɛ)}∈ℱ. Let x1,x2∈U with π(x1)=y1 and π(x2)=y2. Then {n∈ℤ+:d1(fn(x1),fn(x2))>δ}∈ℱ, that is, (X,f) is weakly ℱ-sensitive.


## 4.(ℱ1,ℱ2)-Sensitivity

In this section, we introduce and study the notion of (ℱ1,ℱ2)-sensitivity which links chaos and sensitivity via a couple Furstenberg families (ℱ1,ℱ2).Let(X,f) be a TDS and F∈ℬ. A pair (x,y)∈X×X is called F-proximal if(4.1)lim infF∋n→∞d(fn(x),fn(y))=0.We denote the set of allF-proximal pairs by PF.The following lemma comes from [11].Lemma 4.1.
Let(X,f) be a TDS and F={t1<t2<⋯}∈ℬ. Then
(4.2)PF=⋂n=1∞‍⋃i=1∞(f×f)-tiV1/n.
Letℱ be a Furstenberg family. A pair (x,y)∈X×X is called ℱ-proximal if (x,y)∈ℱ(Vɛ) for any ɛ>0. We denote the set of all ℱ-proximal pairs by Pℱ.
Note that [12]:
(4.3)Pℱ=⋂ɛ>0‍⋃F∈ℱ‍⋂n∈F(f×f)-n(Vɛ)=⋂k=1∞‍⋂F∈kℱ‍⋃n∈F(f×f)-n(V1/k)=⋂k=1∞ℱ(V1/k)=⋂F∈kℱPF.
Suppose thatℱ1 and ℱ2 are Furstenberg families.
A TDS(X,f) is called  (ℱ1,ℱ2)-spatiotemporally chaotic  if every x∈X is a limit of points y∈X such that the pair (x,y) is ℱ1-proximal but not ℱ2-asymptotic. That is, for all x∈X
(4.4)x∈Pℱ1(x)∖Asym(ℱ2)(x)¯.
Whenℱ1=ℱ2=M¯(0)=ℬ, it is the usual spatiotemporal chaos.
A TDS(X,f) is called (ℱ1,ℱ2)-sensitive  if there is a positive ɛ such that every x∈X is a limit of points y∈X such that the pair (x,y) is ℱ1-proximal but not ℱ2-ɛ-asymptotic.
That is, for allx∈X
(4.5)x∈Pℱ1(x)∖Asymɛ(ℱ2)(x)¯=Pℱ1(x)∩ℱ2(X×X∖V¯ɛ)(x)¯.
When ℱ1=ℱ2=M¯(0)=ℬ, (X,f) is the usual Li-Yorke sensitivity.
If the pair(x,y) is M¯(1)-proximal but not M¯(1)-ɛ-asymptotic, then (x,y) is the usual distributively scrambled pair.We will use the following lemmas which comes from [10, 11], respectively.Lemma 4.2.
Letℱ be a full Furstenberg family. If (X,f) is ℱ-mixing, then PF(x) is a dense Gδ set of X for each F∈kℱ and each x∈X.Lemma 4.3.
Let(X,f) be a TDS and ℱ a Furstenberg family. (X,f) is ℱ-transitive if and only if for every F∈kℱ and every opene subset U of X, ⋃{f-t(U):t∈F} is an open and dense subset of X (see [10, Proposition 4.1]).Theorem 4.4.
Let(X,f) be a TDS. Let ℱ1 and ℱ2 be Furstenberg families. If there exists a positive ɛ such that X∖Asymɛ(ℱ2)(x) is a dense Gδ set for every x∈X, and Pℱ1(x) is a dense Gδ set of X for every x∈X, then (X,f) is (ℱ1,ℱ2)-sensitive.Proof.
SincePℱ1(x)∖Asymɛ(ℱ2)(x) is a dense Gδ subset of X. Hence, (X,f) is (ℱ1,ℱ2)- sensitive.Theorem 4.5.
Let(X,f) be a nontrivial TDS and ℱ a full filterdual. Suppose that kℱ is countably generated. If (X,f) is ℱ-mixing, then (X,f) is (ℱ,ℱ)-sensitive.Proof.
Suppose thatkℱ is a proper family countably generated by 𝒜, where 𝒜 is a countable set. Then Pℱ(x)=⋂F∈kℱPF(x)=⋂F∈𝒜PF(x). By Lemmas 4.1 and 4.2, Pℱ(x) is a dense Gδ set of X. Choose ɛ>0 such that X×X∖V¯ɛ is a nonempty open subset of X×X. By Lemma 4.3, ℱ(X×X∖V¯ɛ)=⋂F∈kℱ⋃n∈F(f×f)-n(X×X∖V¯ɛ) is a dense Gδ subset of X×X. By Theorem 3.7, (X,f) is ℱ-sensitive. Thus (X,f) is (ℱ,ℱ)-sensitive by Theorem 4.4.Lemma 4.6.
Let(X,f) be a TDS. Suppose that ℱ is a full Furstenberg family and is compatible with the system (X×X,f×f). If there is a fixed point p of f such that ⋃i=1∞f-i(p) is dense subset of X, then Pℱ is a dense Gδ set of X×X.Proof.
As⋃i=1∞f-i(p) is dense subset of X, it is easy to check that so is kℬ({p}ɛ) for any positive ɛ. Since kℬ(Vɛ)⊃kℬ({(p,p)}ɛ)⊃kℬ({p}δ)×kℬ({p}δ) for some positive δ, then kℬ(Vɛ) is a dense set of X×X. As ℱ is full then kℬ⊂ℱ, this implies that ℱ(Vɛ) is a dense subset of X×X. And since ℱ is compatible with the system (X×X,f×f), ℱ(Vɛ) is a Gδ set of X×X. By Pℱ=⋂k=1∞ℱ(V1/k), then Pℱ is a dense Gδ set of X×X.


## 5.(ℱ1,ℱ2)-Sensitivity of Symbolic Dynamics (ΣN,σ)

Finally, as examples we will discuss theℱ-sensitivity and (ℱ1,ℱ2)-sensitivity of symbolic dynamics.LetE={1,2,…,N}(N≥2) with the discrete topology. Let Ei=E, for all i≥1. Let ΣN=∏i=1∞Ei with the product topology. Then ΣN is a compact metric space. ΣN is called the symbolic space generated by E={1,2,…,N}. Let σ:ΣN→ΣN be the shift which will be defined as σ(x1x2x3⋯)=x2x3⋯ for any x1x2x3⋯ of ΣN. Then (ΣN,σ) is called symbolic dynamics. Let [i1i2⋯in]={x∈ΣN:x1=i1,x2=i2,…,xn=in}.We define a metricd which is compatible with the product topology on ΣN as follows: for all x=x1x2…,y=y1y2⋯∈ΣN,(5.1)d(x,y)={0,x=y,1Nk,k=min{i:xi≠yi}-1.Theorem 5.1.
Letℱ be a full Furstenberg family. Then (ΣN,σ) is ℱ-sensitive.Proof.
Letp=111⋯ and q=222⋯. Then p and q are fixed points of σ, and both ⋃i=1∞σ-i(p) and ⋃i=1∞σ-i(q) are dense subsets of ΣN. By Theorem 3.8, (ΣN,σ) is weakly ℱ-sensitive. Let ɛ be a weakly ℱ-sensitive constant. Now we show that (ΣN,σ) is also ℱ-sensitive. For any x=x1x2x3⋯ of ΣN and for any open neighborhood [x1x2⋯xn] of x, there exist points y=y1y2y3⋯ and z=z1z2z3⋯ of [x1x2⋯xn] such that N((y,z),X×X∖V¯ɛ)∈ℱ. Choose u=u1u2u3⋯ of [x1x2⋯xn] such that ut=xt if yt=zt otherwise ut≠xt. Then N((x,u),X×X∖V¯ɛ)=N((y,z),X×X∖V¯ɛ)∈ℱ, so (ΣN,σ) is ℱ-sensitive.Lemma 5.2.
Suppose thatℱ is a full Furstenberg family and is compatible with the system (ΣN×ΣN,σ×σ), then Pℱ(x) is a dense Gδ set of ΣN for every x of ΣN.Proof.
By Lemma4.6, Pℱ(x) is a Gδ set of ΣN for every x of ΣN.Now we show thatPℱ(x) is dense for every x=x1x2x3⋯ of ΣN. For any y=y1y2y3⋯ of ΣN and for any open neighborhood [y1y2⋯yn] of y. Choose z=z1z2⋯ of [y1y2⋯yn] such that σn(x)=σn(z), then z∈kℬ(Vɛ)(x) for any positive ɛ, this implies that kℬ(Vɛ)(x) is dense. Since ℱ is a full then kℬ⊂ℱ, so ℱ(Vɛ)(x) is dense Gδ set of ΣN. By Pℱ(x)=⋂k=1∞ℱ(V1/k)(x), then Pℱ(x) is a dense Gδ set of ΣN.Lemma 5.3.
Suppose thatℱ is a full Furstenberg family and is compatible with the system (ΣN×ΣN,σ×σ), then there exists a positive ɛ such that for every x∈ΣN, ΣN∖Asymɛ(ℱ)(x) is a dense Gδ set of ΣN.Proof.
Letp=111⋯ and q=222⋯. Then p and q are fixed points of σ, and both ⋃i=1∞σ-i(p) and ⋃i=1∞σ-i(q) are dense subsets of ΣN. By Theorem 3.8 there exists a positive ɛ such that kℬ(ΣN×ΣN∖V¯ɛ) is a dense set of ΣN×ΣN. Now we show for every x∈X, X∖Asymɛ(ℱ)(x) is a dense Gδ set of ΣN. For any y=y1y2y3⋯ of ΣN, and for any open neighborhood [y1y2⋯yn]×[x1x2⋯xn] of (y,x) of ΣN×ΣN, there exists (u,v) of [y1y2⋯yn]×[x1x2⋯xn] such that N((u,v),ΣN×ΣN∖V¯ɛ)∈kℬ. Choose z=z1z2z3⋯ of [y1y2⋯yn] such that, zt=xt if ut=vt otherwise zt≠xt, when t>n. Then N((u,x),ΣN×ΣN∖V¯ɛ)∈kℬ. Since ℱ is full, then kℬ⊂ℱ, this implies that N((u,x),ΣN×ΣN∖V¯ɛ)∈ℱ. Hence ℱ(ΣN×ΣN∖V¯ɛ)(x) is a dense set of ΣN. Since ℱ is compatible with the system (ΣN×ΣN,σ×σ), then ℱ(ΣN×ΣN∖V¯ɛ) is a Gδ set of ΣN×ΣN. Hence ℱ(ΣN×ΣN∖V¯ɛ)(x) is a dense Gδ set of ΣN.By Lemmas5.2 and 5.3, the following theorem holds.Theorem 5.4.
Suppose thatℱ1 and ℱ2 are full, and are compatible with (ΣN×ΣN,σ×σ), then (ΣN,σ) is (ℱ1,ℱ2)-sensitive. In particular, (ΣN,σ) is (M¯(1),M¯(1))-sensitive.



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*Source: 649348-2010-02-22.xml*