Furstenberg Families and Sensitivity

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(𝑓^𝑛(𝑥),𝑓^𝑛(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(𝑓^𝑛(𝑥),𝑓^𝑛(𝑦))<𝛿} belongs to ℱ₁ for any positive 𝛿 but the time set {𝑛∶𝑑(𝑓^𝑛(𝑥),𝑓^𝑛(𝑦))>𝜀} belongs to ℱ₂, where ℱ₁ and ℱ₂ are Furstenberg families.

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 𝜀 such that for every 𝑥∈𝑋 and every open neighborhood 𝑈 of 𝑥, there exist 𝑦∈𝑈 and 𝑛∈ℤ₊ with 𝑑(𝑓^𝑛(𝑥),𝑓^𝑛(𝑦))>𝜀; that is, there exists a positive 𝜀^′ such that in any opene (= open and nonempty) set there are two distinct points whose trajectories are apart from 𝜀^′ (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 positive 𝜀 such that for every 𝑥∈𝑋 and every open neighborhood 𝑈 of 𝑥 there exists 𝑦∈𝑈 with limsup_𝑛→∞𝑑(𝑓^𝑛(𝑥),𝑓^𝑛(𝑦))>𝜀.(3)There exists a positive 𝜀 such that in any opene set 𝑈⊂𝑋 there exist 𝑥,𝑦∈𝑈 and 𝑛∈ℤ₊ with 𝑑(𝑓^𝑛(𝑥),𝑓^𝑛(𝑦))>𝜀.(4)There exists a positive 𝜀 such that in any opene set 𝑈⊂𝑋 there exist 𝑥,𝑦∈𝑈 with limsup_𝑛→∞𝑑(𝑓^𝑛(𝑥),𝑓^𝑛(𝑦))>𝜀.

From Proposition 1.1, we know that a TDS (𝑋,𝑓) is sensitive if and only if there exists a positive 𝜀 such that in any opene set there are two distinct points whose trajectories are infinitely many times apart at least of 𝜀.

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

liminf_𝑛→∞𝑑(𝑓^𝑛(𝑥),𝑓^𝑛(𝑦))=0,limsup_𝑛→∞𝑑(𝑓^𝑛(𝑥),𝑓^𝑛(𝑦))>0. (1.1)

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 𝜀 such that every 𝑥∈𝑋 is a limit of points 𝑦∈𝑋 such that the pair (𝑥,𝑦) is proximal but not 𝜀-asymptotic. That is,

liminf_𝑛→∞𝑑(𝑓^𝑛(𝑥),𝑓^𝑛(𝑦))=0,limsup_𝑛→∞𝑑(𝑓^𝑛(𝑥),𝑓^𝑛(𝑦))>𝜀. (1.2)

We see that Li-Yorke sensitivity clearly implies spatiotemporal chaos, but the latter property is strictly weaker (see [6]).

Let 𝐽⊂ℤ₊. The upper density of 𝐽 is

𝜇(𝐽)=limsup_𝑛→∞♯(𝐽∩{0,1,…,𝑛−1})𝑛, (1.3)

where ♯ denotes the cardinality of the set. The lower density of 𝐽 is

𝜇(𝐽)=liminf_𝑛→∞♯(𝐽∩{0,1,…,𝑛−1})𝑛. (1.4)

A pair (𝑥,𝑦)∈𝑋×𝑋 is distributively scrambled pair [9] if there is positive 𝜀 such that 𝜇({𝑛∈ℤ₊∶𝑑(𝑓^𝑛(𝑥),𝑓^𝑛(𝑦))<𝜀}=0, that is, 𝜇({𝑛∈ℤ₊∶𝑑(𝑓^𝑛(𝑥),𝑓^𝑛(𝑦))≥𝜀}=1, and 𝜇({𝑛∈ℤ₊∶𝑑(𝑓^𝑛(𝑥),𝑓^𝑛(𝑦))<𝛿})=1 for any positive 𝛿.

Let ℤ₊ 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 𝜀 such that {𝑛∈ℤ₊∶𝑑(𝑓^𝑛(𝑥),𝑓^𝑛(𝑦))≥𝜀}∈ℱ, and {𝑛∈ℤ₊∶𝑑(𝑓^𝑛(𝑥),𝑓^𝑛(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 𝜀 such that {𝑛∈ℤ₊∶𝑑(𝑓^𝑛(𝑥),𝑓^𝑛(𝑦))<𝛿}∈ℱ₁ for any positive 𝛿, and {𝑛∈ℤ₊∶𝑑(𝑓^𝑛(𝑥),𝑓^𝑛(𝑦))≥𝜀}∈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 𝜀 such that for every 𝑥∈𝑋 and every open neighborhood 𝑈 of 𝑥 there exists 𝑦∈𝑈 such that {𝑛∈ℤ₊∶𝑑(𝑓^𝑛(𝑥),𝑓^𝑛(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 {𝑛∈ℤ₊∶𝑑(𝑓^𝑛(𝑥),𝑓^𝑛(𝑦))<𝛿} belongs to ℱ₁ for any positive 𝛿 but {𝑛∈ℤ₊∶𝑑(𝑓^𝑛(𝑥),𝑓^𝑛(𝑦))>𝜀} belongs to ℱ₂, where ℱ₁ and ℱ₂ are Furstenberg families.

In Section 2, 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

𝑘ℱ={𝐹∈𝒫∶𝐹∩𝐹^′≠∅,∀𝐹^′∈ℱ}={𝐹∈𝒫∶ℤ₊⧵𝐹∉ℱ}. (2.1)

Clearly, if ℱ 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 family ℱ 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 family ℱ 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 families ℱ₁ 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 every 𝑠∈[0,1], let

𝑀(𝑠)={𝐹∈ℬ∶𝜇(𝐹)≥𝑠}. (2.2)

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

𝑁(𝑈,𝑉)={𝑛∈ℤ₊∶𝑓^𝑛(U)∩𝑉≠∅}. (2.3)

In particular we have 𝑁(𝑥,𝑉)={𝑛∈ℤ₊∶𝑓^𝑛(𝑥)∈𝑉} for 𝑥∈𝑋.

Let 𝐴⊂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,

ℱ(𝐴)=_{𝐹∈ℱ𝑛∈𝐹}𝑓^−𝑛(𝐴)=_{𝐹∈𝑘ℱ𝑛∈𝐹}𝑓^−𝑛(𝐴). (2.4)

Let ℱ⊂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 𝐺_𝛿 set of 𝑋 for each open set 𝑈 of 𝑋.

3. ℱ-Sensitivity

In this section, we introduce and study the concept of ℱ-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 on 𝑋:

Δ={(𝑥,𝑥)∶𝑥∈𝑋},𝑉_𝜀={(𝑥,𝑦)∶𝑑(𝑥,𝑦)<𝜀},𝑉_𝜀={(𝑥,𝑦)∶𝑑(𝑥,𝑦)≤𝜀}. (3.1)

For any subset 𝑅⊂𝑋×𝑋 and any point 𝑥∈𝑋, we write

𝑅(𝑥)={𝑦∶(𝑥,𝑦)∈𝑅}. (3.2)

We define the sets of ℱ-asymptotic pairs

Asym_𝜀(ℱ)={(𝑥,𝑦)∶𝑁((𝑥,𝑦),𝑉_𝜀)∈𝑘ℱ}=𝑘ℱ(𝑉_𝜀),Asym_𝜀(ℱ)(𝑥)={𝑦∶(𝑥,𝑦)∈Asym_𝜀(ℱ)},Asym(ℱ)=_𝜀>0Asym_𝜀(ℱ),Asym(ℱ)(𝑥)=_𝜀>0Asym_𝜀(ℱ)(𝑥). (3.3)

We say that (𝑋,𝑓) is weakly ℱ-sensitive [10] if there is a positive 𝜀—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-𝜀-asymptotic. That is, {𝑛∈ℤ₊∶𝑑(𝑓^𝑛(𝑥),𝑓^𝑛(𝑦))>𝜀}∈ℱ, or 𝑁((𝑥,𝑦),𝑋×𝑋⧵𝑉_𝜀)∈ℱ.

We say that (𝑋,𝑓) is ℱ-sensitive if there exists a positive 𝜀—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-𝜀-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 𝜀>0 there exists a 𝑥∈𝑋 and there exists an open neighborhood 𝑈 of 𝑥 such that 𝑁((𝑥,𝑦),𝑋×𝑋⧵𝑉_𝜀)∉ℱ₂ for each 𝑦∈𝑈. Thus ℤ₊⧵𝑁((𝑥,𝑦),𝑉_𝜀)∉ℱ₂, this implies 𝑁((𝑥,𝑦),𝑉_𝜀)∈𝑘ℱ₂. Since 𝑘ℱ₁⊃𝑘ℱ₂⋅𝑘ℱ₂, by the triangle inequality we have 𝑁((𝑎,𝑏),𝑉_2𝜀)∈𝑘ℱ₁ for any 𝑎 and 𝑏 of 𝑈. Then ℤ₊⧵𝑁((𝑥,𝑦),𝑋×𝑋⧵𝑉_2𝜀)∈𝑘ℱ₁. So 𝑁((𝑥,𝑦),𝑋×𝑋⧵𝑉_2𝜀)∉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 𝐹_𝜎 set of 𝑋 for each closed subset 𝑉 of 𝑋.

Proof. Suppose that 𝑉 is a closed subset of 𝑋, then 𝑥∈𝑘ℱ(𝑉)⇔𝑁(𝑥,𝑉)∈𝑘ℱ⇔ℤ₊⧵𝑁(𝑥,𝑋⧵𝑉)∈𝑘ℱ⇔𝑁(𝑥,𝑋⧵𝑉)∉ℱ⇔𝑥∉ℱ(𝑋⧵𝑉). (3.4)
Hence, 𝑘ℱ(𝑉)=𝑋⧵ℱ(𝑋⧵𝑉). Thus ℱ(𝑋⧵𝑉) is a 𝐺_𝛿 set of 𝑋, if and only if 𝑘ℱ(𝑉) is an 𝐹_𝜎 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 𝜀 such that for every 𝑥∈𝑋, 𝑋⧵Asym_𝜀(ℱ₂)(𝑥) is a dense 𝐺_𝛿 set.

Proof. Since ℱ₂ is compatible with the system (𝑋×𝑋,𝑓×𝑓), then Asym_𝜀(ℱ₂) is an 𝐹_𝜎 set of 𝑋×𝑋 by Lemma 3.3. Suppose that Asym_𝜀(ℱ₂)=⋃^∞_𝑛=1𝐶_𝑛, where every 𝐶_𝑛 is a closed subset of 𝑋×𝑋, then Asym_𝜀(ℱ₂)(𝑥)=⋃^∞_𝑛=1𝐶_𝑛(𝑥). Suppose that for each 𝜀>0 there exists 𝑥∈𝑋 such that Asym_𝜀(ℱ₂)(𝑥) is not first category. By Baire theorem there exists an opene subset 𝑈 of 𝑋 for some 𝑛 such that 𝑈⊂𝐶_𝑛(𝑥). Hence for each 𝑦∈𝑈, 𝑁((𝑥,𝑦),𝑉_𝜀)∈𝑘ℱ₂. Since 𝑘ℱ₁⊃𝑘ℱ₂⋅𝑘ℱ₂, by the triangle inequality we have 𝑁((𝑎,𝑏),𝑉_2𝜀)∈𝑘ℱ₁ for any 𝑎 and 𝑏 of 𝑈. Then ℤ₊⧵𝑁((𝑎,𝑏),𝑋×𝑋⧵𝑉_2𝜀)∈𝑘ℱ₁. So 𝑁((𝑎,𝑏),𝑋×𝑋⧵𝑉_2𝜀)∉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) Let 𝑉 be a closed subset of 𝑋. Suppose that 𝑘ℱ is a proper family countably generated by 𝒜, where 𝒜 is countable set, then
𝑘ℱ(𝑉)=_{𝐹∈𝑘ℱ𝑛∈𝐹}𝑓^−𝑛(𝑉)=_{𝐹∈𝒜𝑛∈𝐹}𝑓^−𝑛(𝑉). (3.5)
Hence, 𝑘ℱ(𝑉) is an 𝐹_𝜎 set.
(2) Suppose that ℱ=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 that 𝑡∈(0,1]. It is easy to see that 𝑘ℱ={𝐹∈ℬ∶𝜇(𝐹)>1−𝑡}𝑘ℱ(𝑉)={𝑥∶liminf_𝑚→∞♯{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₊𝑙,∀𝑚>𝑛,∃𝑙∈{1,2,…,𝑚}∶𝑚>1−𝑡,∃(𝑟₁,𝑟₂,…,𝑟_𝑙)∶1≤𝑟₁≤⋯≤𝑟_𝑙≤𝑚∀𝑖∈{1,2,…,𝑙},𝑓^𝑟𝑖=(𝑥)∈𝑉}_{∞∞𝑛=1𝑚=𝑛+1𝑙∈Θ𝑚(𝑟1,…,𝑟𝑙)∈Λ𝑙,𝑚𝑙𝑖=1}𝑓^−𝑟𝑖(𝑉), (3.6) where Θ_𝑚𝑙={1≤𝑙≤𝑚∶𝑚Λ>1−𝑡},_𝑙,𝑚={(𝑟₁,…,𝑟_𝑙)∶1≤𝑟₁≤⋯≤𝑟_𝑙≤𝑚}. (3.7) Hence 𝑘ℱ(𝑉) is an 𝐹_𝜎 set.

By Lemma 3.3, 3.5 holds.

Example 3.6. Let ℱ₁={𝐹∈ℬ∶𝜇(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 positive 𝜀 such that for every 𝑥∈𝑋, 𝑋⧵Asym_𝜀(ℱ₂)(𝑥) is a dense 𝐺_𝛿 set.
Since ℱ₁={𝐹∈ℬ∶𝜇(𝐹)>0.8}, then 𝑘ℱ₁={𝐹∈ℬ∶𝜇(𝐹)≥0.2}. Since ℱ₂={𝐹∈ℬ∶𝜇(𝐹)≥0.4}, then 𝑘ℱ₂={𝐹∈ℬ∶𝜇(𝐹)>0.6}. For any 𝐹₁,𝐹₂∈𝑘ℱ₂, then
𝜇(𝐹₁∩𝐹₂)=1−𝜇(ℤ₊⧵(𝐹₁∩𝐹₂))=1−𝜇((ℤ₊⧵𝐹₁)∪(ℤ₊⧵𝐹₂))≥1−0.4−0.4=0.2. (3.8)
Hence 𝑘ℱ₁⊃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 𝜀 such that for every 𝑥∈𝑋, 𝑋⧵Asym_𝜀(ℱ₂)(𝑥) is a dense 𝐺_𝛿 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 positive 𝜀 such that Asym_𝜀(ℱ) is a first category subset of 𝑋×𝑋.(3)There exists a positive 𝜀 such that for every 𝑥∈𝑋,  Asym_𝜀(ℱ)(𝑥) is a first category subset of 𝑋.(4)There exists a positive 𝜀 such that for every 𝑥∈𝑋, 𝑥∈𝑋⧵Asym_𝜀(ℱ)(𝑥).(5)There exists a positive 𝜀 such that ℱ(𝑋×𝑋⧵𝑉_𝜀) (3.9) 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 𝜀>0, there exists an opene subset 𝑈 of 𝑋 such that 𝑁((𝑥,𝑦),𝑋×𝑋⧵𝑉_𝜀)∉ℱ for each (𝑥,𝑦)∈𝑈×𝑈, that is, ℤ₊⧵𝑁((𝑥,𝑦),𝑉_𝜀)∉ℱ. Then 𝑁((𝑥,𝑦),𝑉_𝜀)∈𝑘ℱ, this implies (𝑥,𝑦)∈Asym_𝜀(ℱ). Hence, 𝑈×𝑈⊂Asym_𝜀(ℱ). So Asym_𝜀(ℱ) is not of first category.
(3) ⇒ (2). By Lemma 3.3, we know that Asym_𝜀(ℱ)=𝑘ℱ(𝑉_𝜀) is an 𝐹_𝜎 set. Suppose that Asym_𝜀(ℱ)=⋃^∞_𝑖=1𝐶_𝑖, where 𝐶_𝑖 is a closed subset of 𝑋×𝑋. Then Asym_𝜀(ℱ)(𝑥)=⋃^∞_𝑖=1𝐶_𝑖(𝑥). If Asym_𝜀(ℱ) is not first category then by the Baire category theorem some 𝐶_𝑖 has nonempty interior. If 𝑈×𝑉⊂𝐶_𝑖 and 𝑥∈𝑈, then 𝑉⊂𝐶_i(𝑥). So Asym_𝜀(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 𝜀>0 there exists an opene subset 𝑈⊂𝑋 such that 𝑈×𝑈⊂Asym_𝜀(ℱ). Let 𝑥∈𝑈. Then 𝑥∈𝑈⊂Asym_𝜀(ℱ)(𝑥), this implies 𝑥∉𝑋⧵Asym_𝜀(ℱ)(𝑥).
(3) ⇒ (4). If there exists a positive 𝜀 such that for every 𝑥∈𝑋,Asym_𝜀(ℱ)(𝑥) is a first category subset of 𝑋, then 𝑋⧵Asym_𝜀(ℱ)(𝑥) is a dense 𝐺_𝛿 subset of 𝑋. Thus (4) is true.
(2) ⇔ (5). At first, we note that
ℱ(𝑋×𝑋⧵𝑉_𝜀)=𝑋×𝑋⧵𝑘ℱ(𝑉_𝜀)=𝑋×𝑋⧵Asym_𝜀(ℱ). (3.10) 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=1𝑓^−𝑖(𝐴) and ⋃^∞_𝑖=1𝑓^−𝑖(𝐵) are dense subsets of 𝑋, then there exists a positive 𝜀 such that 𝑘ℬ(𝑋×𝑋⧵𝑉_𝜀) is a dense subset of 𝑋×𝑋, and if ℱ is a full Furstenberg family then (𝑋,𝑓) is weakly ℱ-sensitive.

Proof. Since 𝑑(𝐴,𝐵)>0, there exist positive numbers 𝛿 and 𝜀 such that [𝐴]_𝛿×[𝐵]_𝛿⊂𝑋×𝑋⧵𝑉_𝜀. Since ⋃^∞_𝑖=1𝑓^−𝑖(𝐴) and ⋃^∞_𝑖=1𝑓^−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_𝜀)⊃𝑘ℬ([𝐴]_𝛿×[𝐵]_𝛿)⊃𝑘ℬ([𝐴]_𝛿)×𝑘ℬ([𝐵]_𝛿), then 𝑘ℬ(𝑋×𝑋⧵V_𝜀) is a dense subset of 𝑋×𝑋. Since ℱ is full then 𝑘ℬ⊂ℱ, this implies that ℱ(𝑋×𝑋⧵𝑉_𝜀) 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 map 𝜋∶(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. Let 𝜀 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))>𝜀 then 𝑑₁(𝑥,𝑦)>𝛿.
Let 𝑈 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(𝑔^𝑛(𝑦₁),𝑔^𝑛(𝑦₂))>𝜀)}∈ℱ. Let 𝑥₁,𝑥₂∈𝑈 with 𝜋(𝑥₁)=𝑦₁ and 𝜋(𝑥₂)=𝑦₂. Then {𝑛∈ℤ₊∶𝑑₁(𝑓^𝑛(𝑥₁),𝑓^𝑛(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

liminf_{𝐹∋𝑛→∞}𝑑(𝑓^𝑛(𝑥),𝑓^𝑛(𝑦))=0. (4.1)

We denote the set of all 𝐹-proximal pairs by 𝑃_𝐹.

The following lemma comes from [11].

Lemma 4.1. Let (𝑋,𝑓) be a TDS and 𝐹={𝑡₁<𝑡₂<⋯}∈ℬ. Then 𝑃_𝐹=^∞∞_𝑛=1𝑖=1(𝑓×𝑓)^−𝑡𝑖𝑉_1/𝑛. (4.2)
Let ℱ be a Furstenberg family. A pair (𝑥,𝑦)∈𝑋×𝑋 is called ℱ-proximal if (𝑥,𝑦)∈ℱ(𝑉_𝜀) for any 𝜀>0. We denote the set of all ℱ-proximal pairs by 𝑃_ℱ.
Note that [12]:
𝑃_ℱ=_{𝜀>0𝐹∈ℱ𝑛∈𝐹}(𝑓×𝑓)^−𝑛(𝑉_𝜀)=^∞_{𝑘=1𝐹∈𝑘ℱ𝑛∈𝐹}(𝑓×𝑓)^−𝑛(𝑉_1/𝑘)=^∞_𝑘=1ℱ(𝑉_1/𝑘)=_{𝐹∈𝑘ℱ}𝑃_𝐹. (4.3)
Suppose that ℱ₁ 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
𝑥∈𝑃_ℱ1(𝑥)⧵Asym(ℱ₂)(𝑥). (4.4)
When ℱ₁=ℱ₂=𝑀(0)=ℬ, it is the usual spatiotemporal chaos.
A TDS (𝑋,𝑓) is called (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, for all 𝑥∈𝑋
𝑥∈𝑃_ℱ1(𝑥)⧵Asym_𝜀(ℱ₂)(𝑥)=𝑃_ℱ1(𝑥)∩ℱ₂(𝑋×𝑋⧵𝑉_𝜀)(𝑥). (4.5) 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)-𝜀-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. Let ℱ be a full Furstenberg family. If (𝑋,𝑓) is ℱ-mixing, then 𝑃_𝐹(𝑥) is a dense 𝐺_𝛿 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 𝜀 such that 𝑋⧵Asym_𝜀(ℱ₂)(𝑥) is a dense 𝐺_𝛿 set for every 𝑥∈𝑋, and 𝑃_ℱ1(𝑥) is a dense 𝐺_𝛿 set of 𝑋 for every 𝑥∈𝑋, then (𝑋,𝑓) is (ℱ₁,ℱ₂)-sensitive.

Proof. Since 𝑃_ℱ1(𝑥)⧵Asym_𝜀(ℱ₂)(𝑥) is a dense 𝐺_𝛿 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 that 𝑘ℱ is a proper family countably generated by 𝒜, where 𝒜 is a countable set. Then 𝑃_ℱ(𝑥)=⋂_{𝐹∈𝑘ℱ}𝑃_𝐹(𝑥)=⋂_𝐹∈𝒜𝑃_𝐹(𝑥). By Lemmas 4.1 and 4.2, 𝑃_ℱ(𝑥) is a dense 𝐺_𝛿 set of 𝑋. Choose 𝜀>0 such that 𝑋×𝑋⧵𝑉_𝜀 is a nonempty open subset of 𝑋×𝑋. By Lemma 4.3, ℱ(𝑋×𝑋⧵𝑉_𝜀)=⋂_{𝐹∈𝑘ℱ}⋃_𝑛∈𝐹(𝑓×f)^−𝑛(𝑋×𝑋⧵𝑉_𝜀) is a dense 𝐺_𝛿 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=1𝑓^−𝑖(𝑝) is dense subset of 𝑋, then 𝑃_ℱ is a dense 𝐺_𝛿 set of 𝑋×𝑋.

Proof. As ⋃^∞_𝑖=1𝑓^−𝑖(𝑝) is dense subset of 𝑋, it is easy to check that so is 𝑘ℬ({𝑝}_𝜀) for any positive 𝜀. Since 𝑘ℬ(𝑉_𝜀)⊃𝑘ℬ({(𝑝,𝑝)}_𝜀)⊃𝑘ℬ({𝑝}_𝛿)×𝑘ℬ({𝑝}_𝛿) for some positive 𝛿, then 𝑘ℬ(𝑉_𝜀) is a dense set of 𝑋×𝑋. As ℱ is full then 𝑘ℬ⊂ℱ, this implies that ℱ(𝑉_𝜀) is a dense subset of 𝑋×𝑋. And since ℱ is compatible with the system (𝑋×𝑋,𝑓×𝑓), ℱ(𝑉_𝜀) is a 𝐺_𝛿 set of 𝑋×𝑋. By 𝑃_ℱ=⋂^∞_𝑘=1ℱ(𝑉_1/𝑘), then 𝑃_ℱ is a dense 𝐺_𝛿 set of 𝑋×𝑋.

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

Finally, as examples we will discuss the ℱ-sensitivity and (ℱ₁,ℱ₂)-sensitivity of symbolic dynamics.

Let 𝐸={1,2,…,𝑁}(𝑁≥2) with the discrete topology. Let 𝐸_𝑖=𝐸, for all 𝑖≥1. Let Σ_𝑁=∏^∞_𝑖=1𝐸_𝑖 with the product topology. Then Σ_𝑁 is a compact metric space. Σ_𝑁 is called the symbolic space generated by 𝐸={1,2,…,𝑁}. Let 𝜎∶Σ_𝑁→Σ_𝑁 be the shift which will be defined as 𝜎(𝑥₁𝑥₂𝑥₃⋯)=𝑥₂𝑥₃⋯ for any 𝑥₁𝑥₂𝑥₃⋯ of Σ_𝑁. Then (Σ_𝑁,𝜎) is called symbolic dynamics. Let [𝑖₁𝑖₂⋯𝑖_𝑛]={𝑥∈Σ_𝑁∶𝑥₁=𝑖₁,𝑥₂=𝑖₂,…,𝑥_𝑛=𝑖_𝑛}.

We define a metric 𝑑 which is compatible with the product topology on Σ_𝑁 as follows: for all 𝑥=𝑥₁𝑥₂…,𝑦=𝑦₁𝑦₂⋯∈Σ_𝑁,

𝑑(𝑥,𝑦)=10,𝑥=𝑦,𝑁^𝑘,𝑘=min{𝑖∶𝑥_𝑖≠𝑦_𝑖}−1. (5.1)

Theorem 5.1. Let ℱ be a full Furstenberg family. Then (Σ_𝑁,𝜎) is ℱ-sensitive.

Proof. Let 𝑝=111⋯ and 𝑞=222⋯. Then 𝑝 and 𝑞 are fixed points of 𝜎, and both ⋃^∞_𝑖=1𝜎^−𝑖(𝑝) and ⋃^∞_𝑖=1𝜎^−𝑖(𝑞) are dense subsets of Σ_𝑁. By Theorem 3.8, (Σ_𝑁,𝜎) is weakly ℱ-sensitive. Let 𝜀 be a weakly ℱ-sensitive constant. Now we show that (Σ_𝑁,𝜎) is also ℱ-sensitive. For any 𝑥=𝑥₁𝑥₂𝑥₃⋯ of Σ_𝑁 and for any open neighborhood [𝑥₁𝑥₂⋯𝑥_𝑛] of 𝑥, there exist points 𝑦=𝑦₁𝑦₂𝑦₃⋯ and 𝑧=𝑧₁𝑧₂𝑧₃⋯ of [𝑥₁𝑥₂⋯𝑥_𝑛] such that 𝑁((𝑦,𝑧),𝑋×𝑋⧵𝑉_𝜀)∈ℱ. Choose 𝑢=𝑢₁𝑢₂𝑢₃⋯ of [𝑥₁𝑥₂⋯𝑥_𝑛] such that 𝑢_𝑡=𝑥_𝑡 if 𝑦_𝑡=𝑧_𝑡 otherwise 𝑢_𝑡≠𝑥_𝑡. Then 𝑁((𝑥,𝑢),𝑋×𝑋⧵𝑉_𝜀)=𝑁((𝑦,𝑧),𝑋×𝑋⧵𝑉_𝜀)∈ℱ, so (Σ_𝑁,𝜎) is ℱ-sensitive.

Lemma 5.2. Suppose that ℱ is a full Furstenberg family and is compatible with the system (Σ_𝑁×Σ_𝑁,𝜎×𝜎), then 𝑃_ℱ(𝑥) is a dense 𝐺_𝛿 set of Σ_𝑁 for every 𝑥 of Σ_𝑁.

Proof. By Lemma 4.6, 𝑃_ℱ(𝑥) is a 𝐺_𝛿 set of Σ_𝑁 for every 𝑥 of Σ_𝑁.

Now we show that 𝑃_ℱ(𝑥) is dense for every 𝑥=𝑥₁𝑥₂𝑥₃⋯ of Σ_𝑁. For any 𝑦=𝑦₁𝑦₂𝑦₃⋯ of Σ_𝑁 and for any open neighborhood [𝑦₁𝑦₂⋯𝑦_𝑛] of 𝑦. Choose 𝑧=𝑧₁𝑧₂⋯ of [𝑦₁𝑦₂⋯𝑦_𝑛] such that 𝜎^𝑛(𝑥)=𝜎^𝑛(𝑧), then 𝑧∈𝑘ℬ(𝑉_𝜀)(𝑥) for any positive 𝜀, this implies that 𝑘ℬ(𝑉_𝜀)(𝑥) is dense. Since ℱ is a full then 𝑘ℬ⊂ℱ, so ℱ(𝑉_𝜀)(𝑥) is dense 𝐺_𝛿 set of Σ_𝑁. By 𝑃_ℱ(𝑥)=⋂^∞_𝑘=1ℱ(𝑉_1/𝑘)(𝑥), then 𝑃_ℱ(𝑥) is a dense 𝐺_𝛿 set of Σ_𝑁.

Lemma 5.3. Suppose that ℱ is a full Furstenberg family and is compatible with the system (Σ_𝑁×Σ_𝑁,𝜎×𝜎), then there exists a positive 𝜀 such that for every 𝑥∈Σ_𝑁, Σ_𝑁⧵Asym_𝜀(ℱ)(𝑥) is a dense 𝐺_𝛿 set of Σ_𝑁.

Proof. Let 𝑝=111⋯ and 𝑞=222⋯. Then 𝑝 and 𝑞 are fixed points of 𝜎, and both ⋃^∞_𝑖=1𝜎^−𝑖(𝑝) and ⋃^∞_𝑖=1𝜎^−𝑖(𝑞) are dense subsets of Σ_𝑁. By Theorem 3.8 there exists a positive 𝜀 such that 𝑘ℬ(Σ_𝑁×Σ_𝑁⧵𝑉_𝜀) is a dense set of Σ_𝑁×Σ_𝑁. Now we show for every 𝑥∈𝑋, 𝑋⧵Asym_𝜀(ℱ)(𝑥) is a dense 𝐺_𝛿 set of Σ_𝑁. For any 𝑦=𝑦₁𝑦₂𝑦₃⋯ of Σ_𝑁, and for any open neighborhood [𝑦₁𝑦₂⋯𝑦_𝑛]×[𝑥₁𝑥₂⋯𝑥_𝑛] of (𝑦,𝑥) of Σ_𝑁×Σ_𝑁, there exists (𝑢,𝑣) of [𝑦₁𝑦₂⋯𝑦_𝑛]×[𝑥₁𝑥₂⋯𝑥_𝑛] such that 𝑁((𝑢,𝑣),Σ_𝑁×Σ_𝑁⧵𝑉_𝜀)∈𝑘ℬ. Choose 𝑧=𝑧₁𝑧₂𝑧₃⋯ of [𝑦₁𝑦₂⋯𝑦_𝑛] such that, 𝑧_𝑡=𝑥_𝑡 if 𝑢_𝑡=𝑣_𝑡 otherwise 𝑧_𝑡≠𝑥_𝑡, when 𝑡>𝑛. Then 𝑁((𝑢,𝑥),Σ_𝑁×Σ_𝑁⧵𝑉_𝜀)∈𝑘ℬ. Since ℱ is full, then 𝑘ℬ⊂ℱ, this implies that 𝑁((𝑢,𝑥),Σ_𝑁×Σ_𝑁⧵𝑉_𝜀)∈ℱ. Hence ℱ(Σ_𝑁×Σ_𝑁⧵𝑉_𝜀)(𝑥) is a dense set of Σ_𝑁. Since ℱ is compatible with the system (Σ_𝑁×Σ_𝑁,𝜎×𝜎), then ℱ(Σ_𝑁×Σ_𝑁⧵𝑉_𝜀) is a 𝐺_𝛿 set of Σ_𝑁×Σ_𝑁. Hence ℱ(Σ_𝑁×Σ_𝑁⧵𝑉_𝜀)(𝑥) is a dense 𝐺_𝛿 set of Σ_𝑁.

By Lemmas 5.2 and 5.3, the following theorem holds.

Theorem 5.4. Suppose that ℱ₁ and ℱ₂ are full, and are compatible with (Σ_𝑁×Σ_𝑁,𝜎×𝜎), then (Σ_𝑁,𝜎) is (ℱ₁,ℱ₂)-sensitive. In particular, (Σ_𝑁,𝜎) is (𝑀(1),𝑀(1))-sensitive.

Acknowledgments

The authors greatly thank the referees for the careful reading and many helpful remarks. This work was supported by the National Nature Science Funds of China (10771079, 10471049), and Guangzhou Education Bureau (08C016).