Furstenberg Families and Sensitivity

Volume 2010, Article ID 649348,12pages doi:10.1155/2010/649348

Research Article

Furstenberg Families and Sensitivity

Huoyun Wang,

Jincheng Xiong,

and Feng Tan

1Department of Mathematics, Guangzhou University, Guangzhou 510006, China

2Department of Mathematics, South China Normal University, Guangzhou 526061, China

Correspondence should be addressed to Huoyun Wang,[email protected] Received 31 August 2009; Revised 17 November 2009; Accepted 22 January 2010 Academic Editor: Yong Zhou

Copyrightq2010 Huoyun Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We introduce and study some concepts of sensitivity via Furstenberg families. A dynamical systemX, fisF-sensitive if there exists a positiveεsuch that for everyx∈Xand every open neighborhoodUofxthere existsy∈Usuch that the pairx, yis notF-ε-asymptotic; that is, the time set{n : dfⁿx, fⁿy > ε}belongs toF, whereFis a Furstenberg family. A dynamical systemX, f is F1, F2-sensitive if there is a positiveεsuch that every x ∈ X is a limit of pointsy∈Xsuch that the pairx, yisF1-proximal but notF2-ε-asymptotic; that is, the time set {n:dfⁿx, fⁿy< δ}belongs toF1for any positiveδbut the time set{n:dfⁿx, fⁿy> ε}

belongs toF2, whereF1andF2are Furstenberg families.

1. Introduction

Throughout this paper a topological dynamical systemTDSis a pairX, f, whereX is a compact metric space with a metricdandf:X → Xis a continuous surjective map. LetZ

be the set of nonnegative integers.

The phrase—sensitive dependence on initial condition—was first used by Ruelle1, to indicate some exponential rate of divergence of orbits of nearby points. Following the work by Guckenheimer2, Auslander and Yorke3, Devaney4, a TDSX, fis called sensitive if there exists a positiveεsuch that for everyx ∈ X and every open neighborhoodUofx, there existy∈Uandn∈Zwithdfⁿx, fⁿy> ε; that is, there exists a positiveεsuch that in any openeopen and nonemptyset there are two distinct points whose trajectories are apart fromεat least one moment.

Recently, several authors studied the sensitive propertycf. Abraham et al.5, Akin and Kolyada6. The following proposition holds according to6.

Proposition 1.1. LetX, fbe a TDS. The following conditions are equivalent.

1 X, fis sensitive.

2There exists a positiveεsuch that for everyx ∈ X and every open neighborhoodUofx there existsy∈Uwith lim sup_{n→ ∞}dfⁿx, fⁿy> ε.

3There exists a positiveεsuch that in any opene setU⊂Xthere existx, y∈Uandn∈Z withdfⁿx, fⁿy> ε.

4There exists a positive εsuch that in any opene set U ⊂ X there exist x, y ∈ Uwith lim sup_{n→ ∞}dfⁿx, fⁿy> ε.

FromProposition 1.1, we know that a TDSX, fis sensitive if and only if there exists a positive εsuch that in any opene set there are two distinct points whose trajectories are infinitely many times apart at least ofε.

Some authors introduced concepts which link the Li-Yorke versions of chaos with the sensitivity in the recent years. Blanchard et al.7introduced the concept of spatiotemporal chaos. A TDSX, fis called spatiotemporally chaotic if everyx∈Xis a limit of pointsy∈X such that the pairx, yis proximal but not asymptotic; that is, the pairx, yis a Li-Yorke scrambled pair8. That is

lim inf

n→ ∞ d

fⁿx, fⁿ y

0, lim sup

n→ ∞ d

fⁿx, fⁿ y

>0. 1.1

Akin and Kolyada6introduced the concept of Li-Yorke sensitivity. A TDSX, fis called Li-Yorke sensitive if there is a positiveεsuch that everyx∈Xis a limit of pointsy∈X such that the pairx, yis proximal but notε-asymptotic. That is,

lim inf

n→ ∞ d

fⁿx, fⁿ y

0, lim sup

n→ ∞ d

fⁿx, fⁿ y

> ε. 1.2

We see that Li-Yorke sensitivity clearly implies spatiotemporal chaos, but the latter property is strictly weakersee6.

LetJ⊂Z. The upper density ofJis μJ lim sup

n→ ∞

J∩ {0,1, . . . , n−1}

n , 1.3

wheredenotes the cardinality of the set. The lower density ofJis μJ lim inf

n→ ∞

J∩ {0,1, . . . , n−1}

n . 1.4

A pairx, y∈ X×X is distributively scrambled pair9if there is positiveεsuch that μ{n ∈ Z : dfⁿx, fⁿy < ε} 0, that is, μ{n ∈ Z : dfⁿx, fⁿy ≥ ε} 1, and μ{n∈Z:dfⁿx, fⁿy< δ} 1 for any positiveδ.

LetZ be the set of nonnegative integers, and letPbe the collection of all subsets of Z. A subsetFofP is called a Furstenberg family10if it is hereditary upwards; that is, F₁⊂F₂andF₁∈ FimplyF₂∈ F.

In the past few years, some authors11–14investigated proximity, mixing, and chaos via Furstenberg family. In13,F-scrambled pair was defined via a Furstenberg familyF. A pairx, yis calledF-scrambled pair if there is positiveεsuch that{n∈Z:dfⁿx, fⁿy≥ ε} ∈ F, and{n∈Z :dfⁿx, fⁿy< δ} ∈ Ffor any positiveδ. In14,F1,F2-scrambled pair was defined via Furstenberg familiesF1andF2. A pairx, yis calledF1,F2-scrambled pair if there is positiveεsuch that{n∈Z:dfⁿx, fⁿy< δ} ∈ F1for any positiveδ, and {n∈Z:dfⁿx, fⁿy≥ε} ∈ F2.

In this paper we investigate the sensitivity from the viewpoint of Furstenberg families.

A dynamical systemX, fisF-sensitive if there exists a positiveεsuch that for every x ∈ X and every open neighborhood U of x there exists y ∈ U such that {n ∈ Z : dfⁿx, fⁿy> ε}belongs toF, whereFis a Furstenberg family.

A dynamical systemX, fisF1,F2-sensitive if there is a positiveεsuch that every x∈Xis a limit of pointsy∈Xsuch that{n∈Z:dfⁿx, fⁿy< δ}belongs toF1for any positiveδbut{n∈Z:dfⁿx, fⁿy> ε}belongs toF2, whereF1andF2are 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 familiesfor details see 10. For a Furstenberg familyF, the dual family is

kF

F∈ P:F∩F/∅, ∀F∈ F

{F∈ P:Z\F /∈ F}. 2.1

Clearly, ifFis a Furstenberg family then so iskF. LetPbe the collection of all subsets ofZ. It is easy to see thatkP ∅, k∅P. Clearly,kkF FandF1 ⊂ F2 implieskF2 ⊂kF1. Let Bbe the family of all infinite subsets ofZ. It is easy to see thatBis a Furstenberg family and kBis 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 Fis proper if and only if Z ∈ Fand ∅/∈ F. Any subset A ofP can generate a Furstenberg familyA {F∈ P:F⊃Afor someA∈ A}.

A Furstenberg family F is countably generated10, 13 if there exists a countable subsetAofPsuch thatA F. Clearly,kBis a countably generated proper family.

For Furstenberg familiesF1 and F2, let F1 · F2 {F1 ∩F₂ : F₁ ∈ F, F2 ∈ F2}. A Furstenberg familyFis full if it is proper andF ·kF ⊂ B. It is easy to see that a Furstenberg familyFis full if and only ifkB · F ⊂ F. Clearly,kBandBare full. Clearly, ifFis full then kB ⊂ F. A Furstenberg familyFis a filterdual ifFis proper and kF ⊃kF ·kF.

For everys∈0,1, let

Ms

F ∈ B:μF≥s

. 2.2

Clearly,M0 Band everyMsis a full Furstenberg familysee13.

LetX, fbe a TDS andU, V ⊂X. We define the meeting time set

NU, V

n∈Z:fⁿU∩V /∅

. 2.3

In particular we haveNx, V {n∈Z:fⁿx∈V}forx∈X.

LetA⊂Xandx∈X. IfNx, A∈ F,xis called anF-attaching point ofA. The set of allF-attaching points ofAis called the set ofF-attaching ofA, denoted byFA. Clearly,

FA

F∈F

n∈F

f⁻ⁿA

F∈kF

n∈F

f⁻ⁿA. 2.4

LetF ⊂ Bbe a Furstenberg family. Recall that a TDSX, fisF-transitive if for each pair of opene subsets UandV ofX,NU, V ∈ F.X, f isF-mixing if X×X, f ×fis F-transitive.

LetX, fbe a TDS. A Furstenberg familyFis compatible with the systemX, f 13 if the set ofF-attaching ofUis aGδset ofXfor each open setUofX.

3. F-Sensitivity

In this section, we introduce and study the concept ofF-sensitivity. LetX, fbe a TDS and Fa Furstenberg family. Suppose thatA⊂X. LetA_δ {x∈X :dx, A< δ}.Adenotes the closure ofA. A subsetBofXis called invariant forfiffB⊂B.

We will use the following relations onX:

Δ {x, x:x∈X}, Vε x, y

:d x, y

< ε , V_ε

x, y :d

x, y

≤ε .

3.1

For any subsetR⊂X×Xand any pointx∈X, we write

Rx

y: x, y

∈R

. 3.2

We define the sets ofF-asymptotic pairs

AsymεF x, y

:N x, y , Vε

∈kF

kF Vε

,

Asym_εFx

y: x, y

∈Asym_εF ,

AsymF

ε>0

Asym_εF,

AsymFx

ε>0

AsymεFx.

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 subsetU ofX there exist xand yof Usuch that the pairx, yis notF-ε-asymptotic. That is, {n ∈ Z : dfⁿx, fⁿy > ε} ∈ F, or Nx, y, X×X\Vε∈ F.

We say thatX, fisF-sensitive if there exists a positiveε—aF-sensitive constant—

such that for everyx∈Xand every open neighborhoodUofxthere existsy∈Usuch that the pairx, yis notF-ε-asymptotic.

Theorem 3.1. Let X, f be a TDS. LetF1 and F2 be Furstenberg families. Suppose that kF1 ⊃ kF2·kF2. IfX, fis weaklyF1-sensitive, thenX, fthat isF2-sensitive.

Proof. IfX, fis notF2-sensitive, then for each ε > 0 there exists ax ∈ X and there exists an open neighborhoodU ofx such thatNx, y, X×X \Vε/∈ F2 for eachy ∈ U. Thus Z\Nx, y, Vε/∈ F2, this impliesNx, y, Vε∈kF2. SincekF1⊃kF2·kF2, by the triangle inequality we haveNa, b, V2ε∈kF1for anyaandbofU. ThenZ\Nx, y, X×X\V2ε∈ kF1. SoNx, y, X×X\V_2ε/∈ F1, this contradicts theX, fis weaklyF1-sensitive.

Corollary 3.2. LetX, fbe a TDS andFa filderdual. The systemX, fis weaklyF-sensitive if and only if it isF-sensitive.

Lemma 3.3. LetX, fbe a TDS. A Furstenberg familyFis compatible with the systemX, fif and only if the set ofkF-attaching ofV is anFσ set ofXfor each closed subsetV ofX.

Proof. Suppose thatVis a closed subset ofX, then

x∈kFV⇐⇒Nx, V∈kF

⇐⇒Z\Nx, X\V∈kF

⇐⇒Nx, X\V/∈ F

⇐⇒x /∈ FX\V.

3.4

Hence,kFV X\ FX\V. ThusFX\Vis aG_δset ofX, if and only ifkFVis anF_σset ofX.

Lemma 3.4. LetX, fbe a TDS andF1andF2Furstenberg families. Suppose thatF2is compatible with the systemX×X, f ×f, andkF1 ⊃ kF2·kF2. IfX, fis weaklyF1-sensitive, then there exists a positiveεsuch that for everyx∈X,X\Asym_εF2xis a denseG_δset.

Proof. SinceF2is compatible with the systemX×X, f ×f, then AsymεF2is anFσ set of X×XbyLemma 3.3. Suppose that Asym_εF2

_∞

n1C_n, where everyC_nis a closed subset ofX×X, then Asym_εF2x _∞

n1C_nx. Suppose that for eachε > 0 there existsx ∈ X such that AsymεF2xis not first category. By Baire theorem there exists an opene subset UofX for somensuch thatU ⊂ C_nx. Hence for eachy ∈ U,Nx, y, Vε ∈ kF2. Since kF1 ⊃kF2·kF2, by the triangle inequality we haveNa, b, V2ε∈kF1for anyaandbof U. ThenZ\Na, b, X×X\V2ε∈kF1. SoNa, b, X×X\V2ε/∈ F1, this contradicts the X, fthat is weaklyF1-sensitive.

The following lemma is proved in13. We give another proof here for completeness.

Lemma 3.5. LetX, fbe a TDS andFa Furstenberg family. IfkFis a countably generated proper family, orFMt, t∈0,1, thenFis compatible with the systemX, f.

Proof. 1 Let V be a closed subset of X. Suppose that kF is a proper family countably generated byA, whereAis countable set, then

kFV

F∈kF

n∈F

f⁻ⁿV

F∈A

n∈F

f⁻ⁿV. 3.5

Hence,kFVis anF_σ set.

2Suppose thatF Mt, t ∈ 0,1. If t 0, thenF B. SincekBis a countably generated proper family, the result is true by1.

Suppose thatt∈0,1. It is easy to see thatkF{F∈ B:μF>1−t}

kFV

x: lim inf

m→ ∞

j∈ {1,2, . . . , m}:x∈f^−jV

m >1−t

x:∃n∈Z, ∀m > n,

j ∈ {1,2, . . . , m}:x∈f^−jV

m >1−t

x:∃n∈Z, ∀m > n,∃l∈ {1,2, . . . , m}: l

m >1−t,

∃r1, r2, . . . , rl: 1≤r1≤ · · · ≤rl≤m ∀i∈ {1,2, . . . , l}, f^rix∈V

^∞

n1

∞ mn1

l∈Θm

r1,...,rl∈Λl,m

l i1

f^−riV,

3.6

where

Θm

1≤l≤m: l

m >1−t

,

Λl,m{r1, . . . , rl: 1≤r1≤ · · · ≤rl≤m}.

3.7

HencekFVis anFσset.

ByLemma 3.3,3.5holds.

Example 3.6. LetF1{F ∈ B:μF>0.8}andF2 {F ∈ B:μF≥0.4}. IfX, fis weakly F1-sensitive, then

1 X, fisF2-sensitive,

2there exists a positiveεsuch that for everyx∈X,X\Asym_εF2xis a denseGδ

set.

SinceF1{F ∈ B:μF>0.8}, thenkF1 {F ∈ B:μF≥0.2}. SinceF2 {F ∈ B: μF≥0.4}, thenkF2{F∈ B:μF>0.6}. For anyF1, F2 ∈kF2, then

μF1∩F2 1−μZ\F1∩F2 1−μZ\F₁∪Z\F₂

≥1−0.4−0.4 0.2.

3.8

HencekF1 ⊃ kF2 ·kF2. IfX, f is weakly F1-sensitive, then X, f isF2-sensitive byTheorem 3.1. By Lemmas 3.5and3.4ifX, fis weaklyF1-sensitive, then there exists a positiveεsuch that for everyx∈X,X\Asym_εF2xis a denseG_δset.

The following theorem is based on arguments in Huang and Ye15. It is called Huang- Ye equivalences in6. We state it here via Furstenberg families.

Theorem 3.7. LetX, fbe a TDS. IfFis a filterdual and is compatible withX×X, f×f, then the following statements are equivalent.

1 X, fis weaklyF-sensitive.

2There exists a positiveεsuch that Asym_εFis a first category subset ofX×X.

3There exists a positiveεsuch that for everyx∈X, Asym_εFxis a first category subset ofX.

4There exists a positiveεsuch that for everyx∈X,x∈X\Asym_εFx.

5There exists a positiveεsuch that

F X×X\Vε

3.9

is dense inX×X.

6 X, fisF-sensitive.

Proof. 1⇔6. ByCorollary 3.2, it holds.

2 ⇒1. If the system is not weakly F-sensitive then for everyε > 0, there exists an opene subsetUof X such that Nx, y, X×X \V_ε/∈ F for each x, y ∈ U×U, that is,Z\Nx, y, Vε/∈ F. ThenNx, y, Vε ∈ kF, this impliesx, y ∈ AsymεF. Hence, U×U⊂Asym_εF. So AsymεFis not of first category.

3⇒2. ByLemma 3.3, we know that AsymεF kFVεis anFσset. Suppose that AsymεF _∞

i1Ci, whereCiis a closed subset ofX×X. Then AsymεFx _∞

i1Cix.

If Asym_εFis not first category then by the Baire category theorem someC_ihas nonempty interior. IfU×V ⊂Ciandx∈U, thenV ⊂Cix. So AsymεFxis not first category.

1⇒3. ByLemma 3.4, it holds.

Thus, we have proved that1–3are equivalent.

4⇒1. IfX, fis not weaklyF-sensitive, then for anyε > 0 there exists an opene subsetU ⊂ X such that U×U ⊂ AsymεF. Letx ∈ U. Thenx ∈ U ⊂ AsymεFx,this impliesx /∈X\Asym_εFx.

3⇒4. If there exists a positiveεsuch that for everyx∈X,AsymεFxis a first category subset ofX, thenX\Asym_εFxis a denseG_δsubset ofX. Thus4is true.

2⇔5. At first, we note that

F X×X\V_ε

X×X\kF V_ε

X×X\Asym_εF. 3.10

By Baire theorem,2⇔5.

Theorem 3.8. LetX, fbe a TDS. Suppose thatX, fhave two nonempty invariant subsetsAand BofX withdA, B inf{da, b:a∈A, b∈B} >0 such that_∞

i1f⁻ⁱAand_∞

i1f⁻ⁱBare dense subsets ofX, then there exists a positiveεsuch thatkBX×X\V_εis a dense subset ofX×X, and ifFis a full Furstenberg family thenX, fis weaklyF-sensitive.

Proof. SincedA, B > 0, there exist positive numbersδ andεsuch that A_δ×B_δ ⊂ X × X\Vε. Since_∞

i1f⁻ⁱAand _∞

i1f⁻ⁱBare dense subsets ofX, it is easy to check that so arekBA_δandkBB_δ. SincekBX×X\V_ε⊃kBA_δ×B_δ⊃kBA_δ×kBB_δ, thenkBX×X\Vεis a dense subset ofX×X. SinceFis full thenkB ⊂ F, this implies that FX×X\V_εis a dense subset ofX×X. Hence,X, fis weaklyF-sensitive.

A map is semiopen if the image of an opene subset contains an opene subset. A factor mapπ :X, f → Y, gbetween dynamical systems is a continuous surjective mapπ:X → Y such thatg◦π π◦f. The weaklyF-sensitivity can be lifted up by a semi-open factor map.

Theorem 3.9. LetX, fand Y, gbe TDS andπ : X → Y semi-open factor map. Let Fbe a Furstenberg family. IfY, gis weaklyF-sensitive, so isX, f.

Proof. Letεbe a weaklyF-sensitive constant forY, g. Sinceπ is continuous then there is δ >0 such that ifd2πx, πy> εthend1x, y> δ.

LetU be an opene subset ofX. As π is semi-open, πUcontains an opene subset V ofY. Since Y, gis weakly F-sensitive, then there exist y₁ and y₂ of V such that{n ∈ Z : d2gⁿy1, gⁿy2 > ε} ∈ F. Let x1, x2 ∈ Uwith πx1 y1 and πx2 y2. Then {n∈Z:d₁fⁿx1, fⁿx2> δ} ∈ F, that is,X, fis weaklyF-sensitive.

4. F

, F

-Sensitivity

In this section, we introduce and study the notion ofF1,F2-sensitivity which links chaos and sensitivity via a couple Furstenberg familiesF1,F2.

LetX, fbe a TDS andF∈ B. A pairx, y∈X×Xis calledF-proximal if

lim inf

Fn→ ∞d

fⁿx, fⁿ y

0. 4.1

We denote the set of allF-proximal pairs byP_F.

The following lemma comes from11.

Lemma 4.1. LetX, fbe a TDS andF{t1< t₂<· · · } ∈ B. Then

P_F ^∞

n1

∞ i1

f×f_−ti

V_1/n. 4.2

LetFbe a Furstenberg family. A pairx, y∈X×Xis calledF-proximal ifx, y∈ FVε for anyε >0. We denote the set of allF-proximal pairs byP_F.

Note that [12]:

P_F

ε>0

F∈F

n∈F

f×f_−n Vε

^∞

k1

F∈kF

n∈F

f×f_−n V1/k

^∞

k1

FV1/k

F∈kF

P_F.

4.3

Suppose thatF1andF2are Furstenberg families.

A TDSX, fis calledF1,F2-spatiotemporally chaotic if everyx ∈ X is a limit of points y∈Xsuch that the pairx, yisF1-proximal but notF2-asymptotic. That is, for allx∈X

x∈P_F1x\AsymF2x. 4.4

WhenF1F2M0 B, it is the usual spatiotemporal chaos.

A TDSX, fis calledF1,F2-sensitive if there is a positiveεsuch that everyx ∈X is a limit of pointsy∈Xsuch that the pairx, yisF1-proximal but notF2-ε-asymptotic.

That is, for allx∈X

x∈P_F1x\Asym_εF2x P_F1x∩ F2X×X\V_εx. 4.5

WhenF1 F2M0 B,X, fis the usual Li-Yorke sensitivity.

If the pair x, y is M1-proximal but not M1-ε-asymptotic, then x, y is the usual distributively scrambled pair.

We will use the following lemmas which comes from10,11, respectively.

Lemma 4.2. LetFbe a full Furstenberg family. IfX, fisF-mixing, thenPFxis a denseGδset ofXfor eachF ∈kFand eachx∈X.

Lemma 4.3. LetX, fbe a TDS andFa Furstenberg family.X, fisF-transitive if and only if for everyF ∈kFand every opene subsetUofX,

{f^−tU:t ∈F}is an open and dense subset ofX (see [10, Proposition 4.1]).

Theorem 4.4. LetX, fbe a TDS. LetF1andF2be Furstenberg families. If there exists a positiveε such thatX\AsymεF2xis a denseGδset for everyx∈X, andP_F1xis a denseGδset ofXfor everyx∈X, thenX, fisF1,F2-sensitive.

Proof. Since P_F1x\ Asym_εF2x is a dense G_δ subset of X. Hence, X, f is F1,F2- sensitive.

Theorem 4.5. LetX, fbe a nontrivial TDS andFa full filterdual. Suppose thatkFis countably generated. IfX, fisF-mixing, thenX, fisF,F-sensitive.

Proof. Suppose thatkFis a proper family countably generated byA, whereAis a countable set. ThenP_Fx

F∈kFP_Fx

F∈AP_Fx. By Lemmas4.1and4.2,P_Fxis a denseG_δset ofX. Chooseε >0 such thatX×X\V_εis a nonempty open subset ofX×X. ByLemma 4.3, FX×X\Vε

F∈kF

n∈Ff×f⁻ⁿX×X\Vεis a denseG_δsubset ofX×X. ByTheorem 3.7, X, fisF-sensitive. ThusX, fisF,F-sensitive byTheorem 4.4.

Lemma 4.6. LetX, fbe a TDS. Suppose thatFis a full Furstenberg family and is compatible with the systemX×X, f×f. If there is a fixed pointpoffsuch that_∞

i1f⁻ⁱpis dense subset ofX, thenP_Fis a denseG_δset ofX×X.

Proof. As_∞

i1f⁻ⁱpis dense subset ofX, it is easy to check that so iskB{p}_εfor any positive ε. SincekBVε ⊃kB{p, p}_ε ⊃kB{p}_δ×kB{p}_δfor some positiveδ, thenkBVεis a dense set ofX ×X. As Fis full then kB ⊂ F, this implies thatFVε is a dense subset of X×X. And sinceFis compatible with the systemX×X, f×f,FVεis aGδset ofX×X.

ByP_F∞

k1FV1/k, thenP_Fis a denseG_δset ofX×X.

5. F

, F

-Sensitivity of Symbolic Dynamics Σ

, σ

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

LetE {1,2, . . . , N}N ≥2with the discrete topology. LetE_i E, for all i≥ 1. Let ΣN_∞

i1E_iwith the product topology. ThenΣNis a compact metric space.ΣNis called the symbolic space generated byE {1,2, . . . , N}. Letσ :ΣN → ΣN be the shift which will be defined asσx1x₂x₃· · · x₂x₃· · · for anyx₁x₂x₃· · · ofΣN. ThenΣN, σis called symbolic dynamics. Leti1i₂· · ·i_n {x∈ΣN:x₁i₁, x₂i₂, . . . , x_ni_n}.

We define a metricdwhich is compatible with the product topology onΣNas follows:

for allxx₁x₂. . . , yy₁y₂· · · ∈ΣN,

d x, y

⎧⎨

⎩

0, xy, 1

N^k, kmin

i:x_i/y_i

−1. 5.1

Theorem 5.1. LetFbe a full Furstenberg family. ThenΣN, σisF-sensitive.

Proof. Letp111· · · andq222· · ·. Thenpandqare fixed points ofσ, and both_∞

i1σ⁻ⁱp and_∞

i1σ⁻ⁱqare dense subsets ofΣN. ByTheorem 3.8,ΣN, σis weaklyF-sensitive. Let εbe a weaklyF-sensitive constant. Now we show thatΣN, σis alsoF-sensitive. For any x x₁x₂x₃· · · of ΣN and for any open neighborhoodx1x₂· · ·x_nof x, there exist points yy₁y₂y₃· · · andzz₁z₂z₃· · · ofx1x₂· · ·x_nsuch thatNy, z, X×X\V_ε∈ F. Choose uu₁u₂u₃· · · ofx1x₂· · ·x_nsuch thatu_tx_tify_tz_totherwiseu_t/x_t. ThenNx, u, X× X\V_ε Ny, z, X×X\V_ε∈ F, soΣN, σisF-sensitive.

Lemma 5.2. Suppose thatFis a full Furstenberg family and is compatible with the systemΣN × ΣN, σ×σ, thenP_Fxis a denseG_δset ofΣNfor everyxofΣN.

Proof. ByLemma 4.6,P_Fxis aG_δset ofΣNfor everyxofΣN.

Now we show that P_Fx is dense for every x x1x2x3· · · of ΣN. For any y y₁y₂y₃· · · ofΣN and for any open neighborhood y1y₂· · ·y_nof y. Choosez z₁z₂· · · of y1y₂· · ·y_nsuch thatσⁿx σⁿz, thenz∈kBVεxfor any positiveε, this implies that kBVεxis dense. Since F is a full thenkB ⊂ F, soFVεx is denseGδ set of ΣN. By P_Fx _∞

k1FV1/kx, thenP_Fxis a denseG_δset ofΣN.

Lemma 5.3. Suppose thatFis a full Furstenberg family and is compatible with the systemΣN × ΣN, σ×σ, then there exists a positiveεsuch that for everyx∈ΣN,ΣN\Asym_εFxis a dense Gδset ofΣN.

Proof. Letp111· · · andq222· · ·. Thenpandqare fixed points ofσ, and both_∞

i1σ⁻ⁱp and_∞

i1σ⁻ⁱqare dense subsets ofΣN. ByTheorem 3.8there exists a positiveεsuch that kBΣN×ΣN\Vεis a dense set ofΣN×ΣN. Now we show for everyx∈X,X\AsymεFx is a dense Gδ set of ΣN. For any y y1y2y3· · · of ΣN, and for any open neighborhood y1y₂· · ·y_n×x1x₂· · ·x_nofy, xofΣN×ΣN, there existsu, vofy1y₂· · ·y_n×x1x₂· · ·x_n such that Nu, v,ΣN ×ΣN \Vε ∈ kB. Choose z z1z2z3· · · of y1y2· · ·yn such that, z_t x_t if u_t v_t otherwise z_t/x_t, when t > n. Then Nu, x,ΣN × ΣN \ V_ε ∈ kB.

Since F is full, then kB ⊂ F, this implies that Nu, x,ΣN × ΣN \ Vε ∈ F. Hence FΣN×ΣN\V_εxis a dense set ofΣN. SinceFis compatible with the systemΣN×ΣN, σ×σ, thenFΣN×ΣN\V_εis aG_δset ofΣN×ΣN. HenceFΣN×ΣN\V_εxis a denseG_δset of ΣN.

By Lemmas5.2and5.3, the following theorem holds.

Theorem 5.4. Suppose thatF1 and F2 are full, and are compatible with ΣN×ΣN, σ×σ, then ΣN, σisF1,F2-sensitive. In particular,ΣN, σisM1, M1-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 China10771079, 10471049, and Guangzhou Education Bureau08C016.

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