Furstenberg Families and Sensitivity

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


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 .
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 lim sup 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 lim inf 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, 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 where denotes the cardinality of the set.The lower density of J is 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, 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 , F 1 , F 2 -scrambled pair was defined via Furstenberg families F 1 and F 2 .A pair x, y is called F 1 , F 2 -scrambled pair if there is positive ε such that {n ∈ Z : d f n x , f n y < δ} ∈ F 1 for any positive δ, and {n ∈ Z : 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 F 1 , F 2 -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 F 1 for any positive δ but {n ∈ Z : d f n x , f n y > ε} belongs to F 2 , where F 1 and F 2 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 F 1 , F 2 -sensitivity is introduced and investigated, and the sensitivity of symbolic dynamics in the sense Furstenberg families is discussed finally.

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 F 1 ⊂ F 2 implies kF 2 ⊂ kF 1 .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 F 1 and F 2 , let For every s ∈ 0, 1 , let 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 In particular we have N x, V {n ∈ Z : 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.

F-Sensitivity
In this section, we introduce and study the concept of F-sensitivity.Let X, f be a TDS and We will use the following relations on X:

3.1
For any subset R ⊂ X × X and any point x ∈ X, we write We define the sets of F-asymptotic pairs Asym ε F x .

3.3
We say that X, f is weakly F-sensitive 10 if there is a positive ε-a weakly Fsensitive 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 We say that X, f is F-sensitive if there exists a positive ε-a F-sensitive constantsuch 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 F 1 and F 2 be Furstenberg families.Suppose that Proof.If X, f is not F 2 -sensitive, then for each ε > 0 there exists a x ∈ X and there exists an open neighborhood 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 3.4.Let X, f be a TDS and F 1 and F 2 Furstenberg families.Suppose that F 2 is compatible with the system X × X, f × f , and Suppose that for each ε > 0 there exists x ∈ X such that Asym ε F 2 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 ∈ kF 2 .Since kF 1 ⊃ kF 2 • kF 2 , by the triangle inequality we have N a, b , V 2ε ∈ kF 1 for any a and b of 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 Hence, kF V is an 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 : where

3.7
Hence kF V is an F σ set.
Example 3.6.Let F 1 {F ∈ B : μ F > 0.8} and 2 there exists a positive ε such that for every

3.8
Hence If X, f is weakly F 1 -sensitive, then X, f is F 2 -sensitive by Theorem 3.1.By Lemmas 3.5 and 3.4 if X, f is weakly F 1 -sensitive, then there exists a positive ε such that for every x ∈ X, X \ Asym ε F 2 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.

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 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 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 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.
Theorem 3.8.Let X, f be a TDS.Suppose that X, f have two nonempty invariant subsets A and 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 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 d 2 π x , π y > ε then d 1 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 y 1 and y 2 of V such that {n ∈ Z :

F 1 , F 2 -Sensitivity
In this section, we introduce and study the notion of F 1 , F 2 -sensitivity which links chaos and sensitivity via a couple Furstenberg families Let X, f be a TDS and 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 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 .

4.3
Suppose that F 1 and F 2 are Furstenberg families.A TDS X, f is called F 1 , F 2 -spatiotemporally chaotic if every x ∈ X is a limit of points y ∈ X such that the pair x, y is When F 1 F 2 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 F 1 and F 2 be Furstenberg families.If there exists a positive ε such that X \ Asym ε F 2 x is a dense G δ set for every x ∈ X, and 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∈A P F x .By Lemmas 4.1 and 4.2, 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 Proof.As ∞ i 1 f −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 × Finally, as examples we will discuss the F-sensitivity and F 1 , F 2 -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 1 E 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 x n i n }.We define a metric d which is compatible with the product topology on Σ N as follows: for all x x 1 x 2 . . ., y y 1 y 2

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 1 σ −i p and ∞ i 1 σ −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 Lemma 5.2.Suppose that F is a full Furstenberg family and is compatible with the system 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 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 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 1 σ −i p and ∞ i 1 σ −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 y 1 y 2 y 3 • • • of Σ N , and for any open neighborhood 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 By Lemmas 5.2 and 5.3, the following theorem holds.Theorem 5.4.Suppose that F 1 and F 2 are full, and are compatible with Σ N × Σ N , σ × σ , then Σ N , σ is F 1 , F 2 -sensitive.In particular, Σ N , σ is M 1 , M 1 -sensitive.
1 x 2 x 3 • • • of Σ N .For any y y 1 y 2 y 3 • • • of Σ N and for any open neighborhood y 1 y 2 • • • y n of y.Choose z z 1 z 2 • • • of y 1 y 2 • • • y n such that σ n x