F-equicontinuity and an Analogue of Auslander-Yorke Dichotomy Theorem
aa r X i v : . [ m a t h . D S ] O c t F -equicontinuity and an Analogue ofAuslander-Yorke Dichotomy Theorem Hyonhui Ju † , Jinhyon Kim † , Songhun Ri † , Peter Raith ‡ † Faculty of Mathematics,
Kim Il Sung
University, Pyongyang, DPR Korea ‡ Faculty of Mathematics, University of Vienna, Austria
Abstract
In this paper, we introduce an F -equicontinuity and show an analogue of Auslander-Yorke dichotomy theorem for F -sensitivity. Precisely, under the condition that k F is translation invariant, we prove that a transitive system is either F -sensitive oralmost k F -equicontinuous , and so generalize the result of previous work. Also weshow that F -equicontinuity is preserved by an open factor map and consider theimplication between F -equicontinuity and mean equicontinuity. Keywords : equicontinuity, sensitivity, mean equicontinuity, F -sensitivity, mean sen-sitivity, Furstenberg family. A topological dynamical system (
X, T ) means a compact metric space (
X, d ) with acontinuous self-surjection T defined on it. Throughout this paper we are only interestedin a nontrivial topological dynamical system, where the state space is a compact metricspace without isolated points. Here a trivial dynamical system means that the state spaceis a singleton.A dynamical system ( X, T ) is deterministic in the sense that the evolution of the sys-tem is described by a map T , so that the present(the initial state) completely determinesthe future(the forward orbit of the state). Li and Yorke introduced the term “chaos” intomathematics in 1975([9]) and showed that a deterministic system has an unpredictableand complex behavior. And later many definitions of chaos have been introduced intomathematics by several scholars, and although there is no universal mathematical def-inition of chaos, it is generally agreed that a chaotic dynamical system should exhibitsensitive dependence on initial conditions, i.e., minor changes in the initial state lead The first author was supported by the Ernst Mach Follow-Up Grant (EZA) of OeAD for post-docsin Austria.*Corresponding author E-mail address: [email protected]@ryongnamsan.edu.kp
1o completely different long-term behavior. A dynamical system (
X, T ) is called sensi-tive ([3]) if there exists ε > x ∈ X and every neighborhood U x of x ,there exist y ∈ U x and n ∈ N with d ( T n x, T n y ) > ε .The equicontinuity is opposite to the notion of sensitivity. A dynamical system ( X, T )is called equicontinuous if for every ε > δ > d ( x, y ) < δ implies d ( T n x, T n y ) < ε for n = 0 , , , · · · .Recall that a point x ∈ X is equicontinous if for every ε > δ > y ∈ X with d ( x, y ) < δ , d ( T n x, T n y ) < ε for n = 0 , , , · · · . Andrecall that a dynamical system ( X, T ) is called almost equicontinuous if there exist someequicontinuous points.The well-known Auslander-Yorke dichotomy theorem([4]) states that a minimal dy-namical system is either sensitive or equicontinuous, which was supplemented in [2]: atransitive system is either sensitive or almost equicontinuous.In [8], the authors introduced the notions of mean equicontinuity and mean sensitivity.A dynamical system (
X, T ) is called mean equicontinuous if for every ε >
0, there is a δ > d ( x, y ) < δ implies lim sup n →∞ n P n − i =0 d ( T i x, T i y ) < ε and a point x ∈ X iscalled mean equicontinuous if for every ε > δ > y ∈ X with d ( x, y ) < δ , lim sup n →∞ n P n − i =0 d ( T i x, T i y ) < ε . A transitive system ( X, T ) is called almost mean equicontinuous if there is at least one mean equicontinuous point.A dynamical system (
X, T ) is called mean sensitive if there exists ε > x ∈ X and every neighborhood U x of x , there is y ∈ U x withlim sup n →∞ n P n − i =0 d ( T i x, T i y ) > ε .And they showed an analogue of Auslander-Yorke theorem, which states that a tran-sitive dynamical system is either almost mean equicontinuous(in the sense of containingsome mean equicontinous points) or mean sensitive and that a minimal system is eithermean equicontinuous or mean sensitive.Recently, some scholars considered various forms of sensitivity via Furstenberg familysuch as syndetic sensitivity, cofinite sensitivity and thickly sensitivity and so on.([6], [10].[11], [13-15])In [7] the authors considered equcontinuity via a syndetic Furstenberg family and intro-duced a notion of syndetically equicontinuity and showed that an analogue of Auslander-Yorke dichotomy theorem could also be found for some stronger forms of sensitivity.Precisely, they proved that a minimal system is either thickly sensitive or syndeticallyequicontinuous.1. Concerning the study on analogue of Auslander-Yorke dichotomy the-orem, recently we can find more result in [12] where is obtained an AuslanderYorkes typedichotomy theorem for r-sensitivity being stronger version of sensitivity.Through the notion of syndetically equicontinuity in [7], we know that it can begeneralized to F -equicontinuity, where F is a Furstenberg family. Also we know thatthe notion of mean equicontinuity in [8] is related to F -equicontinuity.In this paper we introduce an F -equicontinuity and show an analogue of Auslander-Yorke dichotomy theorem for transitive system(Section 3) which generalize the result ofTheorem 3.4 in [7].We also show that the notion of mean equicontinuity introduced in [8] could be consid-2red as an F -equicontinuity(Section 4) and that F -equicontinuity is preserved by openfactor map.This paper is organized as follows. In section 2, we provide some basic concepts anddefinitions in topological dynamical system. And we introduce the notion of F -equicon-tinuity. In section 3, F -equicontinuity and an analogue of Auslander-Yorke dichotomytheorem are discussed. In section 4, we show that F -equicontinuity is preserved by openfactor maps and discuss the implication between mean equicontinuity and F -equiconti-nuity. In this section we recall some basic concepts related to Furstenberg family(more detailin [1]).Denote by Z + the set of all non-negative integers.Let P be the collection of all subsets of Z + . A collection F ⊂ P is Furstenbergfamily if F ⊂ F and F ∈ F imply F ∈ F .Given a Furstenberg family F , define its dual family k F as follows: k F = { F ∈ P : Z + \ F / ∈ F } = { F ∈ P : for any F ′ ∈ F , F ∩ F ′ = ∅} .Then it is easy to check that F is a Furstenberg family if and only if k F is so, andthat k ( k F ) = F .For i ∈ Z + and F ∈ P , let F + i = { j + i : j ∈ F } and F − i = { j − i : j ∈ F } ∩ Z + .A Furstenberg family F is called translation invariant if for any F ∈ F and any i ∈ Z + , F + i ∈ F and F − i ∈ F .Given two Furstenberg families F and F , define F · F = { F ∩ F : F ∈ F , F ∈ F } . A Furstenberg family F is said to be a filter if it satisfies F · F ⊂ F and it has the Ramsey property if F ∪ F ∈ F implies F ∈ F or F ∈ F .It can be easily checked that Furstenberg family F has the Ramsey property if andonly if k F is a filter.Let B be the collection of all infinite subsets of Z + and F cf be the family of cofinitesubsets, that is, the collection of subsets of Z + with finite complements. It is easy to seethat F cf = k B .Let F t be the collection of the subsets of Z + which contain arbitrary long runs ofpositive integers and denote its dual family by F s . The element of F s is called syndetic set. And then the set F ∈ P is a syndetic set if and only if there is an N ∈ N such that { i, i + 1 , · · · , i + N } ∩ F = ∅ for every i ∈ Z + . Also the element of F t is called thick set.The set F ∈ P is thickly syndetic set if for every N ∈ N the positions where length N runs begin form a syndetic set. 3e recall the upper density of a set F ⊂ Z + by D ( F ) = lim sup n →∞ F ∩ [0 , n − n , where · ) means the cardinality of a set([8]). For every a ∈ [0 , D ( a +) = { F ∈ B : D ( F ) > a } .Similarly, D ( F ), the lower density of F , is defined by D ( F ) = lim inf n →∞ F ∩ { , , · · · , n − } ) n . The upper Banach density BD ∗ ( F ) is defined by BD ∗ ( F ) = lim sup N − M →∞ F ∩ [ M, N ]) N − M + 1 . Similarly, we can define the lower Banach density BD ∗ ( F )([8]). For every a ∈ [0 , BD ∗ ( a +) = { F ∈ B : BD ∗ ( F ) > a } . F -equicontinuity Firstly we recall some concepts of topological dynamical system.A dynamical system (
X, T ) is called transitive if for any nonempty open subsets
U, V ⊂ X , N T ( U, V ) = { n ∈ Z + : U ∩ T − n V = ∅} is nonempty and a point x ∈ X is transitive if its orbit O + ( x ) = { T n ( x ) : n ∈ Z + } is dense in X . Denote by Tran( X, T )the set of all transitive points of (
X, T ). (
X, T ) is transitive if and only if Tran(
X, T )is a dense G δ subset of X .A dynamical system ( X, T ) is called minimal if Tran(
X, T ) = X .Let ( X, d ) be a compact metric space and T : X → X be a continuous map. And let F be a Furstenberg family. For given x ∈ X and a subset G ⊂ X , set N T ( x, G ) = { n ∈ Z + : T n ( x ) ∈ G } . We write as follows:∆ ε = { ( x, y ) ∈ X × X : d ( x, y ) < ε } , B ( x, δ ) = { y ∈ X : d ( x, y ) < δ } , ∆ ε = { ( x, y ) ∈ X × X : d ( x, y ) ≤ ε } . Now we introduce the notion of F -equicontinuity. Definition 2.1.
A dynamical system (
X, T ) is said to be F -equicontinuous if for every ε > δ > x, y ∈ X with d ( x, y ) < δ , N T × T (( x, y ) , ∆ ε ) ∈ F . x ∈ X is called an F -equicontinuous point (or ( X, T ) is F -equicontinuousat x ∈ X ) if for every ε > δ > y ∈ B ( x, δ ), N T × T (( x, y ) , ∆ ε ) ∈ F .A transitive system is called almost F -equicontinuous if there is at least one F -equi-continuous point. The set of all F -equicontinuous points is denoted by Eq F ( T ).Set Eq F ε := { x ∈ X | there is a δ > y, z ∈ B ( x, δ ), N T × T (( y, z ) , ∆ ε ) ∈ F } .And we need more following definitions for our study. Definition 2.2. ([6], [10-14]) A topological dynamical system (
X, T ) is said to be F − sensitive if there exists ε > F − sensitive constant ) such that for any nonempty open subset U of X S f ( U, ε ) = { n ∈ Z + : diam T n ( U ) > ε } ∈ F . In addition, if F is F s ( F cf , F t respectively), then ( X, T ) is called syndeticallysensitive ( cofinitely sensitive , thickly sensitive respectively). Definition 2.3. ([8]) A dynamical system (
X, T ) is said to be mean-L-stable if for every ε > δ > x, y ∈ X with d ( x, y ) < δ , D ( { n ∈ Z + : d ( T n x, T n y ) ≥ ε } ) < ε. Definition 2.4. ([8]) Let X and Y be topological spaces and π : X → Y be a map. Themap π is called open if the image of each nonempty open subset of X is open in Y , and semi-open if the image of each nonempty open subset of X has nonempty interior in Y .And π is said to be open at a point x ∈ X if for every neighborhood U of x , π ( U ) is aneighborhood of π ( x ). F -sensitivity In this section we study on analogues of Auslander-Yorke theorem for F -sensitivity using F -equicontinuity.We need following lemmas for it. Lemma 3.1.
Let ( X, T ) be a dynamical system and F be a translation invariant Fursten-berg family. Then Eq F ε is an open subset of X and T − (Eq F ε ) ⊂ Eq F ε . Moreover if F isa filter then Eq F ( T ) = T ε> Eq F ε .Proof. Assume that F is a translation invariant family. Fix any x ∈ Eq F ε and then thereexists a δ > z, w ∈ B ( x, δ ), N T × T (( z, w ) , ∆ ε ) ∈ F .If y ∈ B ( x, δ/
2) and z, w ∈ B ( y, δ/
2) then z, w ∈ B ( x, δ ).So B ( x, δ/ ⊂ Eq F ε , thus Eq F ε is an open set.5ext, if x ∈ T − (Eq F ε ) then T x ∈ Eq F ε and there is a δ > y ′ , y ′′ ∈ B ( T x, δ ), N T × T (( y ′ , y ′′ ) , ∆ ε ) ∈ F .Since T is continuous, there is a η > y, z ∈ B ( x, η ) implies T y, T z ∈ B ( T x, δ ). So if y, z ∈ B ( x, η ) then N T × T (( T y, T z ) , ∆ ε ) ∈ F .Since F is a translation invariant and 1 + N T × T (( T y, T z ) , ∆ ε ) ⊂ N T × T (( y, z ) , ∆ ε ), N T × T (( y, z ) , ∆ ε ) ∈ F . Therefore x ∈ Eq F ε and T − (Eq F ε ) ⊂ Eq F ε .Finally, we are going to show that if F is a filter then Eq F ( T ) = T ε> Eq F ε .If x ∈ Eq F ( T ) then for any ε > δ > y ∈ B ( x, δ ) implies N T × T (( x, y ) , ∆ ε/ ) ∈ F . So for every y, z ∈ B ( x, δ ), since N T × T (( y, z ) , ∆ ε ) ⊃ N T × T (( x, y ) , ∆ ε/ ) ∩ N T × T (( x, z ) , ∆ ε/ )and F is a filter, N T × T (( y, z ) , ∆ ε ) ∈ F . Therefore x ∈ Eq F ε and Eq F ( T ) ⊂ T ε> Eq F ε .The proof of Eq F ( T ) ⊃ T ε> Eq F ε is clear. So Eq F ( T ) = T ε> Eq F ε . Lemma 3.2.
Let ( X, T ) be a dynamical system and F be a filter. Then ( X, T ) is F -equicontinuous if and only if Eq F ( T ) = X .Proof. The proof of necessity is clear. We will prove the sufficiency.Fix any ε >
0. Since Eq F ( T ) = X , for every x ∈ X there is a δ x > y ∈ B ( x, δ x ) implies N T × T (( x, y ) , ∆ ε/ ) ∈ F .Given y, z ∈ B ( x, δ x ), we have N T × T (( y, z ) , ∆ ε ) ⊃ N T × T (( x, y ) , ∆ ε/ ) ∩ N T × T (( x, z ) , ∆ ε/ )by the triangular inequality. And since F is a filter, we have N T × T (( y, z ) , ∆ ε ) ∈ F .By the compactness of X , there are finite points x , x , · · · , x N ∈ X such that S Ni =1 B ( x i , δ x i /
2) = X . Set δ = min { δ x / , · · · , δ x N / } .Now we are going to prove the F -equicontinuity of ( X, T ).Let u, v ∈ X be two points with d ( u, v ) < δ . Then there exists an i ∈ { , , · · · N } such that u ∈ B ( x i , δ x i / ⊂ B ( x i , δ x i ). Also d ( u, v ) < δ ≤ δ x i / v ∈ B ( x i , δ x i ).So N T × T (( u, v ) , ∆ ε ) ∈ F and therefore ( X, T ) is F -equicontinuous.Next proposition shows a property of the set of F -equicontinuous points for a transi-tive dynamical system. Proposition 3.1.
Let ( X, T ) be a transitive dynamical system and F be a translationinvariant Furstenberg family. Then the set of F -equicontinuous points is either empty orresidual. If in addition ( X, T ) is almost F -equicontinuous then Tran(
X, T ) ⊂ Eq F ( T ) .Moreover if in addition F is a filter, and ( X, T ) is minimal and almost F -equicontinuousthen it is F -equicontinuous. roof. By Lemma 2.1, the set Eq F ε is open and T − (Eq F ε ) ⊂ Eq F ε . If Eq F ε is not emptythen for any nonempty open subset U of X , by the transitivity of ( X, T ), there existsan n ∈ Z + such that ∅ 6 = T − n (Eq F ε ) ∩ U ⊂ Eq F ε ∩ U . So since Eq F ε intersects with anynonempty open subset, Eq F ε is dense in X .Thus by the Baire Category theorem, Eq F ( T ) is empty or residual because Eq F ( T ) = T ε> Eq F ε .If Eq F ( T ) is residual then for any ε > F ε is open and dense. If x ∈ T ran ( X, T )then there exists n ∈ N such that T n ( x ) ∈ Eq F ε . So x ∈ T − n (Eq F ε ) ⊂ Eq F ε . Thus x ∈ T ε> Eq F ε = Eq F ( T ).If ( X, T ) is minimal then Tran(
X, T ) = X and so Eq F ( T ) = X . By Lemma 2.2( X, T ) is F -equicontinuous.Next dichotomy theorem and corollary are anologues of the Auslander-Yorke’s di-chotomy theorem for F -sensitivity. Theorem 3.1.
Let ( X, T ) be a transitive dynamical system and F be a Furstenbergfamily such that its dual family k F is translation invariant. Then ( X, T ) is either F -sensitive and Eq k F ( T ) = ∅ , or almost k F -equicontinuous and Tran(
X, T ) ⊂ Eq k F ( T ) .Proof. It suffices to show that if (
X, T ) is not F -sensitive then Tran( X, T ) ⊂ Eq k F ( T ).Assume that ( X, T ) is not F -sensitive. Then for any ε > U of X such that S T ( U, ε/ / ∈ F . So F = { n ∈ Z + : diam T n ( U ) ≤ ε/ } = Z + \ S T ( U, ε/ ∈ k F . Take any x ∈ U and then there is a δ > B ( x, δ ) ⊂ U .For any y ∈ B ( x, δ ) ⊂ U , since N T × T (( x, y ) , ∆ ε ) ⊃ N T × T (( x, y ) , ∆ ε/ ) ⊃ F , N T × T (( x, y ) , ∆ ε ) ∈ k F . Thus x ∈ Eq k F ( T ) and by Proposition 3.1, ( X, T ) is almost k F -equicontinuous and Tran( X, T ) ⊂ Eq k F ( T ). Remark 1.
Theorem 3.1 coincides with Theorem 3.4 in [7] if the family F is replacedwith thick family F t . So Theorem 3.1 is a generalization of Theorem 3.4 in [7]. Corollary 3.1.
Assume that F has Ramsey property and its dual family k F is translationinvariant. If ( X, T ) is minimal then it is either F -sensitive or k F -equicontinuous. Remark 2.
Corollary 3.1 is a generalization of Auslander-Yorke’s theorem which isobtained by replacing the family F with the family B . F -equiconti-nuity In this section we discuss some relations between mean equicontinuity and F -equiconti-nuity.In [8] is proved that mean equicontinuity is preserved by factor maps.Here we are going to show that F -equicontinuity is preserved by open factor maps.7 emma 4.1. Let ( X, T ) and ( Y, S ) be topological dynamical systems and π : X → Y bea factor map. If x ∈ X is an F -equicontinuous point of T and π is open at x ∈ X then y = π ( x ) is an F -equicontinuous point of S .Proof. Since π is continuous, for any ε > δ > d X ( x, x ′ ) < δ implies d Y ( π ( x ) , π ( x ′ )) < ε . Here d X and d Y respectively denote the metric of X and Y .And we write∆ Xδ = { ( x, x ′ ) ∈ X × X : d X ( x, x ′ ) < δ } , ∆ Yδ = { ( y, y ′ ) ∈ Y × Y : d Y ( y, y ′ ) < δ } . And since x is an F -equicontinuous point of T , for the above δ > δ > x ′ ∈ B ( x, δ ), F = N T × T (( x, x ′ ) , ∆ Xδ ) ∈ F .So if n ∈ F then d X ( T n x, T n x ′ ) < δ and this implies d Y ( π ( T n x ) , π ( T n x ′ )) = d Y ( S n ( π ( x )) , S n ( π ( x ′ ))) < ε. Thus N S × S (( y, π ( x ′ )) , ∆ Yε ) ⊃ F and this implies N S × S (( y, π ( x ′ )) , ∆ Yε ) ∈ F .Since π is open at x ∈ X , π ( B ( x, δ )) is a neighborhood of y = π ( x ) and so there isa δ > B ( y, δ ) ⊂ π ( B ( x, δ )).Therefore for every y ′ ∈ B ( y, δ ), N S × S (( y, y ′ ) , ∆ Yε ) ∈ F , that is, y is an F -equi-continuous point of S . Theorem 4.1.
Let ( X, T ) and ( Y, S ) be transitive dynamical systems and π : X → Y be a semi-open factor map. And let F be a translation invariant family. If ( X, T ) isalmost F -equicontinuous then so is ( Y, S ) .Proof. Since π is semi-open, by Lemma 2.1 in [5], the set { x ∈ X : π is open at x } isresidual in X . So we can take a transitive point x ∈ X such that π is open at x ∈ X . Since( X, T ) is almost F -equicontinuous, by Proposition 3.1 x ∈ X is an F -equicontinuouspoint of T and by Lemma 3.1 y = π ( x ) is also an F -equicontinuous point of S . Thus( Y, S ) is also almost F -equicontinuous.Let ( X, d X ) and ( Y, d Y ) be the metric spaces. The metric on the product space X × Y is defined by d (( x, y ) , ( x ′ , y ′ )) = p ( d X ( x, x ′ )) + ( d Y ( y, y ′ )) . Then the followinglemma holds. Lemma 4.2.
Let ( X, d ) be a compact metric space and U be a nonempty open subset of X × X . Let ∆ X be a diagonal of X × X , that is, ∆ X = { ( x, x ) : x ∈ X } . If ∆ X ⊂ U then there exists a δ > such that ∆ Xδ = { ( x, y ) ∈ X × X : d X ( x, y ) < δ } ⊂ U. Proof.
For every ( x, y ) ∈ X × X , clearly d (( x, y ) , ∆ X ) = d (( y, x ) , ∆ X ). Since ∆ X isclosed in X × X , there is a δ > B (∆ X , δ ) = { ( x, y ) ∈ X × X : d (( x, y ) , ∆ X ) < δ } ⊂ U. If ( x, y ) ∈ ∆ Xδ then δ > d X ( x, y ) = d (( x, y ) , ( y, y )) ≥ d (( x, y ) , ∆ X ) and thisimplies ( x, y ) ∈ B (∆ X , δ ). Therefore ∆ Xδ ⊂ B (∆ X , δ ) ⊂ U .8 heorem 4.2. Let ( X, T ) and ( Y, S ) be topological dynamical systems and π : X → Y be an open factor map. And let F be a Furstenberg family. If ( X, T ) is F -equicontinuousthen so is ( Y, S ) .Proof. Since π is continuous, for every ε > δ > d X ( x, x ′ ) < δ implies d Y ( π ( x ) , π ( x ′ )) < ε . For the above δ >
0, since (
X, T ) is F -equicontinuous,there exists a δ > d X ( x, x ′ ) < δ implies F = N T × T (( x, x ′ ) , ∆ Xδ ) ∈ F . Soif n ∈ F then d X ( T n x, T n x ′ ) < δ and this implies d Y ( π ( T n x ) , π ( T n x ′ )) = d Y ( S n ( π ( x )) , S n ( π ( x ′ ))) < ε. Thus N S × S (( π ( x ) , π ( x ′ )) , ∆ Yε ) ⊃ F .Since π is open, π × π : X × X → Y × Y is also open. So π × π (∆ Xδ ) is open in Y × Y and contains a diagonal of Y × Y , that is, π × π (∆ Xδ ) ⊃ ∆ Y . Then by Lemma 3.2 thereexists a δ > Yδ ⊂ π × π (∆ Xδ ).Thus for every ( y, y ′ ) ∈ ∆ Yδ there exists a pair ( x, x ′ ) ∈ ∆ Xδ such that y = π ( x ) , y ′ = π ( x ′ ). Since d X ( x, x ′ ) < δ , N S × S (( y, y ′ ) , ∆ Yε ) ⊃ F and this implies N S × S (( y, y ′ ) , ∆ Yε ) ∈ F . Therefore (
Y, S ) is also F -equicontinuous.Following lemma 4.3 and 4.4 show some implications between mean equicontinuityand F -equicontinuity. Lemma 4.3. If ( X, T ) is mean equicontinuous then for any a ∈ (0 , , it is kD ( a +) -equicontinuous. Also if ( X, T ) is almost mean equicontinuous, then for any a ∈ (0 , ,it is almost kD ( a +) -equicontinuous.Proof. Assume that there exists an a ∈ (0 ,
1) such that (
X, T ) is not kD ( a +)-equi-continuous. Then there is ε > δ >
0, there exist x, y ∈ X with d ( x, y ) < δ such that N T × T (( x, y ) , ∆ ε /a ) / ∈ kD ( a +). So Z + \ N T × T (( x, y ) , ∆ ε /a ) = { n ∈ Z + : d ( T n x, T n y ) ≥ ε /a } ∈ D ( a +).Set F = { n ∈ Z + : d ( T n x, T n y ) ≥ ε /a } , and thenlim sup n →∞ n P n − i =0 d ( T i x, T i y ) == lim sup n →∞ n P i ∈ [0 , n − ∩ F d ( T i x, T i y ) + P i ∈ [0 , n − ∩ F C d ( T i x, T i y ) ! ≥ lim sup n →∞ , n − ∩ F ) n · ε a ≥ ε . This contradicts to the mean equicontinuity of (
X, T ). The proof of second part issimilar to this. 9 emma 4.4. If ( X, T ) is kD (0+) -equicontinuous then it is mean equicontinuous. Alsoif ( X, T ) is almost kD (0+) -equicontinuous then it is almost mean equicontinuous.Proof. Since (
X, T ) is kD (0+)-equicontinuous, for every ε > δ > x, y ∈ X with d ( x, y ) < δ , N T × T (( x, y ) , ∆ ε ) ∈ kD (0+), that is, { n ∈ Z + : d ( T n x, T n y ) ≥ ε } = Z + \ N T × T (( x, y ) , ∆ ε ) / ∈ D (0+) . So D ( { n ∈ Z + : d ( T n x, T n y ) ≥ ε } ) = 0 < ε , that is, ( X, T ) is mean-L-stable.Thus by Lemma 3.1 in [8], (
X, T ) is mean equicontinuous. The proof of second partis similar to this.Following lemma shows an implication between mean sesitivity and F -sesitivity. Lemma 4.5. If ( X, T ) is mean sensitive with sensitive constant δ > then for any a ∈ h , δ diam( X ) (cid:17) , it is D ( a +) -sensitive.Proof. For any a ∈ h , δ diam( X ) (cid:17) , set δ ′ = δ − a · diam( X ) >
0. Now we are going to showthat (
X, T ) is D ( a +)-senstive with sensitive constant δ ′ > X and set F = S T ( U, δ ′ ). We choose x ∈ U andthen by the mean continuity of ( X, T ), there exists y ∈ U such thatlim sup n →∞ n P n − i =0 d ( T i x, T i y ) > δ .So δ < lim sup n →∞ n P n − i =0 d ( T i x, T i y )= lim sup n →∞ n P i ∈ [0 , n − ∩ F d ( T i x, T i y ) + P i ∈ [0 , n − ∩ F C d ( T i x, T i y ) ! ≤ lim sup n →∞ , n − ∩ F ) n · diam( X ) + δ ′ = D ( F ) · diam( X ) + δ ′ . Therefore D ( F ) > δ − δ ′ diam( X ) = a , that is F = S T ( U, δ ′ ) ∈ D ( a +).As a consequence of the above consideration, the following proposition holds. It isimmediately followed by Theorem 5.4 in [8]. Proposition 4.1.
Let ( X, T ) be a topological dynamical system. If ( X, T ) is transitive,then there is an a > such that ( X, T ) is either D ( a +) -sensitive or kD ( a +) -equiconti-nuous. References [1] E. Akin,
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