Topological transitivity in quasi-continuous dynamical systems
aa r X i v : . [ m a t h . GN ] J u l TOPOLOGICAL TRANSITIVITY IN QUASI-CONTINUOUSDYNAMICAL SYSTEMS † JILING CAO AND AISLING MCCLUSKEY
Abstract.
A quasi-continuous dynamical system is a pair (
X, f ) consistingof a topological space X and a mapping f : X → X such that f n is quasi-continuous for all n ∈ N , where N is the set of non-negative integers. Inthis paper, we show that under appropriate assumptions, various definitionsof the concept of topological transitivity are equivalent in a quasi-continuousdynamical system. Our main results establish the equivalence of topologicaland point transitivity in a quasi-continuous dynamical system. These extendsome classical results on continuous dynamical systems in [3], [10] and [25],and some results on quasi-continuous dynamical systems in [7] and [8]. Introduction
In the literature, two groups of different definitions of chaotic dynamical systemshave been proposed. In the first group, chaos is approached from the measure the-oretic point of view. In the second group, chaos is approached from the non-linearanalysis point of view, where a mapping f : X → X is considered chaotic in X if f has at least sensitive dependence on initial conditions in X . To this require-ment, many authors add topological transitivity, and another condition frequentlyrequired is the existence of a dense orbit, see [11], [12], [13], [20], [24] and [27]. Thelatter property has also been called point transitivity by some authors.As a motivation for the notion of topological transitivity of a system, one maythink of a real physical system, where a state is never given or measured exactly,but always up to a certain error. So, instead of points, one should study (small)open subsets of the phase space and describe how they move in that space. Intu-itively, a topologically transitive mapping f has points that eventually move underiteration from one arbitrarily small neighbourhood to any other. Consequently, thedynamical system cannot be broken down or decomposed into two subsystems (dis-joint sets with nonempty interiors) which do not interact under f , i.e., are invariantunder the mapping.In the study of dynamical systems, it is generally assumed that “topologicaltransitivity” and “point transitivity” are equivalent when X is a compact metricspace and f is continuous, e.g., Proposition 39 of [6]. However, as noted in [18] and[10], these two conditions are independent in general even in compact metric spaces Mathematics Subject Classification.
Primary 37B99; Secondary 54H20.
Keywords and phrases . Dynamical system, isolated point, orbit, point transitive, quasi-continuous,topological transitive. † The paper was partially written when the first author visited the National University of Ireland,Galway, in July 2013 when he was on sabbatical leave. He would like to acknowledge the supportby NUI Galway Millennium Fund and the hospitality of the School of Mathematics, Statistics andApplied Mathematics at NUI, Galway. with continuous mappings. Moreover, there are several different common definitionsof topological transitivity. It had been a part of the folklore of dynamical systemsthat under reasonable assumptions they are equivalent until Akin and Carlson [3]provided a complete description of the relationships among them. In [2], Akin et al.further described various strengthenings of the concept of topological transitivity.A common framework in the study of dynamical systems assumes that the phasespace is compact metric and the self-mapping is continuous. One of the difficultiesin relating the definitions of the two groups, mentioned at the beginning, derivesfrom a topological property of f . This is because non-linear definitions normallyrequire the continuity of f , but measure theoretic definitions may apply to functionswith some type of discontinuity which is not too far from continuity. Motivatedby this, Crannell and Martelli studied dynamics of quasi-continuous mappings in[8]. They showed the equivalence for quasi-continuous mappings of two non-linearanalysis definitions of chaotic dynamical systems due to Wiggins [27] and Martelli[20]. They also extended several well known results in [5] and [26] for continuousdynamical systems to quasicontinuous systems. Note that Crannell and Martelli[8] assumed the phase spaces of their systems to be compact metric.In this paper, we continue the study of dynamics of quasi-continuous systems.Our motivation is to study relationships among various definitions of topologicaltransitivity in quasi-continuous dynamical systems whose phase spaces are generaltopological spaces. The rest of this paper is organized as follows. In Section 2,we introduce notation, definitions and basic relationships among different conceptsof topological transitivity and point transitivity. In Section 3, we introduce quasi-continuous dynamical systems and study some basic properties. We also providetwo results which show how far a quasi-continuous system is from a continuousone. Section 4 is devoted to a study of equivalence among different versions oftopological transitivity, and equivalence of topological and point transitivity in aquasi-continuous dynamical system. In the last section, we discuss what happenswith these equivalences in a quasi-continuous dynamical system when the phasespace contains isolated points.2. Definitions and basic relationships
Let N denote the set of nonnegative integers and let Z denote the set of integers.By a dynamical system, we mean a pair ( X, f ), where X is a topological space(called the phase space ) and f : X → X is a mapping from X into itself (notnecessarily continuous). The dynamics of the system is given by iteration. Toavoid triviality, throughout the paper, we assume that X contains at least twopoints. A point x ∈ X “moves”, with its trajectory being the sequence x , f ( x ), f ( x ), f ( x ), . . . , where f n is the n th iteration of f . The point f n ( x ) is the positionof x after n units of time. The set of points of the trajectory of x under f is calledthe forward orbit of x , denoted by Orb f ( x ), that is, Orb f ( x ) = { f n ( x ) : n ∈ N } .The omega limit set for x under f , denoted by ωf ( x ), is given by ωf ( x ) := \ k ∈ N { f n ( x ) : n ≥ k } , and is precisely the set of all accumulation points of the sequence h f n ( x ) : n ∈ N i .A bi-infinite sequence h x k : k ∈ Z i is called an orbit sequence if f ( x k ) = x k +1 forall k ; and the set { x k : k ∈ Z } of its elements is called an orbit . In addition, we OPOLOGICAL TRANSITIVITY IN QUASI-CONTINUOUS · · · will also call a sequence h x k : k ≥ n i an orbit sequence if n ∈ N and f ( x k ) = x k +1 for all k ≥ n and f − ( x n ) = ∅ ; the set of elements of this sequence is Orb f ( x n ).For subsets A, B ⊆ X , following [3], we define the hitting time sets N ( A, B ) := { n ∈ Z : f n ( A ) ∩ B = ∅} and N + ( A, B ) := N ( A, B ) ∩ N . Note that N ( A, B ) = N + ( A, B ) ∪ N + ( B, A ) . Definition 2.1.
A dynamical system (
X, f ) is called topologically transitive (TT + )if for every pair of nonempty open sets U, V ⊆ X , the set N + ( U, V ) is nonempty.It is easy to see that (
X, f ) is topologically transitive if, and only if, for everynonempty open set U , S n ∈ N f n ( U ) is dense in X . Remark 2.2.
Definition 2.1 is the definition for topological transitivity commonlygiven in the literature e.g., [2], [10], [18] and [25]. Note that Akin and Carlson [3]define the properties TT and IN in the system (
X, f ) as follows:(IN) X is not the union of two proper, closed and +invariant sets, where a set A ⊆ X is called +invariant if f ( A ) ⊆ A .(TT) For every pair of nonempty open sets U, V ⊆ X , the set N ( U, V ) is nonempty.They labelled topological transitivity in Definition 2.1 as the property TT + . In thesame paper, they also defined the property TT ++ as follows:(TT ++ ) For every pair of nonempty open sets U, V ⊆ X , the set N + ( U, V ) is infinite.A point x ∈ X is called a transitive point when for every nonempty open V ⊆ X ,the hitting time set N + ( { x } , V ) is nonempty. This is equivalent to saying thatOrb f ( x ) is dense. The set of transitive points of ( X, f ) is denoted by Trans f .Following [3], we define the following properties:(DO) There is an orbit sequence h x k : k ∈ Z i or h x k : k ≥ n i (for some n ) densein X .(DO + ) There is a point x ∈ X such that Orb f ( x ) is dense.(DO ++ ) There is a point x ∈ X such that ωf ( x ) = X . Definition 2.3. [3] A dynamical system (
X, f ) is called point transitive if DO + holds in ( X, f ).Akin and Carlson [3] established the following implications for any general dy-namical system (
X, f ):DO ++ DO + DOTT ++ TT + TT INDiagram 1.It was shown in Theorem 1.4 of [3] if X is a perfect (i.e. without isolated points) and T space, then DO ++ and DO + are equivalent. Furthermore, if X is a perfect and T space and f is a continuous mapping, then TT ++ , TT + and TT are equivalent,refer to Proposition 4.2 in [3]. J. CAO AND A. MCCLUSKEY Quasi-continuous dynamical systems
Suppose that X and Y are topological spaces and f : X → Y is a mapping. Wesay that f is quasi-continuous at a point x ∈ X if, for each open neighbourhood W of f ( x ) and each open neighbourhood U of x , there exists a nonempty opensubset V of U such that f ( V ) ⊆ W . If f is quasi-continuous at each point of X ,then we say that f is quasi-continuous on X .This notion informally appeared in Baire’s PhD thesis [4], where he indicatedthat it was suggested to him by Volterra. Later, the notion of a quasi-continuousmapping was formally introduced/defined by Kempisty [16] for real-valued functionsof real variables. Quasi-continuity of mappings between general topological spaceswas also studied by Levine [19] under the name of semi-continuity.As noted by Crannell and Martelli in [8], the composite of two quasi-continuousmappings may fail to be quasi-continuous. Thus, when we try to extend results ina dynamical system ( X, f ) where f is continuous, we cannot just simply relax f tobe quasi-continuous. This motivates the following definition. Definition 3.1 ([8]) . A system (
X, f ) is said to be quasi-continuous if for every n ∈ N , f n : X → X is quasi-continuous.There is a quasi-continuous dynamical system which is not continuous, as shownby the following simple example. Example 3.2.
Let X = [0 ,
1] be endowed with the usual topology. Define amapping f : X → X by f ( x ) = ( , if 0 ≤ x ≤ ;1 , if < x ≤ f is continuous at any point x = and is quasi-continuous at x = .So f is quasi-continuous but not continuous on X . Furthermore it can be readilychecked that f n = f for all n ∈ N . Thus we conclude that ( X, f ) is a quasi-continuous but not continuous dynamical system.The following lemma gives a characterization of quasi-continuity which is a usefultool in the study of quasi-continuous dynamical systems. The proof of this lemmais straightforward and can be found in [8] or other references.
Lemma 3.3 ([8]) . Let X and Y be topological spaces and f : X → Y a mapping.Then f is quasi-continuous on X if, and only if, for each pair of nonempty opensets U ⊆ X and V ⊆ Y , either U and f − ( V ) are disjoint or there is a nonemptyopen subset W ⊆ X such that W ⊂ U ∩ f − ( V ) . Our next result is analogous to Lemma 4.1 in [3].
Proposition 3.4.
Let ( X, f ) be a quasi-continuous dynamical system where X isa perfect and Hausdorff phase space. If ( X, f ) satisfies TT , then for any nonemptyopen subset U ⊆ X , N + ( U, U ) is infinite.Proof. Let U ⊆ X be a nonempty open subset. We will define inductively a nestedsequence h U n : n ∈ N i of nonempty open subsets of U and a strictly increasingsequence h k n : n ∈ N i in N such that f k n ( U n ) ⊆ U . Initial step . Let U = U and k = 0. It is trivial that f k ( U ) ⊆ U . OPOLOGICAL TRANSITIVITY IN QUASI-CONTINUOUS · · · Induction step.
Suppose that we have defined a finite sequence U ⊇ U ⊇ · · · ⊇ U n − ⊇ U n of nonempty open subsets in U and a finite sequence of integers k < k < · · · < k n in N such that f k n ( U n ) ⊆ U . Since X is perfect and Hausdorff, we can pick up twodistinct points x n +1 and y n +1 in U n and two disjoint nonempty open subsets V n +1 and W n +1 in U n such that x n +1 ∈ V n +1 and y n +1 ∈ W n +1 . Since ( X, f ) satisfiesTT, there is an integer i n +1 such that i n +1 ∈ N + ( V n +1 , W n +1 ) ∪ N + ( W n +1 , V n +1 ) . Without loss of generality, we assume that i n +1 ∈ N + ( V n +1 , W n +1 ). By Lemma3.3, there is a nonempty open subset U n +1 such that U n +1 ⊆ V n +1 ∩ f − i n +1 ( W n +1 ) . Now it is clear that U n +1 ⊆ U n . Also, as V n +1 and W n +1 are disjoint, we musthave i n +1 >
0. Put k n +1 = k n + i n +1 . Then k n < k n +1 and f k n +1 ( U n +1 ) = f k n (cid:0) f i n +1 ( U n +1 ) (cid:1) ⊆ f k n ( W n +1 ) ⊆ f k n ( U n ) ⊆ U. This completes the induction step.Finally, by the construction of sequences h U n : n ∈ N i and h k n : n ∈ N i , we have k n ∈ N + ( U, U ) for each n ∈ N . We conclude that N + ( U, U ) is infinite. (cid:3)
A natural question is:
How far is a quasi-continuous system ( X, f ) from a con-tinuous system? To study this question, we define C ∞ ( f ) and C ∞ f as follows: C ∞ ( f ) := { x ∈ X : f n is continuous at x for all n ∈ N } and C ∞ f := { x ∈ X : f n ( x ) ∈ C ( f ) for all n ∈ N } , where C ( f ) is the set of points at which f is continuous. If x ∈ C ∞ f , then f iscontinuous at every point along Orb f ( x ), and accordingly, f n is continuous at x forevery n ∈ N , i.e. C ∞ f ⊆ C ∞ ( f ).Let X be a topological space, and let ρ be a metric on X . Then X is saidto be fragmentable by ρ if for all ǫ > A ⊆ X , there is arelatively open non-empty B ⊆ A with ρ -diam( B ) < ǫ . Note that every metrizablespace X is fragmented by some metric ρ . We refer the reader to [15] for moredetails on fragmentability. Recall that a subset R of X is called residual , if X \ R is a countable union of nowhere dense subsets of X , or equivalently, R contains acountable intersection of dense open subsets. Moreover if X is a Baire space, thenany residual subset R in X contains a dense G δ -set of X . Proposition 3.5.
Let ( X, f ) be a quasi-continuous system. If X is fragmented bya metric ρ such that the topology generated by the metric ρ contains the topology ofthe space X , then C ∞ ( f ) is a residual set in X . Furthermore, if X is also a Bairespace, then C ∞ ( f ) contains a dense G δ -set.Proof. For each n ∈ N , since f n is quasi-continuous, by Theorem 1 in [17], C ( f n )is a residual subset of X . Note that C ∞ ( f ) = \ n ∈ N C ( f n ) . Thus, C ∞ ( f ) is also a residual subset of X . (cid:3) J. CAO AND A. MCCLUSKEY
To see when C ∞ f is residual, we need some notion of openness on f . Giventopological spaces Y and Z , recall that a mapping f : Y → Z is feebly open [14]if for every nonempty open set U ⊆ Y , the interior of f ( U ) is nonempty. Crannellet al [7] called a quasi-continuous and feebly open mapping quopen . Theorem 8of [7] asserts that if X is a compact metric space and f is quopen, then C ∞ f is aresidual set in X . To extend this result, we employ the concept of a δ -open mappingintroduced by Haworth and McCoy [14]. Definition 3.6 ([14]) . Given topological spaces Y and Z , a mapping f : Y → Z iscalled δ -open if for every nowhere dense subset N of Z , f − ( N ) is nowhere dense in Y , or equivalently, for every somewhere dense subset A of Y , f ( A ) is a somewheredense subset of Z .The following proposition may be known, but we cannot find it in the literature.For the sake of completeness, we provide a full proof here. Proposition 3.7.
If a mapping f : Y → Z is quasi-continuous and feebly open,then it is δ -open.Proof. Let N ⊆ Z be a nowhere dense subset of Z . To derive a contradiction,we assume that f − ( N ) is somewhere dense. Then there exists a nonempty opensubset U of Y such that U ⊆ f − ( N ). Since f is a feebly open mapping, then V := int( f ( U )) = ∅ . Furthermore, since N is nowhere dense, the set W := V ∩ ( Z \ N )is nonempty open in Z . Now we have a nonempty open subset U of Y and anonempty open subset W of Z with U ∩ f − ( W ) = ∅ . Since f is quasi-continuous,by Lemma 3.3 we have a nonempty open subset U ′ such that U ′ ⊆ U ∩ f − ( W ).On the one hand, U ′ ⊆ U ⊆ f − ( N ) implies that f ( U ′ ) ∩ N = ∅ . On the otherhand, f ( U ′ ) ⊆ W implies that f ( U ′ ) ∩ N = ∅ . We have reached a contradiction.Thus f − ( N ) must be nowhere dense. (cid:3) Our next result extends Theorem 8 in [7].
Proposition 3.8.
Consider a dynamical system ( X, f ) . If (1) f is quasi-continuous and δ -open, and (2) X is fragmented by a metric ρ such that the topology generated by the metric ρ contains the topology of the space X ,then C ∞ f is a residual set in X . Furthermore if X is also a Baire space, then C ∞ f contains a dense G δ -set.Proof. First, as in [7], we can easily show that C ∞ f = \ n ∈ N f − n ( C ( f )) . By Theorem 1 in [17], C ( f ) is residual in X . Since f is δ -open, then f − ( C ( f ))is also residual. By induction, we see that f − n ( C ( f )) is residual for all n ∈ N . Itfollows that C ∞ f is a residual set in X . (cid:3) Propositions 3.5 and 3.8 indicate that in a certain sense, a quasi-continuousdynamical system approximates some continuous dynamical system. In [1] and [7],the dynamics of a quasi-continuous mapping was described in terms of suitable
OPOLOGICAL TRANSITIVITY IN QUASI-CONTINUOUS · · · closed relations, and connected with the continuous dynamics on an invariant G δ -set and with continuous dynamics on the compact space of sample paths. Remark 3.9.
Note that the conclusion of Proposition 3.5 does not hold if weonly require that f is quasi-continuous, even when X is a compact metric space.Indeed Crannell and Sohaib [9] provide an example of a quasi-continuous function f : [0 , → [0 ,
2] such that f is not continuous at any point of [0 , Equivalence theorems in quasi-continuous dynamicalsystems with perfect phase space
In this section, we establish the equivalence between point and topological tran-sitivity in a quasi-continuous dynamical system whose phase space is perfect.The following result is an analogue of Proposition 4.2 of [3].
Theorem 4.1.
Let ( X, f ) be a quasi-continuous dynamical system. If X is a perfectand Hausdorff space, then the following implications hold: TT −→ TT + −→ TT ++ . Proof.
It suffices to show that TT implies TT ++ . Assume that ( X, f ) satisfies TT.To show that (
X, f ) satisfies TT ++ , let V and W be any pair of nonempty opensubsets of X . By the TT property, there exists n ∈ Z such that f n ( V ) ∩ W = ∅ . Case 1. n ≥ . Since f n is quasi-continuous, there is a nonempty open subset U of X satisfying U ⊆ V ∩ f − n ( W ). Then, by Proposition 3.4, N + ( U, U ) is infinite.For any k ∈ N + ( U, U ), we have ∅ 6 = f n (cid:0) f k ( U ) ∩ U (cid:1) ⊆ f n + k ( U ) ∩ f n ( U ) ⊆ f n + k ( V ) ∩ W and hence n + k ∈ N + ( V, W ). Thus, N + ( V, W ) is infinite.
Case 2. n < . Since f − n is quasi-continuous, there is a nonempty open subset U of X satisfying U ⊆ f n ( V ) ∩ W . Then, by Proposition 3.4, N + ( U, U ) is infinite.For any k ∈ N + ( U, U ) with k > − n , we have ∅ 6 = f k ( U ) ∩ U ⊆ f k ( f n ( V )) ∩ W ⊆ f n + k ( V ) ∩ W, and hence n + k ∈ N + ( V, W ). Thus, N + ( V, W ) is infinite. (cid:3)
Let P be a family of nonempty open subsets in a topological space X . We call P a π -base for X if for every nonempty open subset U of X , there exists some P ∈ P such that P ⊆ U . Theorem 4.2.
Let ( X, f ) be a quasi-continuous dynamical system. Suppose that X is a Baire space with a countable π -base { P n : n ≥ } . Then the followingconditions are equivalent: (1) DO ++ . (2) For any non-empty open subset U ⊆ X and any k ∈ N , S n ≥ k int( f − n ( U )) is dense in X . (3) The set { x ∈ X : ωf ( x ) = X } contains a dense G δ -set of X .Proof. Since (3) → (1) is trivial, we need only to prove (1) → (2) and (2) → (3).(1) → (2). Let a nonempty open set U ⊆ X and a k ∈ N be given. Let V be anarbitrary nonempty open subset of X . By the DO ++ property, there exists some n ≥ k such that f n ( x ) ∈ V . In addition, the DO ++ property also implies that U ∩ { f m ( x ) : m ≥ n + k + 1 } 6 = ∅ . J. CAO AND A. MCCLUSKEY
Thus there exists an m > n + k such that f m ( x ) ∈ U . Since f ( m − n ) is quasi-continuous and f n ( x ) ∈ V ∩ f − ( m − n ) ( U ), by Lemma 3.3 there exists a nonemptyopen subset W of X such that W ⊆ V ∩ f − ( m − n ) ( U ). Thus, ∅ 6 = W ⊆ V ∩ int (cid:16) f − ( m − n ) ( U ) (cid:17) ⊆ V ∩ [ n ≥ k int (cid:0) f − n ( U ) (cid:1) , which implies that (2) holds.(2) → (3). Recall that { P n : n ≥ } is a π -base of X . For any n, k ∈ N , let S n,k := [ j ≥ k int( f − j ( P n )) , and S := \ n ∈ N \ k ∈ N S n,k . By (2), S n,k is open and dense in X and thus S is a dense G δ -set in X . Now weshow that ωf ( x ) = X for all x ∈ S . To this end, let x ∈ S and k ∈ N be fixed. Foreach nonempty open subset V ⊆ X , as { P n : n ≥ } is a π -base, we can choose an n ≥ P n ⊆ V . Then x ∈ S n ,k = [ j ≥ k int( f − j ( P n )) ⊆ [ j ≥ k f − j ( P n ) . Thus there is n ≥ k such that x ∈ f − n ( P n ), which implies that f n ( x ) ∈ V . Thismeans that Orb f ( f k ( x )) is dense. It follows that ωf ( x ) = X . (cid:3) Theorem 4.3.
Let ( X, f ) be a quasi-continuous dynamical system. Suppose that X is a space of the second category with a countable π -base P = { P n : n ∈ N } . Then TT + implies that DO + . In addition, if X is a Baire space, then Trans f contains adense G δ -set of X .Proof. Let (
X, f ) satisfy TT + . For each n ∈ N , we define an open subset U n by U n := [ k ∈ N int (cid:0) f − k ( P n ) (cid:1) . Let U := T n ∈ N U n . We first show that each U n is dense in X . To this end, let V be an arbitrary nonempty open subset of X . Since ( X, f ) satisfies TT + , we have N + ( V, P n ) = ∅ and hence there is a k ∈ N such that V ∩ f − k ( P n ) = ∅ . Since f k is quasi-continuous, by Lemma 3.3 there is a nonempty open subset W in X suchthat W ⊆ V ∩ f − k ( P n ). This implies that V ∩ U n = ∅ which proves the claim.Next we show that U ⊆ Trans f . Let x ∈ U . We need to verify that Orb f ( x )is dense in X . Let G be an arbitrary nonempty open subset of X . Since P is a π -base, there must be some n ∈ N such that P n ⊆ G . Then x ∈ U n implies that f k ( x ) ∈ P n ⊆ G for some k ∈ N . Hence x ∈ Trans f .Finally, since X is a space of the second category, U = ∅ . It follows thatTrans f = ∅ and thus ( X, f ) satisfies DO + . In addition, if X is a Baire space,then U is a dense G δ -set of X . (cid:3) Proposition 4.4.
Let ( X, f ) be a quasi-continuous dynamical system whose phasespace X is a perfect and T -space. Then, DO + and DO ++ are equivalent. OPOLOGICAL TRANSITIVITY IN QUASI-CONTINUOUS · · · Proof.
Assume that (
X, f ) satisfies DO + . Then there is a point x ∈ X such thatOrb f ( x ) is dense in X . To show that ωf ( x ) = X , let z ∈ X and k ∈ N , and let U bean open neighborhood of z . Since X is a perfect T -space, U \ { x, f ( x ) , . . . , f k ( x ) } is a nonempty open subset of X , and hence we can take a point y ∈ (cid:0) U \ { x, f ( x ) , . . . , f k ( x ) } (cid:1) ∩ Orb f ( x ) . Since y ∈ Orb f ( x ), y = f m ( x ) for some m ∈ N . Further, since f m ( x ) = y
6∈ { x, f ( x ) , . . . , f k ( x ) } , we have m > k . Thus y ∈ U ∩ { f n ( x ) : n ≥ k } , and hence z ∈ ωf ( x ). Therefore,( X, f ) satisfies DO ++ . (cid:3) As a corollary of Theorems 4.1-4.3 and Proposition 4.4, we obtain the followingresult which extends Theorem 2.1 in [8].
Corollary 4.5.
Let ( X, f ) be a quasi-continuous dynamical system whose phasespace X is perfect and Hausdorff. Assume in addition that X is also a Baire spacewith a countable π -base P = { P n : n ∈ N } . Then TT, TT + , TT ++ , DO, DO + and DO ++ are equivalent. Furthermore, each of these properties is equivalent to eachof the following: (1) Trans f contains a dense G δ -set of X . (2) The set { x ∈ X : ωf ( x ) = X } contains a dense G δ -set of X . To conclude this section, we summarize the relationships between the seven prop-erties IN, TT, TT + , TT ++ , DO, DO + and DO ++ in a quasi-continuous dynamicalsystem by combining Diagram 1, Theorems 4.1 and 4.3 into the following figure.DO ++ DO + DOperfect T TT ++ TT + TT INperfect T perfect T c. π .b.2nd cat.Diagram 2.5. The case of imperfect phase spaces
In this section, we discuss what happens in a quasi-continuous dynamical system(
X, f ), when the phase space X contains isolated points. Throughout this section,we assume that X is a Hausdorff space.Let Iso X be the set of isolated points in X . Similar to the analysis in Section 5of [3], we discuss how Iso X sits in X by analyzing the preimages of isolated pointsand then see what role Iso X plays in the study of topological transitivity. Throughthis analysis, we derive a conclusion similar to that of [3] but our results are anextension of those in Section 5 of [3]. Lemma 5.1. If ( X, f ) satisfies TT , then there is at most one point x ∈ Iso X suchthat f − ( x ) = ∅ .Proof. If f ( X ) is dense in X , then f − ( x ) = ∅ for any x ∈ Iso X . Assume that f ( X )is not dense in X . Then X \ f ( X ) is nonempty open. By an argument similar tothat of Corollary 4.5, we can show that X \ f ( X ) contains only one point x . Then x must be an isolated point such that f − ( x ) = ∅ . (cid:3) The following characterization of a quasi-continuous mapping between two topo-logical spaces is useful.
Lemma 5.2 ([22], [23]) . Given topological spaces X and Y , a mapping f : X → Y is quasi-continuous on X if, and only if, f − ( V ) ⊆ int ( f − ( V )) for any open subset V of Y . Lemma 5.3.
Let ( X, f ) be a quasi-continuous dynamical system satisfying TT . If x ∈ Iso X and | f − ( x ) | > , then x is periodic and | f − ( x ) | = 2 . Thus f − ( x ) is afinite open set consisting entirely of isolated points.Proof. If x ∈ Iso X , by Lemma 5.2 f − ( x ) ⊆ int f − ( x ). In addition, if | f − ( x ) | > y, z ∈ f − ( x ). Since X is Hausdorff, wecan choose two disjoint open subsets U ′ and V ′ in X such that y ∈ U ′ and z ∈ V ′ .Put U := U ′ ∩ int f − ( x ) and V := V ′ ∩ int f − ( x ). Then, U and V are nonemptyopen subsets in X such that U ∩ V = ∅ and U ∪ V ⊆ f − ( x ). Further, as ( X, f )satisfies TT, we can require N + ( U, V ) = ∅ without loss of generality.Let k be the smallest element of N + ( U, V ). Since U ∩ V = ∅ , k >
0. Choose y ∈ U such that f k ( y ) ∈ V . Then, x = f ( y ) and f k ( y ) ∈ f − ( x ). It follows that x = f ( f k ( y )) = f k ( f ( y )) = f k ( x ) . Thus x is periodic. Indeed, by the minimality of k , we conclude that k is the periodof x and the forward orbit of x isOrb f ( x ) = { x, f ( x ) , f ( x ) , · · · , f k − ( x ) } . Suppose | f − ( x ) | >
2. Then, by shrinking U and V if necessary, we can choosea nonempty open subset W of X , which is disjoint from both U and V , such that U ∪ V ∪ W ⊆ f − ( x ) . Since (
X, f ) satisfies TT, there exists an integer m ∈ N such that m ∈ N + ( U, W ) ∪ N + ( W, U ) . Without loss of generality, let m ∈ N + ( U, W ). The disjointness of U and W implies m >
0. We choose a point z ∈ U such that f m ( z ) ∈ W . Similar to the previousargument, we have x = f ( f m ( z )) = f m ( f ( z )) = f m ( x ) . Hence m must be a multiple of k . This implies that f m ( z ) = f m − ( x ) = f k − ( x ) ∈ V, which contradicts the fact that f m ( z ) ∈ W .Finally, since the cardinality of f − ( x ) is 0, 1 or 2, then f − ( x ) is a finite set.If it has cardinality 1 or 2, then int (cid:0) f − ( x ) (cid:1) nonempty in addition to X Hausdorff(in the case of cardinality 2) implies that every point of f − ( x ) is isolated. (cid:3) OPOLOGICAL TRANSITIVITY IN QUASI-CONTINUOUS · · · Lemma 5.4.
Let ( X, f ) be a quasi-continuous dynamical system satisfying TT .Then there is at most one x ∈ Iso X such that | f − ( x ) | = 2 .Proof. Suppose that there are two distinct points x, x ′ ∈ Iso X such that | f − ( x ) | = | f − ( x ′ ) | = 2 . By Lemma 5.3, both x and x ′ are periodic, and Orb f ( x ) = Orb f ( x ′ ) since ( X, f )satisfies TT. Note that there are points y and y ′ such that y ∈ f − ( x ) \ Orb f ( x ) and y ′ ∈ f − ( x ′ ) \ Orb f ( x ′ ). By Lemma 5.3, both y and y ′ are isolated. Since ( X, f )satisfies TT, N ( { y } , { y ′ } ) = ∅ and there is m ∈ N such that f m ( y ) = y ′ withoutloss of generality. Since x = x ′ and y ∈ f − ( x ), we have m > y ′ = f m ( y ) = f m − ( f ( y )) = f m − ( x ) , which contradicts y ′ Orb f ( x ′ ) = Orb f ( x ) . Hence there is at most one isolated point x in X with | f − ( x ) | = 2. (cid:3) Lemmas 5.1, 5.3 and 5.4 tell us that in a quasi-continuous dynamical system(
X, f ) with the property TT, the cardinality of the preimage of any isolated pointis 0, 1 or 2. Further, such a system has at most one isolated point with emptypreimage and at most one isolated point whose preimage contains exactly twopoints.
Proposition 5.5.
Let ( X, f ) be a quasi-continuous dynamical system with isolatedpoints. If ( X, f ) satisfies TT , then Trans f ⊆ Iso X .Proof. Let y ∈ Trans f . Then Orb f ( y ) is dense in X . Let x be any isolated point.Then there is an integer k ∈ N such that x = f k ( y ). If k = 0, then y = x ∈ Iso X .So we assume that k >
0. This implies that y ∈ f − k ( x ). Applying Lemma 5.3 infinitely many steps, we conclude that y ∈ Iso X . Hence Trans f ⊆ Iso X . (cid:3) Proposition 5.5 tells us that we should search for transitive points within thoseisolated points. We shall achieve this in the next result.
Theorem 5.6.
Let ( X, f ) be a quasi-continuous dynamical system with isolatedpoints. Assume that ( X, f ) satisfies TT . (1) If there is a (unique) point x ∈ Iso X such that f − ( x ) = ∅ , then ( X, f ) satisfies DO + and x is the unique transitive point. In this case, none of TT + , TT ++ or DO ++ hold. (2) If | f − ( x ) | = 1 for every x ∈ Iso X , then either all of DO + , TT + , TT ++ and DO ++ hold, or none of them hold. (3) If f − ( z ) = ∅ for every point z ∈ Iso X and there is a (unique) point x ∈ Iso X such that | f − ( x ) | = 2 , then none of DO + , TT + , DO ++ and TT ++ hold.Proof. (1) Since ( X, f ) satisfies TT and f − ( x ) = ∅ , for any nonempty open subset U of X we must have N + ( { x } , U ) = ∅ . This implies that Orb f ( x ) is dense in X and Iso X ⊆ Orb f ( x ). Thus ( X, f ) satisfies DO + and x is a transitive point.Since f − ( x ) = ∅ , we cannot have x ∈ Orb f ( y ) for any y ∈ Iso X and y = x .This means that any isolated point distinct from x (if such a point exists) cannotbe transitive. Hence, x is the unique transitive point. Pick any point y ∈ X but y = x ( y is not necessarily to be an isolated point).Since X is a Hausdorff space, there is an open set U such that y ∈ U and x U .As f − ( x ) = ∅ , we have N + ( U, { x } ) = ∅ . Thus, TT + or any property stronger thanTT + does not hold.(2) We consider two subcases. Subcase 1.
There is a periodic isolated point x ∈ Iso X . ThenOrb f ( x ) = { x, f ( x ) , · · · , f k − ( x ) } for some k >
0. By Lemma 5.3, we have Orb f ( x ) ⊆ Iso X . This and the fact that | f − ( z ) | = 1 for every z ∈ Iso X imply that Orb f ( x ) = { f n ( x ) : n ∈ Z } . Now let U be any nonempty open subset of X . Since ( X, f ) satisfies TT, we have some integer k ∈ N ( { x } , U ). Then f k ( x ) ∈ U ∩ { f n ( x ) : n ∈ Z } = U ∩ Orb f ( x ) . This implies that Orb f ( x ) is dense in X . Since Orb f ( x ) is closed in X , we have X = Orb f ( x ). Thus, in this subcase, all of DO + , TT + , TT ++ and DO ++ hold. Subcase 2.
Iso X does not contain any periodic point. Let x ∈ Iso X . By Lemma5.2, f − ( x ) is open in X . Since x is not a periodic point, N + ( { x } , f − ( x )) = ∅ .Then TT + and thus TT ++ do not hold.To show that Trans f = ∅ , by Proposition 5.5, it suffices to show that y Trans f for every y ∈ Iso X . Let y ∈ Iso X . Since | f − ( y ) | = 1, there is a point y ′ suchthat f ( y ′ ) = y , and y ′ ∈ Iso X by Lemma 5.3. Since y is not a periodic point, y ′ Orb f ( y ). Thus y ′ Orb f ( y ) and hence y Trans f . Therefore Trans f = ∅ .This means that DO + and thus DO ++ do not hold.(3) By Lemma 5.3, x is a periodic point with period k >
0. LetOrb f ( x ) = { x, f ( x ) , · · · , f k − ( x ) } and f − ( x ) = { f k − ( x ) , y } with y = f k − ( x ). Then y ∈ Iso X and Orb f ( x ) ⊆ Iso X .If y = f m ( x ) for some nonnegative integer m < k −
1, then x = f m +1 ( x ), whichcontradicts the fact that k is the period of x . Thus, y / ∈ Orb f ( x ). (As a notationalconvenience, we identify the singleton f − n ( y ) with its single element to allow forease of expression in what follows.) Indeed, by Lemma 5.3, we have that f − n ( y ) ∈ Iso X \ Orb f ( x ) for all n ∈ N . This implies that N + ( { x } , { y } ) = ∅ . Therefore TT + and hence TT ++ do not hold.Next we show that DO + does not hold, that is, Trans f = ∅ . For any point z ∈ Iso X , by the TT property, we have N ( { z } , { y } ) = ∅ . If N + ( { z } , { y } ) = ∅ , then y = f m ( z ) for some m ∈ N . In this case, z ∈ { f − n ( y ) : n ∈ N } . If N + ( { y } , { z } ) = ∅ ,then z = f m ( y ) for some m ∈ N . So, we have either z = y or z = f n ( x ) for some n ∈ N . This means that either z ∈ { f − n ( y ) : n ∈ N } or z ∈ Orb f ( x ) holds. Wehave just verified that Iso X = Orb f ( x ) ∪ { f − n ( y ) : n ∈ N } . Note that no point in Orb f ( x ) is transitive, as y / ∈ Orb f ( x ). Now we consider apoint z ∈ Iso X \ Orb f ( x ). Then z = f − n ( y ) for some n ∈ N . By Lemma 5.3 andassumption, f − ( z ) is an isolated point. Since f − m ( y ) = f − n ( y ) for all m = n and m, n ∈ N , f − ( z ) / ∈ Orb f ( z ). It follows that z Trans f . Hence, Trans f = ∅ byProposition 5.5. Consequently, DO + and hence DO ++ do not hold. (cid:3) Corollary 5.7.
Let ( X, f ) be a quasi-continuous dynamical system with isolatedpoints. Then, DO + , TT + , DO ++ and TT ++ are equivalent. OPOLOGICAL TRANSITIVITY IN QUASI-CONTINUOUS · · · Corollary 5.8.
Let ( X, f ) be a quasi-continuous dynamical system with isolatedpoints. Then, TT and DO are equivalent.Proof. First we know that DO implies TT in any dynamical system. Assume thenthat (
X, f ) satisfies the TT property. We consider the three cases listed in Theorem5.6.In case (1), as shown in the proof of Theorem 5.6, (
X, f ) satisfies DO + and thusDO.In case (2), | f − ( x ) | = 1 for all x ∈ Iso X . Then for any x ∈ Iso X , f − n ( x ) is anisolated point for all n ∈ N . The TT property implies that h f n ( x ) : n ∈ Z i is anorbit sequence dense in X . Thus ( X, f ) satisfies DO.In case (3), let x ∈ Iso X be the unique point such that | f − ( x ) | = 2. Notethat x is a periodic point with period k >
0. Let f − ( x ) = { f k − ( x ) , y } . Then f − n ( y ) ∈ Iso X for all n ∈ N . Again, the TT property implies that h f n ( y ) : n ∈ Z i is an orbit sequence dense in X . Hence ( X, f ) satisfies DO. (cid:3)
Acknowledgement.
The authors would like to thank the referee for her/his thor-ough check of the original manuscript. Her/his valuable comments and suggestionshave improved the presentation of this paper. In particular, Example 3.2 was pro-vided by her/him.
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