Almost automorphy of minimal sets for C^1-smooth strongly monotone skew-product semiflows on Banach spaces
aa r X i v : . [ m a t h . D S ] J a n Almost automorphy of minimal sets for C -smooth stronglymonotone skew-product semiflows on Banach spaces ∗ Yi Wang and Jinxiang Yao † School of Mathematical SciencesUniversity of Science and Technology of ChinaHefei, Anhui, 230026, P. R. China
Abstract
We focus on the presence of almost automorphy in strongly monotone skew-productsemiflows on Banach spaces. Under the C -smoothness assumption, it is shown that anylinearly stable minimal set must be almost automorphic. This extends the celebrated resultof Shen and Yi [Mem. Amer. Math. Soc. 136(1998), No. 647] for the classical C ,α -smoothsystems. Based on this, one can reduce the regularity of the almost periodically forceddifferential equations and obtain the almost automorphic phenomena in a wider range. Keywords : Almost automorphy; Monotone skew-product semiflow; Principal Lyapunovexponents; Exponential Separation; C -smoothness. The notion of almost automorphy, which is a generalization to almost periodicity, was firstintroduced by Bochner [2] in a work of differential geometry. In the terminology of functiontheory, almost periodic and almost automorphic functions can be viewed as natural generaliza-tions to the periodic ones in the strong and weak sense, respectively. From dynamical systemspoint of view, Veech [47–50] first introduced almost automorphic minimal flows. A compactflow ( Y, R ) is called almost automorphic minimal if Y is the closure of the orbit of an almostautomorphic point. Here, a point y ∈ Y is called almost automorphic if any net α ′ ⊂ R hasa subnet α = { t n } such that T α y , T − α T α y exist and T − α T α y = y , where T α is the generalizedtranslation as T α y = lim n y · t n provided that the limit exists (see Section 2). Fundamentalproperties of almost automorphic functions/flows were further investigated in [7, 45, 46], etc. ∗ Supported by NSF of China No.11825106, 11771414 and 11971232, CAS Wu Wen-Tsun Key Laboratory ofMathematics, University of Science and Technology of China. † Corresponding author: [email protected] (J. Yao). C -smooth strongly monotone semiflows, the improved generic convergence was obtained byPol´aˇcik [28] and Smith and Thieme [43]. For strongly monotone discrete-time systems (map-pings), which are usually the Poincar´e mappings associated with periodically forced differentialequations, Pol´aˇcik and Tereˇsˇc´ak [30] proved that the generic convergence to cycles occurs pro-vided that the mapping F is of C ,α -class (i.e., F is a C -map with a locally α -H¨older derivative DF , α ∈ (0 , F , Tereˇsˇc´ak [44] and Wang and Yao [52] succeededin using different approaches to prove the generic convergence to cycles for C -smooth stronglymonotone discrete-time systems.Shen and Yi [37] first discovered that almost automorphic phenomena largely exist in stronglymonotone skew-product semiflows Π( x, y, t ) = ( u ( x, y, t ) , y · t ) on X × Y , t ≥
0, where X is aBanach space, ( Y, R ) is a minimal and distal flow. More precisely, under the assumption that u is C ,α in x ∈ X , they [37] studied the lifting dynamics on minimal sets of the stronglymonotone skew-product semiflow Π, and proved that a linearly stable minimal set must bealmost automorphic and that the generic convergence property failed in almost periodic systems2ven within the category of almost automorphy. Their results have also been applied to showthe existence of almost automorphic dynamics in a large class of almost periodic ordinary,functional and parabolic differential equations. Based on Shen and Yi’s work, Obaya and hiscollaborators [24–26] systematically analyzed the occurrence of almost automorphic dynamicsin monotone skew-product semiflows with applications to functional differential equations.The approach in [37] is based on establishment of the exponential separation (see, e.g. [21,31])along the minimal sets of Π, as well as the idea and techniques for the construction of invariantmeasurable families of submanifolds in the so called Pesin’s Theory (see [27]); and hence, theregularity of α -H¨older continuity of the x -derivative of u cannot be dropped in [37].In the present paper, we shall focus on the presence of almost automorphy in C -smoothstrongly monotone skew-product semiflows. Motivated by our recent work in [52], we will extendthe celebrated result of Shen and Yi [37] by showing that any linearly stable minimal set mustbe almost automorphic for C -smooth strongly monotone skew-product semiflows. Based onour result, one can reduce the regularity of the almost periodically forced equations (inculdingODEs, parabolic equations and delay equations) investigated in [37, Part III], and obtain thealmost auotmorphic phenomena in a wider range.As mentioned above, due to the lack of the α -H¨older continuity, the Pesin’s Theory withthe Lyapunov exponents arguments in [37] can not work any more. Inspired by [44, 52], ournew approach is to introduce a continuous cocycle over the Cartesian square K × K of thelinearly stable minimal set K rather than K itself, and to construct a bundle map T as thehybrid function of the x -derivative of u along K × K . Together with the exponential separationon K with a novel “internal growth control” property (see Proposition 2.5(v)) and a time-discretization technique to the skew-product semiflow, we accomplish our approach by provingthe crucial Propositions 3.3 and 3.4, which enables us to reduce the regularity of the systemsand obtain the almost automorphy of the minimal sets.This paper is organized as follows. In Section 2, we agree on some notations, give relevantdefinitions and preliminary results. We further present the exponential separation theorem (seeProposition 2.5) with the novel additional “internal growth control” property along principalbundles in Proposition 2.5(v), which turns out to be crucial for the proof of our main result. Insection 3, we state our main results and give their proofs. In this section, we first summarize some preliminary materials involved with topologicaldynamics which will appear throughout the paper.Let (
Y, d Y ) be a compact metric space, and σ : Y × R → Y , ( y, t ) y · t be a continuous flowon Y , denoted by ( Y, σ ) or ( Y, R ). A subset M ⊂ Y is invariant if σ t M = M , for each t ∈ R . Anon-empty compact invariant set M ⊂ Y is called minimal if it contains no non-empty, proper,closed invariant subset. We say that ( Y, R ) is minimal if Y itself is a minimal set.3et R + , R − denote the nonnegative, nonpositive reals, respectively. Points y , y ∈ Y arecalled ( positively, negatively ) distal , if inf t ∈ R ( t ∈ R + ,t ∈ R − ) d Y ( y · t, y · t ) >
0. We say that y , y are ( positively, negatively ) proximal if they are not (positively, negatively) distal. A point y ∈ Y is said to be a distal point if it is only proximal to itself. Moreover, ( Y, R ) is a distal flow if every point in Y is a distal point. The ( positive, negative ) proximal relation P ( Y )( P + ( Y ) , P − ( Y )) is a subset of Y × Y defined as follows: P ( Y )( P + ( Y ) , P − ( Y )) = { ( y , y ) ∈ Y × Y | y , y are (positively, negatively) proximal } . P ( Y ) is clearly invariant, reflexive and sym-metric but not transitive in general. Proposition 2.1. ( [37, Part I, Corollary 2.8]).
Suppose that P ( Y ) is an equivalence relation.Then P ( Y ) = P + ( Y ) = P − ( Y ) . For y ∈ Y and a net α = { t n } in R , we define T α y := lim n y · t n , provided that the limitexists. ( Y, R ) is called almost periodic if any nets α ′ , β ′ in R have subnets α , β such that T β y , T α T β y , T α + β y exist and T α T β y = T α + β y for all y ∈ Y , where α + β = { t n + s n } if α = { t n } , β = { s n } . An almost periodic flow is necessarily distal (see, e.g. [37]). A point y ∈ Y is an almost automorphic point if any net α ′ in R has a subnet α = { t n } such that T α y , T − α T α y existand T − α T α y = y , where − α = {− t n } . A flow ( Y, R ) is almost automorphic if there is an almostautomorphic point y ∈ Y with dense orbit. An almost automorphic flow is necessarily minimal(see, e.g. [37]).A flow homomorphism from another continuous flow ( Z, R ) to ( Y, R ) is a continuous map φ : Z → Y such that φ ( z · t ) = φ ( z ) · t for all z ∈ Z , t ∈ R . An onto flow homomorphism is calleda flow epimorphism and an one to one flow epimorphism is referred to as a flow isomorphism .If φ is an epimorphism, then ( Z, R ) is said to be an extension of ( Y, R ). An epimorphism φ is called an N -1 extension for some integer N ≥
1, if card ( φ − ( y )) = N for all y ∈ Y . Let φ : ( Z, R ) → ( Y, R ) be a homomorphism of minimal flows, then φ is an almost automorphicextension if there is a y ∈ Y such that card ( φ − ( y )) = 1. Then, actually φ is an almost 1-1extension , i.e., { y ∈ Y | card ( φ − ( y )) = 1 } is a residual subset of Y . A minimal flow ( Z, R ) is almost automorphic if and only if it is an almost automorphic extension of an almost periodicminimal flow ( Y, R ) (see [48] or [37, Part I, Theorem 2.14]). Proposition 2.2. ( [32] or [37, Part I, Theorem 2.12]).
Let φ : ( Z, R ) → ( Y, R ) be a homomor-phism of distal flows, where ( Y, R ) is minimal. If there is y ∈ Y with card ( φ − ( y )) = N , thenthe following holds: 1) φ is an N -1 extension; 2) ( Z, R ) is almost periodic if and only if ( Y, R ) is. Given a continuous flow ( Y, R ) and a Banach space X , a continuous skew-product semiflow Π : X × Y × R + → X × Y is defined as:Π( x, y, t ) = ( u ( x, y, t ) , y · t ) , ( x, y ) ∈ X × Y, t ∈ R + , ( . )where Π( · , · , t ) can also be written as Π t ( · , · ), for all t ∈ R + and satisfies (i) Π =Id and (ii) the cocycle property : u ( x, y, t + s ) = u ( u ( x, y, s ) , y · s, t ), for each ( x, y ) ∈ X × Y and t, s ∈ R + .4e denote p : X × Y → Y ; ( x, y ) y as the natural projection. A subset M ⊂ X × Y iscalled positively invariant if Π t ( M ) ⊂ M for all t ∈ R + . A compact positively invariant set K ⊂ X × Y is minimal if it does not contain any other nonempty compact positively invariantset than itself.A flow extension of a skew-product semiflow ( X × Y, Π , R + ) is a skew-product flow ( X × Y, ˜Π , R ) such that ˜Π( x, y, t ) = Π( x, y, t ), for each ( x, y ) ∈ X × Y and t ∈ R + . A compactpositively invariant subset is called admits a flow extension if the semiflow restricted to it does.Actually, a compact positively invariant set K ⊂ X × Y admits a flow extension if every pointin K admits a unique backward orbit which remains inside the set K (see [37, Part II]).In this work, we need C -smoothness of the skew-product semiflow Π. Precisely, the skew-product semiflow Π in ( . ) is said to be of class C in x , meaning that u x ( x, y, t ) exists for any t > x, y ) ∈ X × Y ; and for each fixed t >
0, the map ( x, y ) u x ( x, y, t ) ∈ L ( X ) iscontinuous on any compact subset K ⊂ X × Y ; and moreover, for any v ∈ X , u x ( x, y, t ) v → v as t → + uniformly for ( x, y ) in compact subsets of X × Y .Let K ⊂ X × Y be a compact, positively invariant set which admits a flow extension. For( x, y ) ∈ K , we define the Lyapunov exponent λ ( x, y ) as λ ( x, y ) = lim sup t → + ∞ ln k u x ( x,y,t ) k t . Thenumber λ K = sup ( x,y ) ∈ K λ ( x, y ) is called the principal Lyapunov exponent on K . If λ K ≤
0, then K is said to be linearly stable . Proposition 2.3. ( [37, PartII, Corollary 4.2]).
Assume that ( Y, R ) is minimal and Π is ofclass C in x . Assume also that K ⊂ X × Y is a compact, positively invariant set which admitsa flow extension; moreover, K is linearly stable . Then for any ǫ > , there is a C ǫ > suchthat k u x ( x, y, t ) k ≤ C ǫ e εt , for all t ≥ and ( x, y ) ∈ K . A closed convex subset C ⊂ X is called a cone of X if λC ⊂ C for all λ > C ∩ ( − C ) = { } . We call ( X, C ) a strongly ordered
Banach space if C has nonempty interior Int C . Let X ∗ be the dual space of X . C ∗ := { l ∈ X ∗ : l ( v ) ≥ v ∈ C } is called the dual cone of C .If Int C = ∅ , then C ∗ is indeed a closed convex cone in X ∗ (see [4]). Let C ∗ s = { l ∈ C ∗ : l ( v ) > , for any v ∈ C \{ }} . A bounded linear operator L : X → X is strongly positive if Lv ≫ v > X, C ) be a strongly ordered
Banach space. A closed set O + ( X, Y ) := { (( x , y ) , ( x , y )) | x − x ∈ C } induces a (strong) partial ordering ‘ ≥ ’ on each fiber p − ( y ) ( y ∈ Y ) as follows: ( x , y ) ≥ ( x , y ) if (( x , y ) , ( x , y )) ∈ O + ( X, Y ); ( x , y ) > ( x , y ) if ( x , y ) ≥ ( x , y ) , ( x , y ) = ( x , y );( x , y ) ≫ ( x , y ) if (( x , y ) , ( x , y )) ∈ Int O + ( X, Y ), i.e., x − x ∈ Int C . O − ( X, Y ) is the reflec-tion of O + ( X, Y ), that is, O − ( X, Y ) = { (( x , y ) , ( x , y )) | (( x , y ) , ( x , y )) ∈ O + ( X, Y ) } . The set O ( X, Y ) = O + ( X, Y ) ∪ O − ( X, Y ) is referred to as the order relation , that is, ( x , y ) , ( x , y ) are ordered if and only if y = y = y and (( x , y ) , ( x , y )) ∈ O ( X, Y ). The order relation on a min-imal subset K ⊂ X × Y is defined as O ( K ) = { (( x , y ) , ( x , y )) | ( x , y ) , ( x , y ) ∈ K and x − x ∈± C } . 5he skew-product semiflow Π is called strongly order preserving if Π( x , y, t ) ≫ Π( x , y, t )whenever ( x , y ) > ( x , y ) and t >
0. We say that Π is strongly monotone if u x ( x, y, t ) is astrongly positive operator for any ( x, y ) ∈ X × Y, t >
0. Clearly, by virtue of [37, PartII,Theorem 4.3], a strongly monotone skew-product semiflow must be a strongly order preservingskew-product semiflow.
Proposition 2.4.
Assume that ( Y, R ) is minimal and Π is strongly order preserving, and let K ⊂ X × Y be a minimal set of which admits a flow extension. Then (i) there is a residual and invariant set Y ⊂ Y such that for any y ∈ Y , no two elements on K ∩ p − ( y ) are ordered; (ii) If ( x , y ) , ( x , y ) ∈ K are ordered, then they are proximal, that is, the order relationimplies the proximal relation on K .Proof. See [37, PartII, Theorem 3.2 and Corollary 3.3].Before ending this section, we present the following exponential separation theorem for home-omorphisms. One may refer to [21–23,31] for more details and applications of this theorem withthe standard items (i)-(iii). Here, we emphasize a novel “internal growth control” property alongthe principal bundles obtained in item (v) of the following proposition, which turns out to becrucial for the proof of our main results in the next section. A weaker version of such “internalgrowth control” property was obtained in [44,52] for exponential separation for continuous maps.
Proposition 2.5. (Exponential Separation Theorem).
Let ( X, C ) be a strongly ordered Banachspace, F : E → E is a homeomorphism of a compact metric space E , T is a continuous familyof operators { T x ∈ L ( X, X ) : x ∈ E } , and for any x ∈ E , T x is a compact and strongly positiveoperator, then there exist one dimensional continuous bundles E × X x and E × X ∗ x such that: (i) X x =span { v x } and X ∗ x =span { l x } , where k v x k = 1 = k l x k , v x ≫ , l x ∈ C ∗ s , and both l x and v x depend continuously on x ∈ E . (ii) T x X x = X F x , T ∗ x X ∗ F x = X ∗ x . (iii) There are constants
M > and < γ < such that k T nx w k ≤ M γ n k T nx v x k , ( . ) for all x ∈ E , n ≥ and l x ( w ) = 0 with k w k =1, where T nx = T F n − x ◦ T F n − x ◦ · · · ◦ T F x ◦ T x . (iv) If x ∈ E, u ∈ X with l x ( u ) > , then T nx u ∈ Int C for all n sufficiently large. (v) (Internal growth control along principal bundles) For any ǫ > , there is a constant δ > such that, for any δ ∈ [0 , δ ] , x, y ∈ E , m ≥ with d E ( F i x, F i y ) < δ , ≤ i ≤ m , we have k T iy v y k ≤ (1 + ǫ ) i k T ix v x k , ( . ) for all ≤ i ≤ m . roof. For the proof of the standard items (i)-(iii), we refer to [31]. Here we give the proof of(iv)-(v).(iv). Decompose u by u = v + w , with v = l x ( u ) l x ( v x ) v x , l x ( w ) = 0. Then we have k v F n x − T nx u k T nx u k k ≤ k v F n x − T nx v k T nx v k k + k T nx v k T nx v k − T nx ( v + w ) k T nx ( v + w ) k k (ii) = 0 + k T nx v k T nx v k − T nx ( v + w ) k T nx ( v + w ) k k (iii) → , as n → ∞ . Since { v x : x ∈ E } is a compact subset of Int C by (i), T nx u ∈ Int C for all n sufficiently large.This proves (iv).(v). Since T x v x continuously depends on x ∈ E , { T x v x : x ∈ E } is a compact subset ofInt C . Then there exists a constant r > k T x v x k > r , for any x ∈ E . For any ǫ > T x v x uniformly continuously depends on x ∈ E , there exists a constant δ > k T x v x − T x ′ v x ′ k ≤ ǫr < ǫ k T x ′ v x ′ k , for any x, x ′ ∈ E with d E ( x, x ′ ) < δ . Therefore, for any δ ∈ [0 , δ ], x, y ∈ E , m ≥ d E ( F i x, F i y ) < δ , 0 ≤ i ≤ m , we have k T iy v y kk T ix v x k = k T F i − y v F i − y k · · · k T F y v F y k · k T y v y kk T F i − x v F i − x k · · · k T F x v F x k · k T x v x k < (1 + ǫ ) i , ≤ i ≤ m. This proves (v).
In this section, our standing hypotheses are as follows: (H1) ( Y, R ) is minimal and distal, and ( X, C ) is a strongly ordered Banach space. (H2)
Π is a strongly monotone skew-product semiflow on X × Y of class C in x . (H3) K ⊂ X × Y is a minimal set which admits a flow extension.Now we state our main results on the almost automorphy of the minimal set K . Theorem 3.1.
Assume that (H1)-(H3) hold. Assume also the following: (i)
There is τ > such that u x ( x, y, τ ) is compact for all ( x, y ) ∈ ˆ K , where ˆ K = { ( sx + (1 − s ) x , y ) : ( x , y ) , ( x , y ) ∈ K and s ∈ [0 , } . (ii) K is linearly stable.Then there is a minimal flow ( ˜ Y , R ) and flow homomorphisms p ∗ : ( K, R ) → ( ˜ Y , R ) and ˜ p : ( ˜ Y , R ) → ( Y, R ) such that ( ˜ Y , R ) is distal, ˜ p is an N -1 extension for some integer N ≥ , p ∗ is an almost 1-1extension and p = ˜ p ◦ p ∗ , where p : K → Y denotes the natural projection. Moreover, if ( Y, R ) is almost periodic, then ( K, R ) is almost automorphic. emark . Under the assumption that u is C ,α in x , Shen and Yi [37, PartII, Theorem4.5] proved that a linearly stable minimal set must be almost automorphic. As we mentionedin the introduction, the approach in [37] is based on the idea and technique of construction ofinvariant measurable families of submanifolds in the so called Pesin’s Theory (see [27]). So, theregularity of α -H¨older continuity of the x -derivative of u cannot be dropped in [37]. With thehelp of the exponential separation on K × K with the “internal growth control” property alongthe principal bundles and a time-discretization technique, we succeed in reducing the regularity.In the following, we will focus on the proof of Theorem 3.1. Before we proceed further, wegive the following two crucial propositons: Proposition 3.3.
Let K be as in Theorem 3.1. Then there is a δ > such that if ( x , ˜ y ) , ( x , ˜ y ) ∈ K satisfies k x − x k < δ and u ( x , ˜ y, t ) , u ( x , ˜ y, t ) are not ordered (that is, u ( x , ˜ y, t ) − u ( x , ˜ y, t ) / ∈ ± C ) for all t ≥ , then k u ( x , ˜ y, t ) − u ( x , ˜ y, t ) k → , as t → + ∞ . ( . ) Proof.
We write K = { (( x , y ) , ( x , y )) : ( x , y ) , ( x , y ) ∈ K } , on which the metric is defined as d K ((( x , y ) , ( x , y )) , (( x ′ , y ′ ) , ( x ′ , y ′ ))) = q ( d K (( x , y ) , ( x ′ , y ′ ))) + ( d K (( x , y ) , ( x ′ , y ′ ))) , for (( x , y ) , ( x , y )) , (( x ′ , y ′ ) , ( x ′ , y ′ )) ∈ K , where d K (( x i , y ) , ( x ′ i , y ′ )) = p k x i − x ′ i k + d Y ( y, y ′ ) , i = 1 ,
2. Clearly, ( K , d K ) is a compact metric space. We define the continuous map F : K → K ; (( x , y ) , ( x , y )) F (( x , y ) , ( x , y )) , (Π( x , y, τ ) , Π( x , y, τ )) , for any (( x , y ) , ( x , y )) ∈ K . Since K admits a flow extension, F is a homeomorphism. Definethe bundle map T as a hybrid function as: T (( x ,y ) , ( x ,y )) = Z u x ( sx + (1 − s ) x , y, τ ) ds, (( x , y ) , ( x , y )) ∈ K . Recall that u x ( x, y, τ ) is strongly positive and continuous in ( x, y ) ∈ ˆ K . Then, for each(( x , y ) , ( x , y )) ∈ K , T (( x ,y ) , ( x ,y )) is a strongly positive linear operator on X and T (( x ,y ) , ( x ,y )) continuously depends on (( x , y ) , ( x , y )) ∈ K . Moreover, together with the fact that u x ( x, y, τ )is compact for all ( x, y ) ∈ ˆ K , we have for each (( x , y ) , ( x , y )) ∈ K , T (( x ,y ) , ( x ,y )) is a compactoperator on X . Furthermore, one can obtain T n (( x ,y ) , ( x ,y )) = u x ( x , y, nτ ) ( . )and T n (( x ,y ) , ( x ,y )) ( x − x ) = u ( x , y, nτ ) − u ( x , y, nτ ) , ( . )8or any (( x , y ) , ( x , y )) ∈ K and n ∈ N . Here, T n (( x ,y ) , ( x ,y )) = T F n − (( x ,y ) , ( x ,y )) ◦ · · · ◦ T F (( x ,y ) , ( x ,y )) ◦ T (( x ,y ) , ( x ,y )) . In fact, ( . ) is direct from the co-cycle property of Π. While,the definition of T (( x ,y ) , ( x ,y )) entails that T (( x ,y ) , ( x ,y )) ( x − x ) = u ( x , y, τ ) − u ( x , y, τ ) , which implies ( . ) inductively.Now, for the bundle map ( F , T ) on K × X , we utilize Proposition 2.5 to obtain the constants M >
0, 0 < γ < l (( x ,y ) , ( x ,y )) ∈ C ∗ s , v (( x ,y ) , ( x ,y )) ∈ Int C for any (( x , y ) , ( x , y )) ∈ K , such that properties (i)-(v) in Proposition 2.5 hold.Due to the assumption in this Proposition, ( . ) entails that T n (( x , ˜ y ) , ( x , ˜ y )) ( x − x ) = u ( x , ˜ y, nτ ) − u ( x , ˜ y, nτ ) / ∈ ± C, for any n ≥
1. Together with Proposition 2.5(iv), this implies that l (( x , ˜ y ) , ( x , ˜ y )) ( x − x ) = 0 . ( . )Choose an ǫ > γe ǫ τ <
1. Since K is linearly stable, for such ǫ , it follows fromProposition 2.3 that there is a C ǫ > k u x ( x, y, t ) k ≤ C ǫ e ǫ t , for all t ≥ x, y ) ∈ K. ( . )We further choose an ǫ > γe ǫ τ (1 + ǫ ) < . ( . )For such ǫ >
0, by Proposition 2.5(v), there exists a constant δ > . )holds. Let an integer n ≥ C ǫ M ( γe ǫ τ (1 + ǫ )) n < , ( . )where M is from the estimate ( . ) in Proposition 2.5(iii).Due to the continuity of F on K , one can find some δ > d K ( F i (( x , y ) , ( x , y )) , F i (( x ′ , y ′ ) , ( x ′ , y ′ ))) < δ , for any 0 ≤ i ≤ n , ( . )whenever (( x , y ) , ( x , y )) , (( x ′ , y ′ ) , ( x ′ , y ′ )) ∈ K with d K ((( x , y ) , ( x , y )) , (( x ′ , y ′ ) , ( x ′ , y ′ ))) <δ . Now, we claim that, for any ( x , ˜ y ) , ( x , ˜ y ) ∈ K with k x − x k < δ and u ( x , ˜ y, t ) , u ( x , ˜ y, t ) unordered for all t ≥ , k u ( x , ˜ y, iτ ) − u ( x , ˜ y, iτ ) k ≤ C ǫ M ( γe ǫ τ (1 + ǫ )) i k x − x k , for any i ≥ . ( . )9n order to prove the claim, we first prove ( . ) for 1 ≤ i ≤ n . By taking (( x , y ) , ( x , y )) =(( x , ˜ y ) , ( x , ˜ y )) and (( x ′ , y ′ ) , ( x ′ , y ′ )) = (( x , ˜ y ) , ( x , ˜ y )) in ( . ), we have d K ( F i (( x , ˜ y ) , ( x , ˜ y )) , F i (( x , ˜ y ) , ( x , ˜ y ))) < δ , ≤ i ≤ n . ( . )By virtue of ( . ) in Proposition 2.5(v), one has k T i (( x , ˜ y ) , ( x , ˜ y )) v (( x , ˜ y ) , ( x , ˜ y )) k ≤ (1 + ǫ ) i k T i (( x , ˜ y ) , ( x , ˜ y )) v (( x , ˜ y ) , ( x , ˜ y )) k , ≤ i ≤ n . ( . )Therefore, for 1 ≤ i ≤ n , k u ( x , ˜ y, iτ ) − u ( x , ˜ y, iτ ) k ( . ) = k T i (( x , ˜ y ) , ( x , ˜ y )) ( x − x ) k ( . )+( . ) ≤ M γ i k T i (( x , ˜ y ) , ( x , ˜ y )) v (( x , ˜ y ) , ( x , ˜ y )) k · k x − x k ( . ) ≤ M ( γ (1 + ǫ )) i k T i (( x , ˜ y ) , ( x , ˜ y )) v (( x , ˜ y ) , ( x , ˜ y )) k · k x − x k ( . ) ≤ M ( γ (1 + ǫ )) i k u x ( x , ˜ y, iτ ) k · k x − x k ( . ) ≤ C ǫ M ( γe ǫ τ (1 + ǫ )) i k x − x k . (3.12)Next, we will prove ( . ) for 1 ≤ i ≤ n . Choose i = n in (3.12). Then, together with ( . ), k u ( x , ˜ y, n τ ) − u ( x , ˜ y, n τ ) k < k x − x k < δ . Hence, d K ( F n (( x , ˜ y ) , ( x , ˜ y )) , F n (( x , ˜ y ) , ( x , ˜ y ))) < δ . So, we again take (( x , y ) , ( x , y )) = F n (( x , ˜ y ) , ( x , ˜ y ))and (( x ′ , y ′ ) , ( x ′ , y ′ )) = F n (( x , ˜ y ) , ( x , ˜ y )))in ( . ), and obtain d K ( F i (( x , ˜ y ) , ( x , ˜ y )) , F i (( x , ˜ y ) , ( x , ˜ y ))) < δ , for n ≤ i ≤ n . Togetherwith ( . ), we have d K ( F i (( x , ˜ y ) , ( x , ˜ y )) , F i (( x , ˜ y ) , ( x , ˜ y ))) < δ for any 0 ≤ i ≤ n . Again,by ( . ) in Proposition 2.5(v), one has k T i (( x , ˜ y ) , ( x , ˜ y )) v (( x , ˜ y ) , ( x , ˜ y )) k ≤ (1 + ǫ ) i k T i (( x , ˜ y ) , ( x , ˜ y )) v (( x , ˜ y ) , ( x , ˜ y )) k , ≤ i ≤ n . (3.13)Therefore, for 1 ≤ i ≤ n , k u ( x , ˜ y, iτ ) − u ( x , ˜ y, iτ ) k = k T i (( x , ˜ y ) , ( x , ˜ y )) ( x − x ) k≤ M γ i k T i (( x , ˜ y ) , ( x , ˜ y )) v (( x , ˜ y ) , ( x , ˜ y )) k · k x − x k ( . ) ≤ M ( γ (1 + ǫ )) i k T i (( x , ˜ y ) , ( x , ˜ y )) v (( x , ˜ y ) , ( x , ˜ y )) k · k x − x k≤ M ( γ (1 + ǫ )) i k u x ( x , ˜ y, iτ ) k · k x − x k≤ C ǫ M ( γe ǫ τ (1 + ǫ )) i k x − x k . . ) is satisfied for all i ≥
1. Thus,we have proved the claim.By virtue of the claim, we obtain k u ( x , ˜ y, iτ ) − u ( x , ˜ y, iτ ) k →
0, as i → + ∞ . Now, weshow that k u ( x , ˜ y, t ) − u ( x , ˜ y, t ) k → t → + ∞ . To this end, for any ǫ ′ >
0, it followsfrom the uniform continuity of u on K × [0 , τ ] that, there exists δ ′ > k u ( x , y , t ) − u ( x , y , t ) k < ǫ ′ , for ( x , y , t ) , ( x , y , t ) ∈ K × [0 , τ ] with d K × [0 ,τ ] (( x , y , t ) , ( x , y , t )) <δ ′ . For any t >
0, write t = lτ + α , l ∈ N and α ∈ [0 , τ ]. It follows from the claim that, thereexists an integer N > k u ( x , ˜ y, iτ ) − u ( x , ˜ y, iτ ) k < δ ′ , for any i ≥ N . Therefore, k u ( x , ˜ y, t ) − u ( x , ˜ y, t ) k = k u ( u ( x , ˜ y, lτ ) , ˜ y · lτ, α ) − u ( u ( x , ˜ y, lτ ) , ˜ y · lτ, α ) k < ǫ ′ , for any t ≥ N τ .Thus, we have obtained ( . ), which completes the proof. Proposition 3.4.
Let K be as in Theorem 3.1. If ( x , ˜ y ) , ( x , ˜ y ) ∈ K satisfying x − x / ∈ ± C ,then the pair ( x , ˜ y ) and ( x , ˜ y ) are negatively distal.Proof. Suppose on the contrary that there exists a sequence t n → −∞ such that k u ( x , ˜ y, t n ) − u ( x , ˜ y, t n ) k → t n → −∞ . Then, we will obtain a contradiction by showing that x = x .To this purpose, let ǫ > . ). For such ǫ >
0, let δ > F , T ) on K × X ).Now, we claim that, for any < δ < δ , there exists t δ ∈ [ − τ, , such that k u ( x , ˜ y, t δ ) − u ( x , ˜ y, t δ ) k < δ. (3.14)Before we prove the claim, we show that how it implies that x = x . Suppose that x = x .Let 0 < ǫ ′ = k x − x k . Noticing that u is uniformly continuous on K × [0 , τ ], there exists δ ′ > k u ( x , y , t ) − u ( x , y , t ) k < ǫ ′ , whenever ( x , y , t ) , ( x , y , t ) ∈ K × [0 , τ ] with d K × [0 ,τ ] (( x , y , t ) , ( x , y , t )) < δ ′ . For any 0 < δ < min { δ ′ , δ } , it follows from the claim thatthere exists t δ ∈ [ − τ, k u ( x , ˜ y, t δ ) − u ( x , ˜ y, t δ ) k < δ . This implies k x − x k = k u ( u ( x , ˜ y, t δ ) , y · t δ , − t δ ) − u ( u ( x , ˜ y, t δ ) , y · t δ , − t δ ) k < ǫ ′ , which contradicts k x − x k = ǫ ′ .Now, we focus on the proof of the claim. By virtue of Proposition 2.5(i), the set V K , { v (( x ,y ) , ( x ,y )) : (( x , y ) , ( x , y )) ∈ K } ⊂ Int C is compact. So, there exists ǫ > { v ∈ X : d ( v, V K ) < ǫ } ⊂ Int C, (3.15)where d ( v, V K ) = inf w ∈ V K d ( v, w ). We decompose u ( x , ˜ y, t n ) − u ( x , ˜ y, t n ) as u ( x , ˜ y, t n ) − u ( x , ˜ y, t n ) = c n v (Π( x , ˜ y,t n ) , Π( x , ˜ y,t n )) + w n , (3.16)where c n = l (Π( x , ˜ y,tn ) , Π( x , ˜ y,tn )) ( u ( x , ˜ y,t n ) − u ( x , ˜ y,t n )) l (Π( x , ˜ y,tn ) , Π( x , ˜ y,tn )) ( v (Π( x , ˜ y,tn ) , Π( x , ˜ y,tn )) ) and l (Π( x , ˜ y,t n ) , Π( x , ˜ y,t n )) ( w n ) = 0. For each n ≥
1, we write − t n = k n τ + α n with k n ∈ N , α n ∈ [0 , τ ), n = 1 , , · · · . We assert that | c n | ≤ ǫ − M γ k n k w n k , for n ≥ , (3.17)11here M and γ are from ( . ) in Proposition 2.5(iii). In fact, if c n = 0, then we’ve done. If c n = 0, then by ( . ) and (3.16), we write u ( x , ˜ y, − α n ) − u ( x , ˜ y, − α n ) = T k n (Π( x , ˜ y,t n ) , Π( x , ˜ y,t n )) ( u ( x , ˜ y, t n ) − u ( x , ˜ y, t n ))= T k n (Π( x , ˜ y,t n ) , Π( x , ˜ y,t n )) ( c n v (Π( x , ˜ y,t n ) , Π( x , ˜ y,t n )) + w n ) . Noticing that x − x / ∈ ± C , one has u ( x , ˜ y, − α n ) − u ( x , ˜ y, − α n ) / ∈ ± C . So, T k n (Π( x , ˜ y,t n ) , Π( x , ˜ y,t n )) v (Π( x , ˜ y,t n ) , Π( x , ˜ y,t n )) k T k n (Π( x , ˜ y,t n ) , Π( x , ˜ y,t n )) v (Π( x , ˜ y,t n ) , Π( x , ˜ y,t n )) k + T k n (Π( x , ˜ y,t n ) , Π( x , ˜ y,t n )) w n c n k T k n (Π( x , ˜ y,t n ) , Π( x , ˜ y,t n )) v (Π( x , ˜ y,t n ) , Π( x , ˜ y,t n )) k / ∈ ± C. Since T k n (Π( x , ˜ y,t n ) , Π( x , ˜ y,t n )) v (Π( x , ˜ y,t n ) , Π( x , ˜ y,t n )) ∈ V K , (3.15) implies that | c n | ≤ k T k n (Π( x , ˜ y,t n ) , Π( x , ˜ y,t n )) w n k ǫ k T k n (Π( x , ˜ y,t n ) , Π( x , ˜ y,t n )) v (Π( x , ˜ y,t n ) , Π( x , ˜ y,t n )) k . Together with ( . ) in Proposition 2.5(iii), we obtain that | c n | ≤ ǫ − M γ k n k w n k for n ≥ t n → −∞ , we can choose an integer N > M γ k n < ǫ for any n ≥ N . By(3.16)-(3.17), k u ( x , ˜ y, t n ) − u ( x , ˜ y, t n ) k > k w n k , for n ≥ N . (3.18)Fix an integer n ≥ { ǫ − , } C ǫ M ( γe ǫ τ (1 + ǫ )) n < . (3.19)For any δ ∈ (0 , δ ) in (3.14), due to the continuity of F on K , one can choose δ > d K ( F i (( x , y ) , ( x , y )) , F i (( x ′ , y ′ ) , ( x ′ , y ′ ))) < δ, for 0 ≤ i ≤ n , (3.20)whenever (( x , y ) , ( x , y )) , (( x ′ , y ′ ) , ( x ′ , y ′ )) ∈ K with d K ((( x , y ) , ( x , y )) , (( x ′ , y ′ ) , ( x ′ , y ′ ))) <δ . Recall that k u ( x , ˜ y, t n ) − u ( x , ˜ y, t n ) k → n → + ∞ . Then there exists an integer N > t n < − n τ and k u ( x , ˜ y, t n ) − u ( x , ˜ y, t n ) k < δ , for n ≥ N . (3.21)In other words, d K ((Π( x , ˜ y, t n ) , Π( x , ˜ y, t n )) , (Π( x , ˜ y, t n ) , Π( x , ˜ y, t n ))) < δ , for n ≥ N . So, by taking in (3.20) (( x , y ) , ( x , y )) = (Π( x , ˜ y, t n ) , Π( x , ˜ y, t n )) and (( x ′ , y ′ ) , ( x ′ , y ′ )) =(Π( x , ˜ y, t n ) , Π( x , ˜ y, t n )) for some n > max { N , N } , we have d K ( F i (Π( x , ˜ y, t n ) , Π( x , ˜ y, t n )) , F i (Π( x , ˜ y, t n ) , Π( x , ˜ y, t n ))) < δ, ≤ i ≤ n . (3.22)12ogether with ( . ) in Proposition 2.5(v), we obtain k T i (Π( x , ˜ y,t n ) , Π( x , ˜ y,t n )) v (Π( x , ˜ y,t n ) , Π( x , ˜ y,t n )) k ≤ (1 + ǫ ) i k T i (Π( x , ˜ y,t n ) , Π( x , ˜ y,t n )) v (Π( x , ˜ y,t n ) , Π( x , ˜ y,t n )) k , (3.23)for 1 ≤ i ≤ n . Therefore, for 1 ≤ i ≤ n , k u ( x , ˜ y, t n + iτ ) − u ( x , ˜ y, t n + iτ ) k ( . ) = k T i (Π( x , ˜ y,t n ) , Π( x , ˜ y,t n )) ( u ( x , ˜ y, t n ) − u ( x , ˜ y, t n )) k ( . ) = k T i (Π( x , ˜ y,t n ) , Π( x , ˜ y,t n )) ( c n v (Π( x , ˜ y,t n ) , Π( x , ˜ y,t n )) + w n ) k ( . ) ≤ ( | c n | + M γ i k w n k ) · k T i (Π( x , ˜ y,t n ) , Π( x , ˜ y,t n )) v (Π( x , ˜ y,t n ) , Π( x , ˜ y,t n )) k ( . ) ≤ ( ǫ − M γ k n + M γ i ) k w n k · k T i (Π( x , ˜ y,t n ) , Π( x , ˜ y,t n )) v (Π( x , ˜ y,t n ) , Π( x , ˜ y,t n )) k ( . ) ≤ ( ǫ − M γ k n + M γ i ) k w n k · (1 + ǫ ) i k T i (Π( x , ˜ y,t n ) , Π( x , ˜ y,t n )) v (Π( x , ˜ y,t n ) , Π( x , ˜ y,t n )) k ( . ) ≤ ( ǫ − M γ k n + M γ i ) k w n k · (1 + ǫ ) i k u x (Π( x , ˜ y, t n ) , iτ ) k ( . ) ≤ ǫ − M γ k n + M γ i )(1 + ǫ ) i k u x (Π( x , ˜ y, t n ) , iτ ) k · k u ( x , ˜ y, t n ) − u ( x , ˜ y, t n ) k ( . ) ≤ h ǫ − C ǫ M γ k n ( e ǫ τ (1 + ǫ )) i + 2 C ǫ M ( γe ǫ τ (1 + ǫ )) i i · k u ( x , ˜ y, t n ) − u ( x , ˜ y, t n ) k . (3.24)Choose i = n in (3.24). Then by (3.19), (3.21), we have k u ( x , ˜ y, t n + n τ ) − u ( x , ˜ y, t n + n τ ) k < δ , and hence, d K ( F n (Π( x , ˜ y, t n ) , Π( x , ˜ y, t n )) , F n (Π( x , ˜ y, t n ) , Π( x , ˜ y, t n ))) < δ < δ , by which we take in (3.20)(( x , y ) , ( x , y )) = F n (Π( x , ˜ y, t n ) , Π( x , ˜ y, t n )) , (( x ′ , y ′ ) , ( x ′ , y ′ )) = F n (Π( x , ˜ y, t n ) , Π( x , ˜ y, t n )) , and obtain d K ( F i (Π( x , ˜ y, t n ) , Π( x , ˜ y, t n )) , F i (Π( x , ˜ y, t n ) , Π( x , ˜ y, t n ))) < δ, for n ≤ i ≤ n . So, together with (3.22), we obtain d K ( F i (Π( x , ˜ y, t n ) , Π( x , ˜ y, t n )) , F i (Π( x , ˜ y, t n ) , Π( x , ˜ y, t n ))) < δ, ≤ i ≤ n . . ) in Proposition 2.5(v), one has k T i (Π( x , ˜ y,t n ) , Π( x , ˜ y,t n )) v (Π( x , ˜ y,t n ) , Π( x , ˜ y,t n )) k≤ (1 + ǫ ) i k T i (Π( x , ˜ y,t n ) , Π( x , ˜ y,t n )) v (Π( x , ˜ y,t n ) , Π( x , ˜ y,t n )) k , (3.25)for any 1 ≤ i ≤ n . Therefore, similarly as the estimates in (3.24), we obtain from (3.25) that k u ( x , ˜ y, t n + iτ ) − u ( x , ˜ y, t n + iτ ) k≤ h ǫ − C ǫ M γ k n ( e ǫ τ (1 + ǫ )) i + 2 C ǫ M ( γe ǫ τ (1 + ǫ )) i i · k u ( x , ˜ y, t n ) − u ( x , ˜ y, t n ) k , (3.26)for any 1 ≤ i ≤ n .Therefore, by repeating the same arguments, we obtain that k u ( x , ˜ y, t n + iτ ) − u ( x , ˜ y, t n + iτ ) k≤ h ǫ − C ǫ M γ k n ( e ǫ τ (1 + ǫ )) i + 2 C ǫ M ( γe ǫ τ (1 + ǫ )) i i · k u ( x , ˜ y, t n ) − u ( x , ˜ y, t n ) k , (3.27)for all 1 ≤ i ≤ l n · n , where the integer l n comes from the expression − t n = l n · n τ + β n , with β n ∈ [0 , n τ ). Clearly, k n ≥ l n · n for n ≥ i = l n · n in (3.27). Note that k n ≥ l n · n , again by (3.19), (3.21), we have k u ( x , ˜ y, t n + l n · n τ ) − u ( x , ˜ y, t n + l n · n τ ) k≤ (2 ǫ − C ǫ M γ k n ( e ǫ τ (1 + ǫ )) l n · n + 2 C ǫ M ( γe ǫ τ (1 + ǫ )) l n · n ) δ ≤ (2 ǫ − C ǫ M γ k n ( e ǫ τ (1 + ǫ )) k n + 2 C ǫ M ( γe ǫ τ (1 + ǫ )) n ) δ ≤ (2 ǫ − C ǫ M ( γe ǫ τ (1 + ǫ )) n + 2 C ǫ M ( γe ǫ τ (1 + ǫ )) n ) δ < δ , and hence, d K ( F l n · n (Π( x , ˜ y, t n ) , Π( x , ˜ y, t n )) , F l n · n (Π( x , ˜ y, t n ) , Π( x , ˜ y, t n ))) < δ < δ . Finally, again, we take in (3.20)(( x , y ) , ( x , y )) = F l n · n (Π( x , ˜ y, t n ) , Π( x , ˜ y, t n )) , (( x ′ , y ′ ) , ( x ′ , y ′ )) = F l n · n (Π( x , ˜ y, t n ) , Π( x , ˜ y, t n )) , and obtain d K ( F i (Π( x , ˜ y, t n ) , Π( x , ˜ y, t n )) , F i (Π( x , ˜ y, t n ) , Π( x , ˜ y, t n ))) < δ, (3.28)for any l n · n ≤ i ≤ ( l n + 1) · n . In particular, one find an integer i satisfying l n · n ≤ i ≤ ( l n + 1) · n such that t n + i τ ∈ [ − τ, t δ = t n + i τ ∈ [ − τ, F i (Π( x , ˜ y, t n ) , Π( x j , ˜ y, t n )) = (Π( x , ˜ y, t δ ) , Π( x j , ˜ y, t δ )), j = 3 ,
4. Then (3.28) directly impliesthat k u ( x , ˜ y, t δ ) − u ( x , ˜ y, t δ ) k < δ . Thus, we have proved the claim, which completes ourproof. 14 emark . Proposition 3.3 and Proposition 3.4 play very crucial roles in proving our mainTheorem. For C ,α -smooth skew-product semiflows, these two Propositions were proved in Shenand Yi [37, PartII, Lemma 4.6]. Proposition 3.6.
Let K be as in Theorem 3.1. Then (a) the proximal relation P ( K ) on K is an equivalence relation; (b) P ( K ) = O ( K ) .Proof. The proof of this proposition is analogous to that in [37, PartII, Lemma 4.7 and Lemma4.8]. We give the detail for the sake of completeness.(a) Since ( Y, R ) is distal, P ( K ) = { (( x , y ) , ( x , y )) ∈ K | ( x , y ) , ( x , y ) are proximal } . Weonly need to check the transitivity. Let (( x , y ) , ( x , y )) , (( x , y ) , ( x , y )) ∈ P ( K ). One of thefollowing alternatives must occur:(i). There is a t ≥ x , y, t ) , Π( x , y, t )) ∈ O ( K ) and (Π( x , y, t ) , Π( x , y, t )) ∈ O ( K ).(ii). There is a t ≥ x , y, t ) , Π( x , y, t )) ∈ O ( K ) but (Π( x , y, t ) , Π( x , y, t )) / ∈ O ( K ) for all t ≥ t ≥ x , y, t ) , Π( x , y, t )) ∈ O ( K ) but (Π( x , y, t ) , Π( x , y, t )) / ∈ O ( K ) for all t ≥ t ≥
0, (Π( x , y, t ) , Π( x , y, t )) / ∈ O ( K ), (Π( x , y, t ) , Π( x , y, t )) / ∈ O ( K ).If (i) holds, then denote ( x ∗ i , y ∗ ) = Π( x i , y, t ), i = 1 , ,
3. Let y ∈ Y in Proposition2.4(i). Since ( Y, R ) is minimal, there exists a sequence { t n } such that y ∗ · t n → y . Bytaking a subsequence, if necessary, we assume that Π( x ∗ , y ∗ , t n ) → ( ˆ x , y ), Π( x ∗ , y ∗ , t n ) → ( ˆ x , y ). Since O ( K ) is a closed relation, ( ˆ x , y ) and ( ˆ x , y ) are ordered. By Proposition2.4(i), one has ( ˆ x , y ) = ( ˆ x , y ). Hence, d K (Π( x ∗ , y ∗ , t n ) , Π( x ∗ , y ∗ , t n )) →
0. Similarly, bytaking a subsequence if necessary, we have d K (Π( x ∗ , y ∗ , t n ) , Π( x ∗ , y ∗ , t n )) →
0. Consequently, d K (Π( x ∗ , y ∗ , t n ) , Π( x ∗ , y ∗ , t n )) →
0, that is, d K (Π( x , y, t n + t ) , Π( x , y, t n + t )) →
0. So,(( x , y ) , ( x , y )) ∈ P ( K ).If (ii) holds, then take y ∈ Y in Proposition 2.4(i). Let ( x , y ) ∈ K , since K is minimal,there exists a sequence t n → ∞ such that y · t n → y . By repeating the same argument in (i),there is a subsequence, still denoted by t n , such that d K (Π( x , y, t n ) , Π( x , y, t n )) → x , y ) , ( x , y )) ∈ P ( K ), there exists t ∈ R such that d K (Π( x , y, t ) , Π( x , y, t )) < δ ( δ is in Proposition 3.3). Then it follows from Proposition 3.3 that k u ( x , y, t ) − u ( x , y, t ) k →
0, as t → + ∞ . Therefore, we have k u ( x , y, t n ) − u ( x , y, t n ) k →
0, as t → + ∞ , that is,(( x , y ) , ( x , y )) ∈ P ( K ).The proof of (iii) is analogous, we omit it.Finally, if (iv) holds, then (Π( x , y, t ) , Π( x , y, t )) / ∈ O ( K ) for all t ∈ R . Since (( x , y ) , ( x , y )) ∈ P ( K ), there exists ζ ∈ R such that k u ( x , y, ζ ) − u ( x , y, ζ ) k is sufficiently small. It thenfollows from Proposition 3.3 that k u ( x , y, t ) − u ( x , y, t ) k →
0, as t → + ∞ . Similarly, we15btain k u ( x , y, t ) − u ( x , y, t ) k →
0, as t → + ∞ . Therefore, k u ( x , y, t ) − u ( x , y, t ) k →
0, thatis, (( x , y ) , ( x , y )) ∈ P ( K ).(b) By Proposition 2.4(ii), O ( K ) ⊂ P ( K ). Now, we prove P ( K ) ⊂ O ( K ). Suppose that(( x , y ) , ( x , y )) ∈ P ( K ) \ O ( K ). Then Proposition 3.4 implies that ( x , y ) , ( x , y ) are negativelydistal. Therefore, ( x , y ) , ( x , y ) are both proximal and negatively distal. But this is impossibleby Proposition 2.1 and Proposition 3.6(a).Now, we are ready to prove Theorem 3.1. Proof of Theorem 3.1.
By Proposition 3.6(b), P ( K ) = O ( K ) are invariant and closed. Let˜ Y = K/P ( K ) = K/O ( K ). Then, ( K, R ) induces a flow ( ˜ Y , R ) by the invariance of P ( K ).Clearly, ( ˜ Y , R ) is distal. Let p : K → Y ; ( x, y ) y be the natural projection. Denote by ˜ p : ˜ Y → Y ; [( x, y )] y the projection induced by p ; and denote by p ∗ : K → ˜ Y = K/P ( K ) the naturalprojection to ˜ Y as p ∗ ( x, y ) = [( x, y )] , ( x, y ) ∈ K . So, p = ˜ p ◦ p ∗ . By the closeness of P ( K ), ˜ p and p ∗ are continuous. Let Y be the residual set given by Proposition 2.4(i) and fix a y ∈ Y . SinceProposition 2.4(i) implies no two points on p − ( y ) are ordered, card ( p − ( y )) = card (˜ p − ( y )).Now, if card (˜ p − ( y )) = ∞ , then there is an accumulation point ( x ∗ , y ) ∈ p − ( y ). Choose a( x , y ) ∈ p − ( y ) such that ( x , y ) = ( x ∗ , y ) and k x − x ∗ k < δ , where δ is in Proposition 3.3.Since ( x , y ) , ( x ∗ , y ) are not ordered, Proposition 2.4(i) implies that u ( x , y , t ) , u ( x ∗ , y , t ) arenot ordered for all t ≥
0. Hence, by Proposition 3.3, k u ( x , y , t ) − u ( x ∗ , y , t ) k → t → + ∞ ,which implies that ( x , y ) and ( x ∗ , y ) are proximal, a contradiction to Proposition 3.6(b).Thus, there is an integer N ≥ card (˜ p − ( y )) = N . By Proposition 2.2, ˜ p is an N -1extension.Next, for any y ∈ Y and any [( x, y )] ∈ ˜ p − ( y ), since ( x ′ , y ) and ( x, y ) are not ordered for any( x ′ , y ) = ( x, y ), one has p ∗− [( x, y )] = { ( x, y ) } , that is, card ( p ∗− [( x, y )]) = 1. Since Y is residualin Y , one has ˜ Y = { [( x, y )] ∈ ˜ p − ( y ) | y ∈ Y } is residual in ˜ Y . Therefore, p ∗ : ( K, R ) → ( ˜ Y , R )is an almost 1-1 extension.Now, if ( Y, R ) is almost periodic, then by Proposition 2.2, ( ˜ Y , R ) is also almost periodic, and( K, R ) is almost automorphic, since p ∗ is an almost 1-1 extension. (cid:3) Before ending this paper, we give the following two additional remarks.
Remark . Under the C -smoothness assumption, we show in Theorem 3.1 the almost auto-morphy of linearly stable minimal set for strongly monotone skew-product semiflows. The resultwas obtained by Shen and Yi [37, PartII, Theorem 4.5] for C ,α -systems. As a consequence, onecan apply our theoretical result (Theorem 3.1) to obtain all the same results in [37, Part III],under the lower C -regularity (instead of C ,α ), for time-almost periodic differential equations,including ODEs, parabolic equations and delay equations. Remark . It deserves to point out that, under C ,α -smoothness, Novo et al. [26] showed thatassumptions (i)-(ii) in Theorem 3.1 imply that K admits a flow extension automatically. It isan interesting question whether it remains true under the weaker C -smoothness hypothesis.16 eferences [1] S. Aubry, The devil’s staircase transformation in incommensurate lattices, The Riemannproblem, complete integrability and arithmetic applications, Lecture Notes in Math., 925Springer Verlag, 1982, 221-245.[2] S. Bochner, Curvature and Betti numbers in real and complex vector bundles, Univ. ePolitec. Torino. Rend. Sem. Mat. 15(1955/56), 225-253.[3] I. Chueshov, Monotone Random Systems-Theory and Applications, Lecture Notes in Math-ematics, Vol. 1779, Springer-Verlag, Berlin, 2002.[4] K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin-Heidelberg-New York1985.[5] A. Denjoy, Sur les courbes d´efinies par les ´equations diff´erentielles `a la surface du tore, J.de Math´e Pures et Appli 11(1932), 333-375.[6] T. Downarowicz and Y. Lacroix, Almost 1-1 extensions of Furstenberg-Weiss type andapplications to Toeplitz flows, Studia Math. 130(1998), 149-170.[7] H. Furstenberg and B. Weiss, On almost 1-1 extensions, Israel J. Math. 65(1989), 311-322.[8] M. W. Hirsch, The dynamical systems approach to differential equations, Bull. Amer. Math.Soc. 11(1984), 1-64.[9] M. W. Hirsch, Stability and convergence in strongly monotone dynamical systems, J. ReineAngew. Math. 383(1988), 1-53.[10] M. W. Hirsch, Systems of differential equations which are competitive or cooperative I:limit sets, SIAM J. Appl. Math. 13(1982), 167-179.[11] M. W. Hirsch, Systems of differential equations which are competitive or cooperative II:convergence almost everywhere, SIAM J. Math. Anal. 16(1985), 423-439.[12] M. W. Hirsch and H. L. Smith, Monotone Dynamical Systems, Handbook of DifferentialEquations: Ordinary Differential Equations, Vol. 2, Elsevier, Amsterdam, 2005.[13] W. Huang and Y. Yi, Almost periodically forced circle flows, J. Funct. Anal. 257(2009),832-902.[14] R. A. Johnson, On a floquet theory for almost periodic two-dimensional linear systems, J.Differ. Equ. 37(1980), 184-205.[15] R. A. Johnson, A linear almost periodic equation with an almost automorphic solution,Proc. Amer. Math. Soc. 82(1981), 199-205.1716] R. A. Johnson, Bounded solutions of scalar almost periodic linear equations, Illinois J.Math. 25(1981), 632-643.[17] R. A. Johnson, On almost-periodic linear differential systems of Millionˇsˇcikov and Vinograd,J. Math. Anal. Appl. 85(1982), 452-460.[18] N. G. Markley and M. E. Paul, Almost automorphic symbolic minimal sets without uniqueergodicity, Israel J. Math. 34(1979), 259-272(1980).[19] J. Mather, Existence of quasi-periodic orbits for twist homeomorphisms of the annulus,Topology 21(1982), 457-467.[20] H. Matano, Strongly order-preserving local semi-dynamical systems-Theory and Applica-tions, Res. Notes in Math., 141, Semigroups, Theory and Applications (H. Brezis, M. G.Crandall, and F. Kappel, eds.), Vol. 1, Longman Scientific and Technical, London, 1986,178-185.[21] J. Mierczy´nski, Flows on ordered bundles, unpublished.[22] J. Mierczy´nski, The C property of carrying simplices for a class of competitive systems ofODEs, J. Differ. Equ. 111(1994), 385-409.[23] J. Mierczy´nski and W. Shen, Spectral Theory for Random and Nonautonomous ParabolicEquations and Applications, Chapman Hall/CRC Monogr. Surv. Pure Appl. Math., Vol.139, CRC Press, Boca Raton, FL, 2008.[24] S. Novo, C. N´u˜ n ez and R. Obaya, Almost automorphic and almost periodic dynamicsfor quasimonotone non-autonomous functional differential equations, J. Dyn. Differ. Equ.17(2005) 589-619.[25] S. Novo, and R. Obaya, Non-autonomous functional differential equations and applications,Stability and bifurcation theory for non-autonomous differential equations, Lecture Notesin Math., Vol. 2065, Springer, Heidelberg, 2013, 185-263.[26] S. Novo, R. Obaya and A. M. Sanz, Topological dynamics for monotone skew-productsemiflows with applications, J. Dyn. Differ. Equ. 25(2013), 1201-1231.[27] Y. B. Pesin, Families of invariant manifolds corresponding to nonzero characteristic expo-nents, Math. USSR, Izv., 10(1976), 1261-1305.[28] P. Pol´aˇcik, Convergence in smooth strongly monotone flows defined by semilinear parabolicequations, J. Differ. Equ. 79(1989), 89-110.[29] P. Pol´aˇcik, Parabolic equations: asymptotic behavior and dynamics on invariant manifolds,Handbook on Dynamical Systems, Vol. 2, Elsevier, Amsterdam, 2002, 835-883.1830] P. Pol´aˇcik and I. Tereˇsˇc´ak, Convergence to cycles as a typical asymptotic behavior in smoothstrongly monotone discrete-time dynamical systems, Arch. Ration. Mech. Anal., 116(1992),339-360.[31] P. Pol´aˇcik and I. Tereˇsˇc´ak, Exponential separation and invariant bundles for maps in orderedBanach spaces with applications to parabolic equations, J. Dyn. Differ. Equ. 5(1993), 279-303.[32] R. J. Sacker and G. R. Sell, Lifting properties in skew-product flows with applications todifferential equations, Mem. Amer. Math. Soc. 11(1977).[33] W. Shen and Y. Yi, Dynamics of almost periodic scalar parabolic equations, J. Differ. Equ.122(1995), 114-136.[34] W. Shen and Y. Yi, Asymptotic almost periodicity of scalar parabolic equations with almostperiodic time dependence, J. Differ. Equ. 122(1995), 373-397.[35] W. Shen and Y. Yi, On minimal sets of scalar parabolic equations with skew-productstructure, Trans. Amer. Math. Soc. 347(1995), 4413-4431.[36] W. Shen and Y. Yi, Ergodicity of minimal sets in scalar parabolic equations, J. Dyn. Differ.Equ. 8(1996), 299-323.[37] W. Shen and Y. Yi, Almost automorphic and almost periodic dynamics in skew-productsemiflows, Mem. Amer. Math. Soc. 136(1998), No. 647.[38] W. Shen, Y. Wang and D. Zhou, Structure of ω -limit sets for almost-periodic parabolicequations on S with reflection symmetry, J. Differ. Equ. 261(2016), 6633-6667.[39] W. Shen, Y. Wang and D. Zhou, Long-time behavior of almost periodically forced parabolicequations on the circle, J. Differ. Equ. 266(2019), 1377-1413.[40] W. Shen, Y. Wang and D. Zhou, Almost automorphically and almost periodically forcedcircle flows of almost periodic parabolic equations on S , J. Dyn. Differ. Equ. 32(2020),1687-1729.[41] H. L. Smith, Monotone Dynamical Systems, an Introduction to the Theory of Competitiveand Cooperative systems, Math. Surveys and Monographs, Vol. 41, Amer. Math. Soc.,Providence, RI, 1995.[42] H. L. Smith, Monotone dynamical systems: reflections on new advances and applications,Discrete Contin. Dyn. Syst. 37(2017), 485-504.[43] H. L. Smith and H. Thieme, Convergence for strongly order-preserving semiflows, SIAM J.Math. Anal. 22(1991), 1081-1101. 1944] I. Tereˇsˇc´ak, Dynamics of C smooth strongly monotone discrete-time dynamical systmes,preprint, Comenius University, Bratislava, 1994.[45] R. Terras, On almost periodic and almost automorphic differences of functions, Duke Math.J. 40(1973), 81-91.[46] R. Terras, Almost automorphic functions on topological groups, Indiana U. Math. J.21(1971/72), 759-773.[47] W. A. Veech, Almost automorphy and a theorem of Loomis, Arch. Math. (Basel) 18(1967),267-270.[48] W. A. Veech, Almost automorphic functions on groups, Amer. J. Math. 87(1965), 719-751.[49] W. A. Veech, On a theorem of Bochner, Ann. Math. 86(1967), 117-137.[50] W. A. Veech, Point-distal flows, Amer. J. Math. 92(1970), 205-242.[51] Y. Yi. On almost automorphic oscillations, Fields Inst. Commun. 42(2004), 75-99.[52] Y. Wang and J. Yao, Dynamics alternatives and generic convergence for C1