Dispersive Semiflows on Fiber Bundles

This paper studies dispersiveness of semiflows on fiber bundles. The main result says that a right invariant semiflow on a fiber bundle is dispersive on the base space if and only if there is no almost periodic point and the semiflow is dispersive on the total space. A special result states that linear semiflows on vector bundles are not dispersive.


Introduction
A right invariant semiflow µ t on the total space Q of a principal bundle Q → B induces a semiflow on the base space B and a semiflow on the associated bundle E = Q× G F .Moreover, the semiflow µ t generates a semigroup action on the typical fiber F of the associated bundle E. The relationship between these dynamics has been extensively studied in the literature.Morse decompositions and chain recurrence for semiflows on fiber bundles were investigated in [4], [5], [6], [7], [9], [10], and [13].Lyapunov stability and attraction for semiflows on fiber bundles were studied in [11].A generalization of Lyapunov stability and attraction for semigroup actions on fiber bundles was recently published ( [3]).The present paper contributes to the studies of recursiveness and dispersiveness for semiflows on fiber bundles.
A topological method of studying dynamical concepts on nonmetric spaces was formulated by Patrão and San Martin [8].They introduced the notion of admissible family of open coverings to study chain recurrence and Morse decomposition of semiflows on topological spaces.The concept of admissible family of open coverings was reformulated in [12] by adding the direct set property.A topological space that admits an admissible family of open covering is called an admissible space.Recently, one discoveries that a topological space is uniformizable if and only if it is admissible ( [1]).In the case of a locally trivial bundle, an admissible family of open coverings of the total space can be constructed by means of the admissible family of open coverings of the base space, in such a way that the projection become uniformly continuous ( [9]).
If π : Q → B is a principal bundle with structure group G and µ is a right invariant semiflow on Q, the collection of the limit sets and the prolongational limit sets is G-invariant (Proposition 3.3).It follows that every point in the fiber qG is forward Poisson stable (nonwandering) whenever q is forward Poisson stable (nonwandering).Therefore, if Q is a homogeneous space of G, all the points in Q are forward Poisson stable (nonwandering) if there is a forward Poisson stable point (nonwandering point).In particular, if Q is a compact homogeneous space of G, then every point in Q is Poisson stable.On the other hand, the semiflow is dispersive if J (q) = ∅ for some q ∈ Q, where J (q) means the prolongational limit set of q.By speaking on the associated bundle E = Q × G F , a forward Poisson stable point v in the fiber F induces forward Poisson stable points [q, v] in E and π (q) in B (Theorem 3.8).This fact implies that linear semiflows on vector bundles are not dispersive.Similarly, a nonwandering point v in F induces nonwandering points [q, v] in E and π (q) in B. If the semiflow induced on the base space B is dispersive, then all the semiflows on Q, E, and F are dispersive.On the other hand, if there is no almost periodic point in B and the semiflow is dispersive on E then the semiflow on B is dispersive (Theorem 3.18).

Recursiveness and dispersiveness
This section contains the basic definitions and properties of recursiveness and dispersiveness for semiflows on admissible spaces.
Let µ : R + × M → M be a continuous semiflow on the Hausdorff space M .We denote by µ t : M → M the map defined by µ t (x) = µ (t, x).For t ∈ R + and X ⊂ M we define the sets The forward and backward orbits of X are the sets µ + (X) = µ + 0 (X) and µ − (X) = µ − 0 (X), respectively.We say that X is forward invariant if µ + (X) ⊂ X; it is backward invariant if µ − (X) ⊂ X; and it is invariant if it is forward and backward invariant.The ω-limit set of X is defined as and the ω * -limit set of X is defined by In general, ω (X) is forward invariant and cls (µ A set N ⊂ M is a (forward) minimal set if N is nonempty, closed, forward invariant, and N has no proper subset with these properties.It is well-known that a set N is a minimal set if and only if cls µ + t (x) = N for all x ∈ N and t ∈ R + .Moreover, if X ⊂ M is a compact forward invariant set then there is a compact minimal set in X.A point x ∈ M is said to be almost periodic if cls (µ + (x)) is a minimal set.In particular, singular points and periodic points are almost periodic.
Note that the singular points, the periodic trajectories, and the almost periodic trajectories of the semiflow lie in the limit sets.This property of the limit sets reveals their aspect of stability, which is stated as follows.
Definition 2.1.A nonempty subset X ⊂ M is said to be forward recursive with respect to the nonempty subset Y ⊂ M if for every t > 0, µ + t (Y ) ∩ X = ∅; the set X is said to be backward recursive with respect to the set Y if for every t > 0, µ − t (Y ) ∩ X = ∅.A point x ∈ M is said to be forward Poisson stable if every neighborhood V of x is forward recursive with respect to the single set {x}, the point x is said to be backward Poisson stable if every neighborhood V of x is backward recursive with respect to the set {x}.The point x is Poisson stable if x is forward and backward Poisson stable.
It is easily seen that the point x ∈ M is forward Poisson stable if and only if x ∈ ω (x), which is equivalent to ω (x) = cls µ + t (x) for every t > 0. The singular points, the periodic points, and the almost periodic points are Poisson stable.
Let U, V be two open coverings of M .We write V U if V is a refinement of U.This relation is a preorder on the set of all open coverings of M .We write Definition 2.2.A family O of open coverings of M is said to be admissible if it satisfies the following properties: The space M is said to be admissible if it admits an admissible family of open coverings.
As mentioned in the introduction, the topological space M is admissible if and only if it is uniformizable.The stars St [x, U], for U in the admissible family, form a neighborhood base at x in the topology of M .
From now on, we assume that M is an admissible space endowed with the admissible family O of open coverings of M .Definition 2.3.Let x ∈ M and t ∈ R + .The forward and backward t-prolongations of x are respectively the sets We denote D + (x) = D + (x, 0) and D − (x) = D − (x, 0) for all x ∈ M .The forward and backward prolongations of a set X ⊂ M are defined as Definition 2.4.The forward prolongational limit set of x ∈ M is defined as and the backward prolongational limit set of x as The forward and backward prolongational limit sets of a set X ⊂ M are defined as J (X) = x∈X J (x) and J * (X) = x∈X J * (x), respectively.Since the admissible family O is directed by refinements we have y ∈ M : there are nets t λ → +∞ and x λ → x such that µ (t λ , x λ ) → y , J * (x) = y ∈ M : there are nets t λ → +∞ and y λ → y such that µ (t λ , y λ ) → x .
In particular, y ∈ J (x) if and only if x ∈ J * (y).This description shows that the concept of prolongational limit set does not depend on the admissible family of open coverings.Note that the forward prolongational limit set J (x) contains the limit set ω (x), while the backward prolongational limit set J * (x) contains the limit set ω * (x).The following theorem is proved in [2, Theorem 4.3, Chapter II] for the case of dynamical systems on metric spaces.The proof for semiflows on admissible spaces follows analogously by using nets.
Theorem 2.5.For any x ∈ M , 2. If µ t is an open map for every t, then J * (x) is backward invariant.
3. Both J (x) and J * (x) are forward invariant.
The following result is proved in [11, Proposition 2.2] and describes the limit sets and the prolongational limit sets under conjugation.Proposition 2.6.Let σ : R + × N → N be a semiflow on the admissible space N .Assume that f : M → N is a continuous semiconjugation between µ and σ, that is, f (µ (t, x)) = σ (t, f (x)) for all x ∈ M and t > 0. For X ⊂ M , the following inclusions hold: Another aspect of recursiveness is represented in terms of the nonwandering points.There are situations in which a point is Poisson unstable but lies in its prolongational limit set.Definition 2.7.A point x ∈ M is nonwandering if every neighborhood U of x is forward recursive with respect to itself, that is, µ + t (U ) ∩ U = ∅ for all t > 0. By [14, Theorem 4.1], the following statements are equivalent: The dispersiveness is characterized by the absence of recursiveness, as follows.Definition 2.9.The semiflow µ is said to be: Poisson unstable, and Poisson unstable whenever it is both forward and backward Poisson unstable.
2. completely unstable if every x ∈ M is wandering.
3. dispersive if for every pair of points x, y ∈ M there are neighborhoods U x of x and U y of y such that U x is not forward recursive with respect to U y , that is, there is t ≥ 0 such that The following result shows how the prolongational limit sets play in dispersive semiflows.
2. For each pair of points x, y ∈ M , y / ∈ J (x).
3. For any point x ∈ M , J (x) = ∅.We may also describe the dispersive semiflows by means of the Lagrange instability.
Definition 2.11.Let x ∈ M .The motion µ x is said to be forward Lagrange stable if cls (µ + (x)) is a compact set; the motion µ x is called forward Lagrange unstable if cls (µ + (x)) is not compact.The semiflow µ is said to be forward Lagrange unstable if for each x ∈ M the motion µ x is forward Lagrange unstable.
Theorem 2.12.The semiflow µ is dispersive if and only if µ is forward Lagrange unstable and D + (x) = µ + (x), for every x ∈ M .Proof.Suppose that µ is forward Lagrange unstable and D + (x) = µ + (x), for every x ∈ M .If the semiflow was not dispersive, then a periodic trajectory µ x (t) would exist.By taking τ > 0 such that µ x (τ ) = x, it would follow that cls (µ + (x)) = µ x ([0, τ ]) is compact, and therefore µ x would be forward Lagrange stable.As to the converse, suppose that µ is dispersive.Then D + (x) = µ + (x), for every x ∈ M , and there is no minimal set in M .As the forward orbits are forward invariant sets, it follows that there is no Lagrange stable motion.✷

Semiflows on fiber bundles
This section contains the main results of the paper.We present a criteria for dispersive semiflow on fiber bundle.
Let G be a topological group acting on the right on a topological space Q.We denote by (q, g) ∈ Q × G → qg ∈ Q the right action of G on Q, and assume that the action is free.A principal bundle is a quadruplet (Q, π, B, G) where π : Q → B is an open and surjective map such that π (qg) = π (q) for all q ∈ Q and g ∈ G.The space B is the base space, the space Q is the total space, and G is the structure group.Assume that G acts on the left on a topological space F .Then G acts on the right on Q × F by (q, v) g = qg, g −1 v .The quotient space E = Q × G F is called bundle associated to the principal bundle (Q, π, B, G), and is indicated by (E, π, B, F, Q).We denote an element of E as [q, v].The projection π E : E → B defined by π ([q, v]) = π (q).Each fiber E x = π −1 E (x) of the associated bundle is homeomorphic with the topological space F , which is called the typical fiber of the associated bundle.
A principal bundle (Q, π, B, G) is locally trivial, that is, there is an open covering {U i } i∈I of the base space B such that, for each i ∈ I, there is a homeomorphism is a homeomorphism.Thus, the associated bundle is locally trivial and the family Let µ : R + × Q → Q be a semiflow on the total space Q of the principal bundle (Q, π, B, G) which commutes with the right action of G, that is, µ t (qg) = µ t (q) g for all q ∈ Q, t ∈ R + , and g ∈ G. Then µ is a right invariant semiflow and each map µ t : Q → Q is called an endomorphism of Q.The semiflow µ induces the semiflow µ B : R + × B → B on the base space defined by µ B (t, π (q)) = π (µ (t, q)), and µ induces the semiflow µ E : R + × E → E on the associated bundle In particular, the projection π is a continuous conjugation between the semiflows µ E and µ B .We often indicate both the semiflows µ B and µ E by µ, if there is no confusion.
The semiflow µ induces a semigroup action on the fiber F of the associated bundle E. For q ∈ Q we define the set which is a subsemigroup of G.We often denote by g t the element of R + q such that µ (t, q) = qg t .Then g 0 = 1 ∈ R + q , where 1 is the identity of G.Moreover, g t g s = g t+s for all g t , g s ∈ R + q .There are situations where the semigroup R + q is an one-parameter subgroup of G.
Example 3.1.Let G be a topological group and H ⊂ G a closed subgroup of G.The quotient map π : G → G/H of G onto the homogeneous space G/H is a principal bundle with structural group H. Let γ : R → H be a one-parameter subgroup of H, that is, a homomorphism of R into H.Define µ : R × G → G by µ (t, g) = γ (t) g.Then µ is a right invariant flow on the principal bundle Example 3.2.Let π : V → B be an n-dimensional real vector bundle.For x ∈ B, a frame σ x on x is an invertible linear map σ x : R n → V x where V x is the fiber of V above x.The set of all frames is denoted by BV .The bundle of frames of V is the bundle p : BV → B defined as p (σ x ) = x.The structural group of BV is open neighborhood of p.For t > 0, we have U g −1 ∩ µ − t (q) = ∅, which implies that U ∩ µ − t (qg) = ∅.Hence, pg ∈ cls µ − t (qg) .It follows that pg ∈ ω * (qg), and therefore ω * (q) g ⊂ ω * (qg).The inclusion ω * (qg) ⊂ ω * (q) g is easy of checking.Now, take p ∈ J (q) and g ∈ G.There are nets t λ → +∞ and q λ → q such that µ (t λ , q λ ) → p.Hence, q λ g → qg and µ (t λ , q λ g) → pg, and therefore pg ∈ J (qg).Thus, J (q) g = J (qg).Finally, if p ∈ J * (q), then q ∈ J (p).Hence, qg ∈ J (pg), and therefore pg ∈ J * (qg).It follows that J * (q) g = J * (qg).✷ In other words, the collection of the limit sets and the prolongational limit sets is invariant by G.This property implies the following results.
Corollary 3.4.Let µ : R + × Q → Q be a right invariant semiflow on the total space Q of the principal bundle (Q, π, B, G).The following statements hold: 1.If q ∈ Q is forward (backward) Poisson stable, then all points in the fiber qG are forward (backward) Poisson stable; 2. If q ∈ Q is nonwandering, then all points in the fiber qG are nonwandering.
Corollary 3.5.Assume that G acts transitively on the right on Q (Q is a homogeneous space of G) and let µ : R + × Q → Q be a right invariant semiflow.
1.If there exists a forward (backward) Poisson stable point in Q, then all the points in Q are forward (backward) Poisson stable; 2. If there exists a nonwandering point in Q, then all the points in Q are nonwandering.
3. If Q is compact, then every point in Q is Poisson stable.
4. If there exists a forward (backward) Poisson unstable point in Q, then the semiflow is forward (backward) Poisson unstable 5.If there exists a wandering point in Q, then the semiflow is completely unstable.
6.If J (q) = ∅ for some q ∈ Q, then the semiflow is dispersive.
Example 3.6.Consider the differential equation g ′ = Ag on G = GL (2, R) with A skew-symmetric.The flow associated to this equation is given by µ (t, g) = exp (tA) g, where exp (tA) ∈ SO (2, R).Hence, the trajectories of this flow are periodic, and therefore every point in G is Poisson stable.
Example 3.7.Consider the differential equation g ′ = Ag on G = GL (n, R) where A is a diagonalizable matrix with non-negative eigenvalues.The flow associated to this equation is given by µ (t, g) = exp (tA) g, where exp (tA) → 0 as t → −∞.Let 1 be the identity matrix.If x ∈ J * (1), then there are sequences t n → −∞ and x n → 1 such that exp (t n A) x n → x.As exp (t n A) → 0, it follows that x = 0. Hence, J * (1) = ∅, and therefore J * (g) = ∅ for all g ∈ G. Thus, the flow is dispersive.
and take u ∈ J (v).Then v ∈ J * (u) = {0}.Thus, J (v) = ∅ if and only if v = 0.Moreover, We now discuss the situation in what a dispersive semiflow on the associated bundle yields a dispersive semiflow on the base space of a fiber bundle.Assume that G acts on the left on a compact metric space (F, d) and let π E : E → B be the bundle associated to π, where We denote O Ψ (E) the family of all Ψ-adapted coverings.The family O Ψ (E) is admissible (see [9, Section 3.2]).
Lemma 3.15.For any q ∈ Q and U ε ∈ O Ψ (E), one has the inclusions where V refines both U and {U i } i∈I . Proof.
1.The point x is called forward Poisson unstable if x / ∈ ω (x), backward Poisson unstable whenever x / ∈ ω * (x), and Poisson unstable if it is both forward and backward Poisson unstable.2. The point x is called wandering whenever x / ∈ J(x).
The total space E is locally compact, since it is locally trivial and F is compact.Let O be the family of all open coverings of B. Since B is paracompact, O is admissible.Given ε > 0 and U ∈ O, a Ψ-adapted covering of E is defined as