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Discrete and Continuous Topological Dynamics: Fields of Cross Sections and Expansive Flows

In this article we consider the general problem of translating definitions and results from the category of discrete-time dynamical systems to the category of flows. We consider the dynamics of homeomorphisms and flows on compact metric spaces, in particular Peano continua. As a translating tool, we construct continuous, symmetric and monotonous fields of local cross sections for an arbitrary flow without singular points. Next, we use this structure in the study of expansive flows on Peano continua. We show that expansive flows admit no stable point and that every point contains a non-trivial continuum in its stable set. As a corollary we obtain that no Peano continuum with an open set homeomorphic with the plane admits an expansive flow. In particular compact surface admits no expansive flow without singular points.


Introduction
In the study of dynamical systems time is usually modeled as discrete or continuous. A discrete-time dynamical system can be understood as an action of the integers, induced by a homeomorphism or a diffeomorphism. The continuous time can be represented by the real numbers, and the dynamics is generated by a vector field or a flow. Both categories are strongly related and several definitions and results can be translated. For example, expansivity is defined in both categories.
Recall that a homeomorphism f : X → X of a compact metric space (X, dist) is an expansive homeomorphism if there is r > 0 such that dist(f n (x), f n (y)) ≤ r for all n ∈ Z implies x = y. For flows, the definition of [4] is a good translation in terms of the results that it allows to recover. It saids that a flow φ : R × X → X is an expansive flow if for all ε > 0 there is δ > 0 such that if dist(φ h(t) (y), φ t (x)) < δ for all t ∈ R with h : R → R an increasing homeomorphism such that h(0) = 0 then there is t ∈ (−ε, ε) such that y = φ t (x). The complexity of the definition makes the translations very difficult, but several techniques were developed. See for example [10,[14][15][16][17].
The purpose of the present article is to develop the technique of local cross sections for translating some definitions and results from discrete to continuous topological dynamical systems on compact metric spaces. In order to describe our results let us consider a non-singular smooth flow φ t : M → M , t ∈ R, on a compact manifold M with a transverse foliation. Given a point x ∈ M we can take a compact disc H r (x) of radius r > 0 contained in the leaf of x. If r is small we have that: (1) for each r > 0 and x ∈ M , H r (x) is a local cross section containing x, (2) the map (r, x) → H r (x) is continuous, 1 (3) (monotonous) for t = 0 small H r (x) is disjoint from H r (φ t (x)), 1 In a smooth category, we can have continuity in the space of C 1 embeddings. In the context of metric spaces we will consider the Hausdorff metric between compact subsets of X.
(4) (symmetric) if x, y are close then y ∈ H(x) if and only if x ∈ H(y). Such a map H, depending on r and x, is an example of what we call field of cross sections and it is illustrated in Figure 1. Then, if the flow is defined on a manifold and it admits a transverse foliation the construction is trivial. But it is known that there are smooth flows without a transverse foliation, see for example [5]. For these cases we have an alternative construction. Suppose that Y is the velocity field of the flow. On a compact smooth manifold we can consider a Riemannian metric and define where exp is the exponential map of the Riemannian metric and r > 0 is small. This definition can be found for example in [11,16]. Unfortunately, in this setting items 3 and 4 above may not hold. For continuous (non-smooth) flows on compact manifolds the task is even harder. A related construction is given in [16] for studying expansive flows of three-manifolds. On metric spaces local cross sections were constructed in [18]. In [4,10,15,16] local cross sections are used in the study of nonsmooth expansive flows on manifolds and metric spaces. In [4,10,15] they consider a finite cover of flow boxes and the corresponding return maps. This approach is a good translating tool but it has some technical difficulties that makes it hard to handle. For example, the return maps may not be continuous at the boundary of the cross sections.
In this article we consider regular flows (no singular points) on compact metric spaces. In Section 2 we develop a theory of fields of local cross sections. In Theorem 2.51 we prove that every regular flow on a compact metric space admits a semicontinuous, monotonous and symmetric field of cross sections. If in addition we have that the space is a Peano continuum, in Theorem 2.53 we conclude that the cross sections can be assumed to vary continuously, each one being a connected set. Precise definitions are given in Section 2.
As a translating tool, one may think of the cross section H r (x) as the ball B r (x) in the discrete time case. To show how this works let us translate the definition of wandering point. Recall that if f : X → X is a homeomorphism of a metric space then x ∈ X is a wandering point if: (1) there is r > 0 such that f n (B r (x)) ∩ B r (x) = ∅ for all n > 0. If φ is a flow on X a point x ∈ X is said to be a wandering point if: (2) there are r, t 0 > 0 such that φ t (B r (x))∩B r (x) = ∅ for all t ≥ t 0 . In this definition there is a parameter t 0 that is not present in (1). Using a local cross section we have that x is a wandering point of the flow φ if and only if: (3) is obtained from (1) by changing f n → φ t and B → H. This is a task that a translating machine can do. This machine needs to be programmed and the present paper is intended to be a contribution in this direction.
In Section 3 we show how this machine should work in a more complex situation. We consider expansive flows on Peano continua and we translate some results of expansive homeomorphisms. As said in the abstract, we prove that expansive flows do not admit stable points. We show that every point contains a non-trivial continuum in its stable set and as a corollary we prove that no compact surface admits an expansive flow without singular points.

Fields of cross sections
In this section we introduce the fields of compact sets, which assigns to each point a compact subset through the point. This concept is related with the coselections defined in [7]. Fields of cross sections of a flow are a special instance of such fields.
In Section 2.3 we show that every regular flow admits a field of cross sections. For this purpose we extend the techniques of [18] while introducing topological 1forms. In Section 2.4 we consider flows on Peano continua. In this case we show that every regular flow admits a continuous field of connected cross sections. In Section 2.5 we introduce the transposition of fields of compact sets and 1-forms. These techniques are used in Section 2.6 to construct monotonous fields of cross sections. In Section 2.7 we show that every regular flow admits a symmetric field of cross sections. The main results of this section are Theorems 2.51 and 2.53.
2.1. Fields of compact sets. Let (X, dist) be a compact metric space. As usual, define the closed ball for all r ≥ 0 and x ∈ X. Also define B r (K) = ∪ x∈K B r (x) if K is a subset of X. Denote by K(X) the set of compact subsets of X equipped with the Hausdorff distance. If K, L are compact subsets of X then the Hausdorff distance is defined as It is known that (K(X), dist H ) is a compact metric space and a proof can be found in [7].
Example 2.5 (The field of closed balls). Given r > 0 consider the field of closed balls B r : X → K(X) defined by (1). It is a field of neighborhoods.
The following is an important example in dynamical systems.
Example 2.6 (Local stable sets). Given a homeomorphism f : X → X we define W s r , W u r : X → K(X) as W s r (x) = {y ∈ X : dist(f n (x), f n (y)) ≤ r for all n ≥ 0} and W u r (x) = {y ∈ X : dist(f n (x), f n (y)) ≤ r for all n ≤ 0}. We have that W s r , W u r are fields of compact sets. Note the dependence with respect to f that is omitted in the notation. In Section 3.1 this definition will be extended for flows.
We wish to remark that fields of compact sets are only assumed to be semicontinuous and in the next definition we require continuity. As before, the topology of K(X) is the one induced by the Hausdorff metric. (1) N r is a field of neighborhoods for all r > 0 and (2) N 0 (x) = {x} and N 1 (x) = X for all x ∈ X.
Remark 2.8. The ball operator may not be continuous and consequently B r may not define a one-parameter field of neighborhoods. It is not continuous with respect to r, for example, if X is a finite set. Moreover, even if X is an arc B r may not be continuous. See Figure 2.1. Figure 2. If X is a circular arc in the plane with the Euclidean metric, B r (x) is discontinuous with respect to x for some values of r. Of course, the arc admits a metric making B r (x) continuous.

X
Recall that X is a Peano continuum if it is a connected, compact and locally connected metrizable space. A metric dist in X is convex if given x, z ∈ X, x = z, and α ∈ (0, dist(x, z)) there is y ∈ X such that dist(x, y) = α and dist(y, z) = dist(x, z) − α.
Remark 2.9. It is known that a compact metric space admits a convex metric defining its topology if and only if it is a Peano continuum. See [3,7,13] for more on this subject. Define The following result is essentially from [3,13].
Theorem 2.10. For a compact metric space X the following statements are equivalent: (1) there is a one-parameter field of neighborhoods N : [0, 1] × X → C(X), (2) X is a Peano continuum. Proof.
(1 → 2). The local connection of X follows because N is continuous, N 0 (x) = {x} and each N r (x), with r > 0, is a connected neighborhood of x. So, each point has a base of connected neighborhoods. It only rests to note that X = N 1 (x) ∈ C(X) to conclude that X is connected.
(2 → 1). If X is a Peano continuum, then there is a convex metric dist defining the topology of X (Remark 2.9). Therefore, the ball operator associated to dist is a one-parameter field of neighborhoods (see [7] for more details).
Let us give some natural definitions and remarks.
Definition 2.11. Given two fields h 1 , h 2 : For example, W s r and W u r are subfields of the ball field B r . Definition 2.12. Given two fields h 1 , h 2 define h 1 ∩h 2 and h 1 ∪h 2 as ( Remark 2.13. If h 1 , h 2 : X → K(X) are fields then h 1 ∩ h 2 and h 1 ∪ h 2 are fields.
Remark 2.14. We have that a homeomorphism f : for all s, t ∈ R and for all x ∈ X. We say that φ is a regular flow if for all x ∈ X there is t ∈ R such that φ t (x) = x, i.e., it has no equilibrium points.
Let φ : R × X → X be a regular flow. Given a field h : X → K(X) and a compact subset I ⊂ R, 0 ∈ I, define the field φ I (h) by We say that C ∈ K(X) is a (local) cross section through x ∈ C if there are τ > 0 and γ > 0 such that for all x ∈ X, and consequently it is a homeomorphism onto its image.
This implies that t = s. This proves the injectivity of the considered restriction of φ. The continuity of the inverse follows by the continuity of φ and the compactness of for all x ∈ X then we say that H is a field of cross sections of time τ .
From the definitions we are assuming that our fields H are semicontinuous. Then, it can be the case that x n → x, r > 0, y ∈ H(x) and B r (y) ∩ H(x n ) = ∅ for all n ≥ 1. In this case we could say that y is discontinuity point (a very confusing terminology that will not be used in the sequel). The following lemma means that if H is a field of cross sections then the discontinuity points in H(x) are uniformly far from x.
Proof. Arguing by contradiction, assume that there are ε > 0, γ ∈ (0, γ), x n , y n → p and z n ∈ H(y n ) ∩ B γ (y n ) such that Since dist(x n , y n ) → 0 we can suppose that dist(x n , y n ) < γ − γ . In this way, By (2) we have that |t n | > ε. Now we take limit as n → ∞. Assume that t n →t with |t| ∈ [ε, τ ], z n → z an recall that x n , y n → p. Since H is semicontinuous and z n ∈ H(y n ) we have that z ∈ H(p). Also, φt(z) ∈ H(p). Since |t| ≤ τ and t = 0 we have a contradiction with the definition of cross section and the lemma is proved.
is a field of neighborhoods for all ε > 0.
is a field of neighborhoods for all ε > 0.
Proof. By contradiction assume that no B ρ is contained in N ε for some fixed ε > 0.
Then we can take x n , y n such that dist(x n , y n ) → 0 and y n / ∈ N ε (x n ) for all n ≥ 1.
. Also assume that y n ∈ B δ (x n ) for all n ≥ 1. Suppose that H is a field of cross sections of time τ . If ε and δ are small we can assume that Then, for each n, there are t n , s n such that z n = φ sn (y n ) ∈ H(φ −ε (x n )) and φ sn (y n ) ∈ H(x n ). We can assume that 0 < s n ≤ t n . By Lemma 2.19 we have that t n → 0. Then s n → 0. Taking limit n → ∞ we obtain: lim z n = φ ε (lim x n ) and lim z n ∈ H(φ −ε (lim x n )) which is a contradiction.
Proposition 2.22. If N is a field of neighborhoods and H is a field of cross sections then N ∩ H is a field of cross sections.
Proof. The semicontinuity of N ∩ H follows by Remark 2.13. Let τ > 0 be a time ) and the proof ends.
For future reference we state the following result. Proof. It is direct from the definitions.

Constructing cross sections. Given a field of neighborhoods
Definition 2.24 (Topological forms). A 1-form is a continuous function Define the field of compact sets ker(ω) : X → K(X) by Given a flow φ on X and a 1-form ω definė Remark 2.25. It could be useful to think of U (N ) as Milnor's tangent microbundle as defined in [12].
The following results are based in [18,Section 29]. They are stated here in such a way that they can be used and extended later.

Proof.
Since ω x is continuous we have that H ρ (x) is a compact set and since ω x (x) = 0 we have that x ∈ H ρ (x). Let a, ρ 1 > 0 be such that B ρ1 ⊂ N and |ω To show that H ρ is semicontinuous take y n ∈ H ρ (x n ), x n → x, and y n → y. We will show that y ∈ H ρ (x). Since ω is continuous we see that ω x (y) = ω x (x). Since dist(x n , y n ) ≤ ρ n for all n ≥ 1 we have that dist(x, y) ≤ ρ. Therefore y ∈ H ρ (x).
Theorem 2.28. If X is a compact metric space then every regular flow admits a field of cross sections.
Proof. If φ is a regular flow then there ist > 0 such that φt(x) = x for all x ∈ X.
Define v x (y) = dist(x, y) for all x, y ∈ X. The result follows by Propositions 2.26 and 2.27.

2.4.
Continuous fields of connected cross sections. In this section we will obtain continuous fields of connected cross sections assuming that X is a Peano continuum. For this, we will need the flow projection from a flow box to a cross section.  Definition 2.31. Given the field of flow boxes F associated to H define the flow projection π x : F (x) → H(x) by π x (y) = φ t (y) ∈ H(x) with |t| ≤ τ . Given a field of compact sets M ⊂ F define π(M ) by (π(M ))(x) = π x (M (x)). Proposition 2.32. If H is a field of cross sections with F the associated field of flow boxes and N ⊂ F is a field of compact sets then: (1) π(N ) is a field of compact sets, if N is continuous then π(N ) is continuous and (4) if N is a field of neighborhoods then π(N ) is a field of cross sections.
Proof. Define J = π(N ). Item 1. We have to prove that J is semicontinuous. Take x n → x and y n ∈ J(x n ) with y n → y. Consider z n ∈ N (x n ) with π xn (z n ) = φ tn (z n ) = y n , z n → z and t n → t. Since N is semicontinuous we have that z ∈ N (x). Also, φ t (z) = y, t ∈ [−τ, τ ] and since y n ∈ J(x n ) ⊂ H(x n ) and H is semicontinuous we have that y ∈ H(x). Then y = π x (z) and y ∈ J(x).
Item 2. It follows because π x is continuous. Item 3. We know that J is semicontinuous. Assume that it is not continuous. Then there are ε > 0 and x, x n ∈ X such that x n → x, J(x) B ε (J(x n )). Then there is z n ∈ J(x) such that z n / ∈ B ε (J(x n )). Take u n ∈ N (x) and t n ∈ [−τ, τ ] such that φ tn (u n ) = z n . Since N is continuous there is v n ∈ N (x n ) such that dist(u n , v n ) → 0. Take s n ∈ [−τ, τ ] such that z n = φ sn (v n ) ∈ J(x n ). We have that dist(z n , z n ) > ε. Taking limit (and subsequences) n → ∞ we can assume that t n → t, s n → s, z n → z, z n → z and u n , v n → u. By the semicontinuity of J we have that z, z ∈ J(x). Also dist(z, z ) ≥ ε. By continuity we have φ t (u) = z and φ s (v) = z . Therefore, z = φ t−s (z ). Since z, z ∈ J(x) and |t − s| ≤ 2τ we have a contradiction because τ is a time for H. Item 4. By Proposition 2.22 we know that N ∩ F is a field of cross sections. Notice that N ∩ F ⊂ J ⊂ H. Since we have proved that J is semicontinuous we can apply Proposition 2.23 to conclude that J is a field of cross sections.
The following result gives us continuous fields of connected cross sections. Proposition 2.33. If X is a Peano continuum and φ is a regular flow then every field of cross sections H : X → K(X) admits a continuous subfield of cross sections H : X → C(X).
Proof. Let H be a field of cross sections. Since X is a Peano continuum we know from Theorem 2.10 that there is a continuous field of neighborhoods N : X → C(X) with N ⊂ F . By Proposition 2.32 we know that H = π(N ) : X → C(X) is a continuous field of cross sections satisfying H ⊂ H. Theorem 2.34. If X is a Peano continuum then every regular flow admits a continuous field of connected cross sections H : X → C(X).
Proof. By Theorem 2.28 we know that there is a field of cross sections H . By Theorem 2.10 there is a continuous field of neighborhoods N ⊂ φ [−τ,τ ] (H ) with N : X → C(X). By Proposition 2.32 we have that H = π(N ) is a continuous field of cross sections, moreover, each H(x) is connected.

2.5.
Transpose fields and 1-forms. The formalism introduced in this section will be applied in the next section to study the monoticy of fields of cross section. Definition 2.35 (Transpose field). Given a field h : X → K(X) define its transpose h T : X → K(X) by h T (x) = {y ∈ X : x ∈ h(y)}.
We say that h is symmetric if h T = h.
Proposition 2.36. The following statements hold: (1) h T T = h for every field h, (2) h T is a field of compact sets if and only if h is a field of compact sets, (3) the field of balls B ρ is symmetric, N is a field of neighborhoods if and only if N T is a field of neighborhoods, (2) Assume that h is semicontinuous and take y n ∈ h T (x n ) with x n → x and y n → y. Then x n ∈ h(y n ). Since h is semicontinuous we have that x ∈ h(y). Then y ∈ h T (x). The converse follows by this and item 1.
So, x ∈ h 2 (y) and y ∈ h T 2 (x). The converse follows by this and item 1.
Remark 2.40. Every subfield of cross sections of a monotonous field of cross sections is monotonous.

Proposition 2.41. A field of cross sections H is monotonous if and only if H T is a monotonous field of cross sections.
Proof. First assume that H is monotonous. By Proposition 2.36 we know that H T is a field of compact sets. By Proposition 2.21 we know that is a field of neighborhoods for all ε > 0. For ε > 0 fixed we can take ρ > 0 such that Since H is monotonous there is ε > 0 such that H(x)∩H(φ t (x)) = ∅ for all x ∈ X and t ∈ [−ε, ε], t = 0. Let us show that H T (x) ∩ φ [−ε,ε] (y) = {y} for all x ∈ X and y ∈ H T (x). Take z ∈ H T (x) ∩ φ [−ε,ε] (y). Then x ∈ H(z) and there is s ∈ [−ε, ε] such that z = φ s (y) with y ∈ H T (x), i.e., x ∈ H(y). Then x ∈ H(z) ∩ H(φ s (z)). Since H is monotonous, this implies that s = 0 and y = z as we wanted to prove.
Since H is a field of cross sections there is τ > 0 such that H(y)∩φ [−τ,τ ] (x) = {x} for all y ∈ X and x ∈ H(y). We will prove that This implies that t = 0 and the first part of the proof ends.
The converse follows because H T T = H.
The notationv T means the derivative of v T (it is not the transpose ofv).
is a monotonous field of cross sections for ρ small.
Proof. By Proposition 2.26 we know that H ρ is a field of cross sections becausev = 0. Now suppose that y ∈ H(x) ∩ H(φ t (x)). This implies that v x (y) = v φt(x) (y) = 0.
Then v T y (x) = v T y (φ t (x)) = 0. Sincev T = 0 we have that t = 0 and H ρ is monotonous.
is a monotonous field of cross sections for ρ small.

Notice that
Theorem 2.45. If X is a compact metric space then every regular flow on X admits a monotonous field of cross sections.
Proposition 2.48. If ω is anti-symmetric then ker(ω) is a symmetric field of compact sets.
Proof. It is a direct consequence of the definitions. It could also be derived from Proposition 2.38.

Proposition 2.49.
If ω is anti-symmetric andω = 0 then H = B ρ ∩ ker ω is a monotonous and symmetric field of cross sections if ρ is small. Proof.
Since ω is symmetric we know by Proposition 2.48 that ker(ω) is a symmetric field of compact sets. By Proposition 2.36 we conclude that H is a symmetric field of compact sets. Since ω is anti-symmetric we have thatω T = −ω. Then, we can apply Proposition 2.42 to conclude that H is a monotonous field of cross sections.
Proof. By Propositions 2.44 and 2.43 there is v such thatv > 0 andv T < 0. Define Theorem 2.51. If X is a compact metric space then every regular flow on X admits a symmetric and monotonous field of cross sections.
Proof. It follows by Propositions 2.50 and 2.49.
Definition 2.52. We say that a field of compact sets H is locally symmetric if there is δ > 0 such that B δ ∩ H is symmetric.
Theorem 2.53. If X is a Peano continuum and φ is a regular flow on X then there are a symmetric and monotonous field of cross sections H and a continuous one-parameter field H : [0, r] × X → C(X) satisfying: (1) H ε : X → C(X) is a monotonous, continuous and locally symmetric field of connected cross sections for all ε ∈ (0, r], , Proof. The field of cross sections H is given by Theorem 2.51. Let F be the field of flow boxes associated with H en denote by π its flow projection. Consider a metric in X such that the field of balls B ε is continuous. Take r > 0 such that The result now is direct.

Expansive flows
In this section we will apply our constructions to the study of expansive flows. In Section 3.1 we define the sectional flow, which is similar to the Poincaré return map (or the linear flow). It is also some kind of holonomy. The definition of the sectional flow does not depend on the expansivity of the flow. In Sections 3.2 and 3.3 we show our results on expansive flows.
3.1. Sectional flow. Let (X, dist) be a Peano continuum and denote by φ a regular flow on X. Consider a continuous, monotonous and locally symmetric oneparameter field of connected cross sections H ε : X → C(X) given by Theorem 2.53. Lemma 3.1. Given x, y ∈ X and a connected set I ⊂ R with 0 ∈ I there is at most one continuous function h : I → R such that h(0) = 0 and φ h(t) (y) ∈ H(φ t (x)) for all t ∈ I. In this case h is strictly increasing.
Proof. Suppose that h 1 , h 2 : I → R are continuous, h 1 (0) = h 2 (0) = 0 and for all t ∈ I it holds that φ hi(t) (y) ∈ H(φ t (x)), for i = 1, 2. Define J = {t ∈ I : h 1 (t) = h 2 (t)}. We have that J is closed because h 1 , h 2 are continuous. If J is not open then there are s ∈ J and s n → s with s n / ∈ J for all n ≥ 1. Then φ hi(sn) ∈ H(φ sn (x)) for i = 1, 2. This contradicts that H(φ sn (x)) is a cross section. Then J is closed and open, since I is connected, I = J and h 1 = h 2 .
We have that h is increasing because H is monotonous.
In case such a function h : I → R exists for x, y define the sectional flow for t ∈ I. Define R + (resp. R − ) as the set of all homeomorphisms of [0, +∞) (resp. If h n : I → R are continuous, h n (0) = 0, x n → x, y n → y satisfy φ hn(t) (y n ) ∈ H(φ t (x n )) for all n ≥ 1 and for all t ∈ I then h n uniformly converges to some h : I → R satisfying φ h(t) (y) ∈ H(φ t (x)) for all t ∈ I.
Proof. Suppose that H has a time τ > 0 and that h n does not uniformly converge. Then there are ε > 0, n k , m k → +∞, t n ∈ I such that |h n k (t k ) − h m k (t k )| ≥ ε for all k ≥ 1. Since h n (0) = h m (0) = 0 for all m, n ≥ 1 we can assume that h n k (t k ) − h m k (t k ) = s ∈ (−τ, τ ), s = 0, for all k ≥ 1. Then φ hn k (t k ) (y n k ) ∈ H(φ t k (x n k )) and φ hm k (t k ) (y m k ) ∈ H(φ t k (x m k )) for all k ≥ 1. Assume that φ t k (x n k ), φ t k (x m k ) → p and φ hn k (t k ) (y n k ) → q. Then φ hm k (t k ) (y m k ) → φ s (q). We have a contradiction because q and φ s (q) are in H(p) and 0 < |s| < τ . (Symmetry). If y ∈ W s ε (x) then there is h ∈ R + such that φ h(t) (y) ∈ H ε (φ t (x)) for all t ≥ 0. We have that h −1 ∈ R + and φ h −1 (t) (x) ∈ H ε (φ t (y)) for all t ≥ 0. We have used that H ε is locally symmetric. This proves that W s ε is symmetric. The proof for W u ε is similar.
3.2. Expansive flows. Let φ : R × X → X be a regular flow on a compact metric space X. Recall that φ is an expansive flow if for all ε > 0 there is δ > 0 such that if dist(φ h(t) (y), φ t (x)) < δ for all t ∈ R with h : R → R an increasing homeomorphism such that h(0) = 0 then there is t ∈ (−ε, ε) such that y = φ t (x). In this case δ is called expansive constant.
Proposition 3.4. The following statements are equivalent: Proof. (1 → 2). Consider that H ρ (x) is a local cross section of time τ for all x ∈ X.
Let us recall that a flow is positive expansive if for all ε > 0 there is (1) φ is positive expansive, (2) there is δ > 0 such that W s δ (x) = {x} for all x ∈ X, (3) X is a finite union of circles.
Proof. The proof of the equivalence of 1 and 2 is analogous to the proof of Proposition 3.4. The equivalence of 1 and 3 is shown in [1].

Stable points.
The following definitions should be understood as Lyapunov stability of trajectories allowing time lags.
. We say that x ∈ X is asymptotically stable if it is stable and there is δ > 0 such that for all y ∈ H δ (x) there is h ∈ R + such that φ h(t) (y) ∈ H ε (φ t (x)) for all t ≥ 0 and dist(φ h(t) (y), φ t (x)) → 0 as t → +∞.
In the sequel we will say that δ > 0 is an Proposition 3.7. If φ is expansive with expansive constant δ and x ∈ X then Proof. By contradiction assume that there are γ ∈ (0, δ] and t n → +∞ such that diam(Φ tn (x, W s δ (x))) ≥ γ. Denote by x n = φ tn (x) and suppose that x n → y. We can also assume that Φ tn (x, W s δ (x)) converges (in the Hausdorff metric) to a compact set Y contained in H δ (y). In this way we have that diam(Y ) ≥ γ > 0 and Y ⊂ W s δ (y) ∩ W u δ (y). This contradicts the expansivity of φ and finishes the proof.
A similar result holds for W u δ (x).
Proof. By contradiction assume that there are ε > 0, sequences x n ∈ C n , 0 ≤ t n ≤ s n with C n a continuum in H(x n ), diam(C n ) → 0, diam(Φ sn (x n , C n )) → 0 and diam(Φ tn (x n , C n )) ≥ ε. As in the previous proof, a limit continuum of Φ tn (x n , C n ) contradicts expansivity of φ.
Define the inverse flow φ −1 t (x) = φ −t (x) and the sets: L -(x) = {y ∈ X : ∃t n → −∞ such that φ tn (x) → y} L + (x) = {y ∈ X : ∃t n → +∞ such that φ tn (x) → y} They are usually called α-limit and ω-limit sets respectively. Proposition 3.9. If X is a Peano continuum and φ is an expansive flow on X with a stable point for φ or φ −1 then X is a circle.
Proof. Assume that x ∈ X is a stable point of φ −1 . Take ε > 0. Since x is a stable point we know that there is δ > 0 such that H δ (x) ⊂ W u ε (x). By Lemma 3.8 we have that there is σ > 0 such that if 0 ≤ s ≤ t then Φ −s (φ t (x), H σ (φ t (x))) ⊂ H δ (φ t−s (x)).
This implies that every point in L + (x) is stable for φ −1 . Then, φ restricted to L + (x) is a positive expansive flow. Since L + (x) is a connected set, by Theorem 3.5, we have that L + (x) is a circle. Since x is a stable point we have that x ∈ L + (x). We have proved that every stable point is periodic. Since the set of stable points is open we conclude that X is a periodic orbit and consequently, X is a circle.
In this context, a Peano continuum is trivial if it is a circle. Recall that X cannot be a singleton because we are assuming that it admits a flow without singular points. Proposition 3.10. If φ is expansive with expansive constant ε and X is a nontrivial Peano continuum then there is δ > 0 such that for all x ∈ X there is a continuum C ⊂ W s ε (x) such that x ∈ C and diam(C) ≥ δ. Proof. For the expansive constant ε consider δ from Lemma 3.8. For x ∈ X consider z ∈ L + (x) and take t k → ∞ such that x k = φ t k (x) → z. We can assume that z ∈ H δ (x k ) for all k ≥ 1. By Proposition 3.9 we know that z is not stable for φ −1 . Then, there is T > 0 such that diam(Φ −T (x k , H δ (x k ))) > ε. Suppose that t k > T for all k ≥ 1. Consider C k ⊂ H δ (x k ) such that C k is a continuum, x k ∈ C k , diam(Φ −t (x k , C k )) ≤ ε for all t ∈ [0, t k ] and diam(Φ −s k (x k , C k )) = ε for some s k ∈ [0, t k ]. By Lemma 3.8 we know that diam(Φ −t (x k , C k )) ≥ δ for all t ≥ s k . Notice that x ∈ Φ −t k (x k , C k ) for all k ≥ 1. Then a limit continuum C of the sequence Φ −t k (x k , C k ) satisfies the thesis of the proposition.
I learned the argument of the following proof from J. Lewowicz in the setting of expansive homeomorphisms. Proof. Let X be a Peano continuum with a non-singular flow φ. Suppose that x ∈ X has a neighborhood U that is homeomorphic with R 2 . In this case we have that a connected cross section H ε (x) is a compact arc. By Proposition 3.10 we have that W s ε (x) contains a non-trivial continuum. But, since the cross section is an arc, we have that it contains an interior point (with respect to the topology of the arc). Then there are stable points. This contradicts Proposition 3.9.  Proof. It follows by Theorem 3.11 recalling that a homeomorphism is expansive if and only if its suspension is an expansive flow (see [4]).