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On partial maps derived from flows

The first-return map, or the Poincar\'e map, is a fundamental concept in the theory of flows. However, it can generally be defined only partially, and additional conditions are required to define it globally. Since this partiality reflects the dynamics, the flow can be described by considering the domain and behavior of such maps. In this study, we define the concept of first-exit maps and first-return maps, which are partial maps derived from flows, to enable such analysis. Moreover, we generalize some notions related to the first-return maps. It is shown that the boundary points of an open set can be classified based on the behavior of these maps, and that this classification is invariant under topological equivalence. Further, we show that some dynamical properties of a flow can be described in terms of the types of boundary points. In particular, if the flow is planar and the open set has a Jordan curve as its boundary, a more detailed analysis is possible, and we present results on the conditions which restrict possible forms of the first-exit maps . Finally, as an application of the results obtained, we consider the relationship between flows and a class of hybrid systems.


Introduction
The first-return map, or the Poincaré map, is a fundamental tool in the study of continuous-time dynamical systems. It has a wide range of applications from the classical proof of the Poincaré-Bendixon theorem to the analysis of chaotic dynamical systems [14,4], and is arguably one of the most valuable concepts in the theory of flows.
However, the first-return map is not necessarily defined globally. To ensure its global existence, additional assumptions are needed in general. For example, the compactness of the phase space and the smoothness of the flow ensure the existence of a global Poincaré section, which yields a globally defined first-return map [2]. However, there is still no guarantee that the domain of the resulting maps is suitable to analyze the dynamics. Since the suspension flow of the first-return map is topologically equivalent to the flow from which the first-return map is derived [9,19,16], the existence of a domain with good topological properties may restrict the possible motion of the flow.
Further, it is inherently difficult to make a first-return map global in applications, even if we ignore the difficulty originating from the undecidability of the domain of the solution of an ordinary differential equation [8]. When a first-return map is not defined at a point, we may infer that the orbit starting from that point never returns. This is a piece of information on the asymptotic behavior of the system and we cannot ignore it. This is a problematic issue if we only consider constructing a global first-return map. However, if we allow partial maps, this observation could be exploited to study the dynamics.
Another point to consider is other partial maps from a flow. The Poincaré half map and the half-return map can be regarded as partial versions of the first-return map [17,5,12]. The first-exit (time) map is another example, although it is not widely used [10,13].
The main aim of this study is to consider partial maps derived from a flow in a manner similar to the first-return map, and to apply them to the analysis of flows. Concretely, there are two objectives: the first is to define a concept that subsumes important and frequently-used examples, and the second is to show that it is possible to describe the dynamics of a flow using such a framework. Further, we aim to apply the results obtained here to a converse problem of representing a partial map by a flow. This will enable us to consider some class of hybrid systems, defined by a combination of discrete and continuous dynamical systems, in terms of a flow.
The main results of this study are as follows. First, we introduce the concept of first-exit and first-return maps, which are generalizations of the first-return map in the usual sense (Definition 3.1). For an open set A, the first-exit map moves a point on ∂A to the first intersection point of the boundary and its forward orbit other than itself . This is a direct generalization of the Poincaré half map. Similarly, the first-return map is defined for a closed set in terms of the first-return point. These can be defined for general continuous flows on a topological space, and therefore, it is not necessary to assume the smoothness. Boundary points of a regular open set are classified into several types according to whether these maps are defined and whether they are fixed under these maps. These types are invariant under topological equivalence, and therefore, can be used to describe the dynamics. Further, we show that various dynamical properties of a flow can be described in terms of these types of boundary points. For example, the invariance of an open set can be expressed by specifying the types of boundary points, which may be regarded as a generalization of the result that a set is invariant if a vector field points inward on the boundary.
For planar flows and open sets with Jordan curves as boundaries, we can obtain a more convenient representation of the first-exit maps in terms of parametrization. Since the first-exit map is defined by a flow, its possible forms are restricted. In particular, we have the following monotonicity result.
Main Theorem B. If a parametric representation of an exit map F E is continuous at s and F E (s) = s,, there exists δ > 0 such that F E (t) > F E (s) whenever s − δ < t < s, and F E (t) < F E (s) whenever s < t < s + δ.
Conversely, we can represent a map by a flow if it is sufficiently wellbehaved .
Using this result, we can represent a class of hybrid systems in terms of flows. Moreover, we may specify the map part of the system.
Main Theorem D. Let (P, Φ, Φ s ) be an impacting system induced by local flows. If P is total, continuous, and not identity, then (P, Φ, Φ s ) is topologically conjugate with another impacting system This article is organized as follows. In Section 2, we introduce basic terms and some preliminary results. In Section 3, we define the notion of first-exit maps and first-return maps, study their basic properties, introduce the concept of types of boundary points, and apply them to the description of dynamical properties. In Section 4, we consider the first-exit maps of planar flows and their parametric representation. In Section 5, we apply the results obtained to the study of a class of hybrid systems.

Preliminaries
In this section, we describe some basic definitions and results used throughout this article.
First, we introduce the main objects of our consideration here, i.e., flows and partial maps.
For each x ∈ X,, the forward orbit of x is the set Details on partial maps can be found in [1]. Definition 2.2 (Partial map). Let X and Y be topological spaces. A partial map is a pair (D, f ) of subset D ⊂ X and a map f : D → Y. For a partial map (D, f ), the set D is called the domain, and is denoted by dom f . As a convention , we denote a partial map (D, f ) by f : X → Y. The image of a partial map is the set im f := f (dom f ) .
A partial map f : The next lemma is trivial, but useful in constructing a flow with prescribed properties.
Lemma 2.4. Let X and Y be topological spaces and Φ : R × X → X a continuous flow. If h : X → Y is a homeomorphism, there exists a unique flow Ψ : for all x ∈ X and t ∈ R. Example 1. Let Φ : R × R 2 → R 2 be a continuous flow. Then, it can be shown that Φ can be extended to a continuous flowΦ on the Riemann sphereĈ by settingΦ(t, ∞) = ∞ for all t ∈ R. If M A :Ĉ →Ĉ is the Möbius transformation defined by the matrix A, there exists another floŵ Ψ on the Riemann sphere such thatΨ(t, M A (z)) = M A (Φ(t, z)). If ∞ is an equilibrium point ofΨ,Ψ is an extension of a continuous flow on R 2 .

First exit maps and first return maps
In this section, we first introduce the notion of first-exit maps and firstreturn maps and study their basic properties. By considering the domain and behavior of these maps, we define the types of boundary points of an open set that can be used to describe the dynamics of the flow from which the map was derived.
3.1. Definition and basic properties. First, we define first-exit maps and first-return maps as follows.
Definition 3.1 (First exit maps and first return maps). Let (X, Φ) be a flow.
(1) For an open set A ⊂ X, the first exit map E Φ A : ∂A → ∂A is a partial map defined by (2) For a closed set B ⊂ X, the first return map P Φ B : ∂B → ∂B is a partial map defined by For notational convenience, we drop the index for the flow and denote it as E A .
We need to check that the first exit map is well-defined.
The first exit map is, in a sense, a dual concept to the first return map. This is observed by the next lemma, which follows immediately from the definition. Therefore, we will mainly consider the first exit map in what follows. For the unit disc D = {(x, y) | x 2 + y 2 < 1}, the first exit map and the first return maps are given by ( 7 4 π < θ < 2π) and ( 5 4 π < θ < 7 4 π) undefined ( 7 4 π ≤ θ < 2π) respectively, where S 1 = ∂D is parametrized by the angle θ.
There exists a strong restriction on the possible form of a first-exit map, since the orbits of a flow are disjoint. Proof. First, we show that E A (x 1 ) = E A (x 2 ) and x 1 = x 2 imply that x 1 or x 2 is a fixed point of E A . We assume Without loss of generality, we may assume that Consequently, if y ∈ ∂A and E −1 A (y) have two different elements x 1 and x 2 ,, either of them is y. Therefore, the number of elements in E −1 A (y) cannot be greater than two.
Remark 3.7. The first exit map can be one-to-one when A is backward invariant.
Although the first exit maps and the return maps are only defined partially, we have the following result. Recall that an open set A is regular if ∂A = ∂Ā.

Types of boundary points.
In what follows, we assume that the open set A is always normal.
According to Lemma 3.8, each x ∈ ∂A can be classified to one of the following types. ( A. Further, type A can be divided into three subclasses. ( These classifications are topological. To prove this theorem, we present a few lemmas.
It is sufficient to show the results for the first-exit maps, as we may use Lemma 3.5 to obtain results for the first-return maps. The case for equilibrium points is obvious. Therefore, by Lemma 3.10, we may assume that the map h maps each forward orbit injectively.
To demonstrate a contradiction, we assume T 2 < T ′ 1 . In this case, we have This contradicts the definition of T 1 . Thus, ). Proof of Theorem 3.9. By Lemma 3.11, types A, B, and C are easily preserved. The subtypes of type A are also preserved, because we have . Now we consider the relationship between the types and behavior of orbits. To avoid the problem of tangency, we introduce the following notion.
(1) S is of codimension one and locally flat.
(2) There exists a neighborhood U of x in X and a homeomorphism φ : A submanifold S ⊂ X without a boundary is forward topologically transversal to Φ if it is forward topologically transversal at every point on S. Remark 3.13. If a submanifold S ⊂ X without a boundary is topologically transversal to Φ at x ∈ S as defined in [16], then Φ and its time reversal Ψ are forward topologically transversal.
Remark 3.14. If a submanifold S ⊂ X without a boundary is forward topologically transversal to Φ at x ∈ S, x is not an equilibrium point.
The difference between forward topological transversality and topological transversality can be observed in the next example.
Example 3. We consider the following map from R × R 2 to R 2 : This map is a continuous flow. We can check that the homeomorphism h : where the flow Ψ is defined by Ψ(t, x, y) = (x + t, y). Then, the plane y = 0 is forward topologically transversal to Φ at the origin, but not topologically transversal.
Proof. Let x ∈ ∂A be fixed, and U be an open neighborhood of x in the definition of forward topological transversality. First, we observe that The conclusion follows from the inclusion U \Ā ⊂ int (X\A) and the definition of forward topological transversality.
There are two possibilities of the behavior in Lemma 3.15. If the types of boundary points are known, we can determine which of them are feasible. Theorem 3. 16. Let X be a topological manifold, Φ be a flow on X, and A ⊂ X an open set with the boundary being a locally flat manifold of codimension one. If ∂A is forward topologically transversal to Φ, we have the following: Proof. We show the statement (1) because the proof of (2) is similar. Let x ∈ ∂A is of type A-2 or C. By Lemma 3.15, there are two possibilities regarding the behavior of the forward orbit of x. To prove by contradiction , we assume that Type A-3 may be regarded as a degenerate case. The next lemma shows that other types imply forward topological transversality. Proof. If Φ is not forward topologically transversal to ∂A at x ∈ ∂A, we may find a sequence t n > 0 with Φ(t n , x) ∈ ∂A and t n → 0 as n → ∞. Then, E A (x) = PĀ(x) = x by definition.
If type A-3 occurs at a boundary point with forward topological transversality, it should be a part of a periodic orbit. This is an analogue of the classical result that the fixed points of first-return maps correspond to periodic points of the original flow. Proof. Let x ∈ ∂A be of type A-3 and T 1 := T e A (x) T 2 := T r A (x). By Lemma 3.15, T 1 = 0 and T 2 > 0 or T 1 > 0 and T 2 = 0. We now consider the former case.
As in Remark 3.14, x is not an equilibrium point. Therefore, x = PĀ(x) = Φ(T 2 , x) is a periodic point. By the definition of T 2 , we have Φ(t, x) ∈Ā for all t ∈ (0, T 2 ). Thus, we obtain O(x)\{x} ⊂ X\Ā. The proof for the other case is similar.
Conversely, let x be a periodic point with O(x)\{x} ⊂ A, and the minimal period be T > 0. Then, we have Φ(t, x) ∈ A for all t ∈ (0, T ), and consequently, Therefore, x ∈ ∂A is of type A-3. The proof for the case O(x)\{x} ⊂ X\Ā is similar.
The invariance of open subsets can be expressed in terms of the type of points on the boundary. This can be seen as a generalization of similar results regarding smooth manifolds and smooth flows. Conversely, we assume that all points on ∂A are of type B. If there exists x ∈ X\A with Φ(T, x) ∈ A for some T > 0, there is t 0 ∈ (0, T ) such that y := Φ(t 0 , x) ∈ ∂A by Lemma 2.3. This is contradictory because y is of type B, and consequently, Φ(T, x) = Φ(T − t 0 , y) ∈ A. Therefore, A is backward invariant.
Due to the problem of tangency, a similar characterization of forward invariance is more complicated. Proof. Let A be forward invariant and ∂A be forward topologically transversal to Φ. We fix x ∈ ∂A. SinceĀ is forward invariant, there exists δ > 0 with Φ((0, δ), x) ⊂ A by Lemma 3.15. By the invariance of A, it follows that Φ(t, x) ∈ A for all t > 0. Therefore, x is of type C.
Conversely, let all points on ∂A be of type C. By Lemma 3.17, Φ is forward topologically transversal to ∂A. If there exists x ∈ A with Φ(T, x) ∈ A for some T > 0, there is t 0 ∈ (0, T ] with y := Φ(t 0 , x) ∈ ∂A by Lemma 2.3. Since y is of type C, Φ(T, x) = Φ(T − t 0 , y) ∈ A. This is a contradiction, and therefore, A is forward invariant.
Since the first-exit map is defined only partially, it is of interest to consider the topological properties of the domain. If the assumption of forward topologically transversality is imposed, we may obtain some information regarding this point.

By the continuity of Φ, we may take an open neighborhood
Next, let us first consider the case where T > 0. Then, by the hypothesis, Proof. Because E A is idempotent, each point in im E A is of type A-1, A-3, or B. By Theorem 3.18, the type A-3 is incompatible with the assumption that there are no periodic points. Therefore, each point in im E A is of type A-1 or B.
If we further impose the assumption of topological transversality, the continuity of first-exit maps can be shown.

Theorem 3.23. Let X be a topological manifold, Φ be a flow on X, A ⊂ X an open set with boundary being a locally flat manifold of codimension one.
If ∂A is topologically transversal to Φ in the sense of the definition in [16] at x and E A (x), then, E A is continuous at x.
Proof. The continuity follows from Main Theorem B in [16] since ∂A is a local section of Φ at x and E A (x).

First-exit maps of planar flows
In this section, we consider the restriction for the first exit maps of planar flows. If the boundary of an open set is a Jordan curve, each parametrization induces a sequence of types, which can be used to study the dynamics around boundary points. Further, the properties of the first-exit maps can be described in terms of their parametrized representations.    Proof. First, we show that PĀ(c(x)) ∈ c(I) for all x ∈ I. Since this is obvious if c(x) is type B, we may assume that c(x) is type A-1. To obtain a contradiction, we assume PĀ(c(x)) = c(s) with s ∈ I. By the definition of the first-return map and the assumption of A-1, each sufficiently small τ > 0 has a corresponding neighborhood V τ of c(s) such that We now consider a sequence τ n with τ n → 0 as n → ∞. Then, we may define sequences {z n } with z n ∈ V τn ∩ A and {σ n } such that Note that σ n > 0, because A is open and z n ∈ A. Since Φ(−τ n , z n ) ∈ A, σ n ≤ τ n . Therefore, σ n → 0 as n → ∞. Moreover, we have Φ(−σ n , z n ) ∈ ∂A lim n→∞ Φ(−σ n , z n ) = c(s) by the continuity of Φ. Further, Φ(−σ n , z n ) is not type A-1 or B, since we have PĀ(Φ(−σ n , z n )) = Φ(−σ n , z n ) by the choice of σ n . Therefore, c(s) is in the closure of ∂A\c(I) in ∂A, which contradicts the assumption c(s) ∈ c(I).
Next, we show that for all δ > 0 and an interval U = (a, q), we have T r A (c(x)) ≤ δ for some x ∈ U . We assume that there exists δ > 0 and an interval U = (a, q) such that Then, by the continuity of Φ, we have for all t ∈ (0, δ). Therefore, q cannot be of type A-2 or C. Hence, we may construct sequences x n and t n > 0 such that x n → q, t n → 0 and PĀ(c(x n )) = Φ(t n , c(x n )) ∈ ∂A. By assuming type A-1, we find s n ∈ (0, t n ) with Φ(s n , c(x n )) ∈ A.
By considering the orbits, it follows that PĀ(c[x n , q)) ⊂ c(J). Let W ′′ be the open set encircled by the forward orbit of c(x n ) and ∂A.
We remark that type sequences define a topological invariant. In particular, two flows cannot be topologically equivalent if there is a combination of types that appears only for one of the two.
Example 4 (Sink and source are not topologically equivalent). Let us consider two flows defined by Φ(t, x, y) = (xe t , ye t ) and Ψ(t, x, y) = (xe −t , ye −t ). Another restriction can be described in terms of F E . Since the orbits of a flow are disjoint, F E satisfies a monotonicity condition. First, we consider this in terms of the first-exit map.
Proof. Let γ 1 be a Jordan curve defined by , c(t 0 )) (t ∈ [1/2, 1)) and A 1 be the interior of the domain encircled by γ 1 . Then, A\A 1 is also encircled by a Jordan curve and c(s 0 ) ∈ A\A 1 .
Proof. The first statement is a direct consequence of Lemma 4.5. The second statement can be obtained from the consideration of another parametrizationc This is a substantial restriction, as we can observe in the next theorem, which is a generalization of the classical result on the monotonicity of onedimensional first-return maps.
Theorem 4.7 (Main Theorem B). If F E is continuous at s and F E (s) = s, there exists δ > 0 such that F E (t) > F E (s) whenever s − δ < t < s and F E (t) < F E (s) whenever s < t < s + δ.
Proof. Let us first consider the case where F E (s) > s. By the continuity of Without loss of generality, we may also assume δ < ǫ and If t ∈ (s, s+δ), we have t < s+δ < s+ǫ = F E (s). Therefore, t ∈ (s, F E (s)), which implies F E (t) < F E (s).
The proof for F E (s) < s is similar. I = (a, b), F E is identity on I, except at most countable points.

Corollary 4.8. If F E monotonically increases on an interval
Proof. By monotonicity, F E is continuous on I, except at most countable points. By applying Theorem 4.7 to continuous points, we obtain this result.  Proof. First, we show that if there exists t ∈ [0, 1) with F E (t) < t, F E (s) < F E (t) for all s ∈ (t, 1). We assume that there exists s ∈ (t, 1) with F E (s) ≥ F E (t). By applying Theorem 4.7 at t, we see that F E takes a minimum at some t ′ ∈ (t, s). By Corollary 4.9, F E (t ′ ) = t ′ . Thus, we obtain which contradicts the assumption of t. By a similar argument, we have F E (s ′ ) < F E (s) for all s ∈ (t, 1) and Thus, if we set α := inf{t ∈ [0, 1) | F E (t) = t}, β := sup{t ∈ [0, 1) | F E (t) = t}, then F E monotonically decreases on [0, α) and [β, 1), and monotonically increases on [α, β).

Remark 4.11.
If F E is continuous on [0, 1) and not equal to identity, F E can be modified to be unimodal. Let α and β be as in Corollary 4.10. We define another parametrizationc bỹ c(t) := c(t + α).

4.3.
Realization of first-exit maps. We now consider a converse question: given a partial map on the boundary, can we find a flow such that the first exit map coincides with it?
Then, it can be checked that Φ(t, x, y) := H Ψ(t, H −1 (x, y)) is the flow with the desired properties. Therefore, we may construct a flow Ψ such that the first-exit map is Q. As in Example 1, we may find another flow Φ with M (Φ(t, z)) = Ψ(t, M (z)). By Lemma 3.11, we have for the first-exit map of Φ.
Although it appears possible to construct a flow with prescribed type sequences by pasting the flows with known types, it requires a consideration of the behavior, and therefore, it is not clear whether it is always feasible.

Application to the study of hybrid systems
As an application of the results obtained earlier, we now consider a class of hybrid systems and consider their relationship with the flows.
For a local flow Φ, we define the orbit of x ∈ X by Remark 5.2. Our notion of local flow differs from the one appearing in literature in that we do not require the domain of each orbit to be wellbehaved [15]. If the non-extendability condition is imposed, this definition essentially coincides with a local dynamical system [18,11]. Here, we consider the next class of hybrid systems, which is an adaptation of the definition appearing in [3].
Definition 5.5. An impacting system is a system defined by a triple of a local flow Φ defined in the neighborhood of H + ⊂ R 2 , a partial map P : R → R such that Im E H + ⊂ dom P × {0}, and a local flow Φ s : R × R s → R s , where R s := {(x, 0) | P (x) = x and E H + (x, 0) = (x, 0)}.
Remark 5.6. The local flow Φ s describes the sliding mode of the system.
The definition for orbits of an impacting system requires an additional notion. Let S and S ′ ⊂ S be well-ordered sets such that succ (S ′ ) ⊂ S, where succ is the successor function. Then, a hybrid time domain is a set T of the form where t : S ′ → R ≥0 ∪ {∞} is an event time sequence defined to be an orderpreserving function, such that t(min S ′ ) = 0 and t(succ (n)) = ∞ implies n is the maximal element of S ′ . In addition, we assume [t(n), t(succ (n)) = {t(n)} if t(n) = t(succ (n)). This definition of hybrid time domain is a modification of the formulation in [6].
(2) For each s ∈ S ′ , the dynamics on [t(s), t(succ s)) is described either by Φ or Φ s , i.e., if γ(s, t(s)) ∈ R s , for all τ ∈ [t(s), t(succ s)), and if γ(s, t(s)) ∈ R s , γ(s, τ ) = Φ(τ − t(s), γ(s, t(s))) for all τ ∈ [t(s), t(succ s)). This definition of forward trajectory is an adaptation of the concept of the solution in [7]. The assumption that the index set S is well-ordered is essential for a rule-based description of a model, as it enables specifying the dynamics in terms of updating rules.
Remark 5.7. In our definition of forward trajectories, uniqueness is not guaranteed. For example, it is possible for trajectories starting from a point on R × {0} to move either by a flow or a map.
Remark 5.8. There is no simultaneous multiple jumping in an impact system. If such an event occurs at (x, 0), it follows that P (x) = x and E H + (x, 0) = (x, 0). Therefore (x, 0) ∈ R s and its orbit remains in R s for some time, because the domain of Φ s is required to be open.
To consider the connection between hybrid systems and local flows, we introduce the following notion. Definition 5.9. A pair of local flows Φ 1 and Φ 2 induces an impacting system if Φ 1 and Φ 2 are defined in the neighborhoods of H − and H + respectively, and (P, Φ 2 , Φ s ) is an impacting system. Here, E H + = (P (x), 0) is the firstexit map of Φ 1 and Φ s is the restriction of Φ 2 to R s . Impacting systems of physical origin are often induced by local flows. By describing the transient dynamics of the switching using a flow, we may obtain a representation of the partial map of reset.
Example 5. A typical example of an impacting system is the impact oscillator, which is a harmonic oscillator with the influence of floor considered.
However, it is not necessary that every flow induces an impact system. is a homeomorphism. Let Φ be the flow induced from H and Ψ(t, x, y) := (x + t, y) by Lemma 2.4. Then, it does not induce an impact system. This is because the timing of the switching is not well-ordered for the Zeno trajectory of the origin, and consequently, the "next" point of the origin cannot be specified by a map.
Remark 5.10. The composition of the first exit map and the reset map can be regarded as a Poincaré map. In particular, it is easily observed that the fixed points of the composite map corresponds to equilibrium or periodic points of the impacting system. Proof. This follows immediately from Lemma 3.11.
The importance of impact oscillators can be understood by the following theorem.
Theorem 5.12 (Main Theorem D). Let (P, Φ, Φ s ) be an impacting system induced by local flows. If P is total, continuous, and not identity, then (P, Φ, Φ s ) is topologically conjugate with another impacting system (Q, Ψ, Ψ s ), Proof. Let the map (P, id 0 ) be the first-exit map of a local flowΦ. By an argument similar to that in Corollary 4.13, we can ascribeΦ to another local flow defined in a neighborhood of the unit disc so that infinity is mapped to 1, which is an equilibrium. Then, we apply Corollary 4.10 and Remark 4.11 to conclude that P is a unimodal map. By flipping the x-axis, we may assume that P decreases monotonically on (−∞, α] for some α ∈ R and identity on [α, ∞). By shifting the plane by α, we may assume that α = 0 without loss of generality.
Let us define a homeomorphism h : R → R by where P −1 denotes the negative branch. Then, H(x, y) := (h(x), y) gives the desired conjugation.
When an impacting system is induced by flows, Lemma 4.6 restricts the possible behavior.
Theorem 5.13. Let (P, Φ, Φ s ) be an impacting system induced by local flows, P (x) = y and E H + (y, 0) = (z, 0). If z lies between x and y, then all intersections of the forward orbit of (z, 0) with R × {0} lie between x and y.
Proof. This is an immediate consequence of Lemma 4.6.

Concluding Remarks
We have defined the concept of the first-exit and first-return maps and considered its basic properties. Although the discussion here will serve as a proof of concept, it is clear that there are issues to be considered.
In particular, the global restriction of type sequences in two-dimensional flows poses interesting questions. It is important to identify whether there is any prohibited combination of types other than BC, because it gives us information on the dynamics of planar flow in general. If this exists, such restriction will be global in nature, and a detailed analysis will be required.