Poincar\'e maps and suspension flows: a categorical remark

Poincar\'e maps and suspension flows are examples of fundamental constructions in the study of dynamical systems. This study aimed to show that these constructions define an adjoint pair of functors if categories of dynamical systems are suitably set. First, we consider the construction of Poincar\'e maps in the category of flows on topological manifolds, which are not necessarily smooth. We show that well-known results can be generalized and the construction of Poincar\'e maps is functorial, if a category of flows with global Poincar\'e sections is adequately defined. Next, we consider the construction of suspension flows and its functoriality. Finally, we consider the adjointness of the constructions of Poincar\'e maps and suspension flows. By considering the naturality, we can conclude that the concepts of topological equivalence or topological conjugacy of flows are not sufficient to describe the correspondence between map dynamical systems and flows with global Poincar\'e sections. We define another category of flows with global Poincar\'e sections and show that the suspension functor and the Poincar\'e map functor form an adjoint equivalence if these categories are considered. Hence, a categorical correspondence between map dynamical systems and flows with global Poincar\'e sections is obtained. This will enable us to better understand the connection between map dynamical systems and flows.


Introduction
Poincaré map and suspension flow constructions are fundamental tools employed in the study of dynamical systems. They are used to reduce a problem concerning continuous-time systems to one of discrete-time systems or vice versa, thereby connecting the two major types of dynamical systems [6,8,13].
Results on their relationship are scattered across the literature, and systematic treatments are scarce. However, by collecting these results, we can easily observe that a categorical relationship may exist between them. For example, the following properties are known: • If two diffeomorphisms are topologically conjugate, then their suspensions are topologically conjugate (Proposition 5.38 in [7]). • A flow with a Poincaré section is locally topologically equivalent to the suspension of its Poincaré map (Theorem 5.40 in [7]).
• Every diffeomorphism on a compact manifold is topologically conjugate with the Poincaré map of its suspension (Proposition 3.7 in [10]).
In the case of flows with global sections, stronger properties hold because Poincaré maps can be defined globally: • Topological equivalence of two flows can be determined in terms of Poincaré maps (Theorem 1 in [1], Proposition 1.11 in [14]). • A flow with a global section is topologically equivalent to the suspension of its Poincare map (Theorem 3.1 in [15]).
In loose terms, these results can be summarized as follows: isomorphisms are preserved under the constructions of Poincaré maps and suspension flows, and a Poincaré map of a suspension or a suspension of a Poincaré map can be identified with the original map or flow. These statements suggest the existence of categorical equivalence between a category of map dynamical systems and one of flows . Some categorical aspects of these constructions have been considered in the case of isomorphisms with topological conjugacy [5]. However, their relation remains unclear because it depends on the choice of categories. For example, some of the results mentioned above are not true if one uses topological conjugacy instead of topological equivalence to define isomorphisms.
This study aimed to perform a categorical treatment of the constructions of Poincaré maps and suspension flows in order to describe the exact relationship between them. This will enable us to unify the known results listed above and also "prove" the folklore correspondence of various notions between discrete-time and continuous-time systems, such as that of topological conjugacy and topological equivalence.
The rest of this paper is organized as follows. In Section 2, we define several categories of dynamical systems. In Section 3, we first introduce the notion of topological transversality for topological manifolds and continuous flows. We show that Poincaré maps can be defined analogously to the smooth case. Then, we define categories of flows with global Poincaré sections to show that the construction of Poincaré maps is functorial. In Section 4, we study the categorical relationship between Poincaré maps and suspension flows. We show that these two form a pair of adjoint equivalence if the categories are selected properly. Finally, in Section 5, we present some concluding remarks.

Categories of dynamical systems
In this section, we define various categories of dynamical systems to set up for the discussion later.
In what follows, topological manifolds are assumed to be second countable and Hausdorff. For the definitions of the concepts and basic results of category theory, we refer to [9,4,12]. Definition 2.1. A map dynamical system is a pair (f, X) of a topological manifold (without boundary) X and a homeomorphism f : Definition 2.2. A flow is a pair (Φ, X) of a topological manifold (without boundary) X and a continuous map Φ : A weak morphism (h, τ ) : (Φ, X) → (Ψ, Y ) between flows is a pair of a continuous map h : X → Y and a map τ : for all x ∈ X and t ∈ R.
Lemma 2.3. Each of the following forms a category if the composition of morphisms is defined by the composition of maps.
(1) Map dynamical systems and their morphisms.
We call the above a category of map dynamical systems Map, a category of flows Flow, and a category of flows with weak morphisms WFlow,, respectively. We note that Flow can be regarded as a subcategory of WFlow, as there is an obvious inclusion functor defined by (Φ, X) → (Φ, X) and

Isomorphisms in Map and
Flow are called topological conjugacies and isomorphic objects are called topologically conjugate. In WFlow, isomorphism is called topological equivalence and isomorphic objects are called topologically equivalent. These definitions coincide with the usual ones.
Remark 2.4. The categories in [5] correspond to Map or Flow in this paper.
Remark 2.5. Each of the categories defined above has a weakly initial element similar to the "universal dynamical system" of [12]. For example, the system (σ, Z) defined by σ(n) = n + 1 for all n ∈ Z is weakly initial in the category Map. Further, the set Map ((σ, Z), (f, X)) is isomorphic to the set of all orbits of (f, X). In particular, a morphism h : (σ, Z) → (f, X) corresponds to an orbit with period m ∈ N if and only if h admits the following factorization: Similar constructions can be carried out for Flow or WFlow.

Topological transversality and global Poincaré section
In this section, we define the concept of topological transversality for continuous flows on topological manifolds. Based on this definition, we show that Poincaré maps can be defined in a manner similar to the classical smooth case. Additionally, we introduce categories of flows with global Poincaré sections to consider the functoriality of the construction of Poincaré maps.
We adopt the definition of topological transversality given in [14,2] with a certain modification.
Definition 3.1. Let (Φ, X) be a flow, where X is an n-dimensional topological manifold. A submanifold S ⊂ X without a boundary is topologically transversal to Φ if (1) S is codimension one and locally flat.
(2) For each x ∈ S, there exists a neighborhood U of x in X and a homeomorphism φ : U → B ⊂ R n , where B is the unit ball such that Φ (U ∩ S) = B ∩ R n−1 × {0}. Further, there exist δ + (x) > 0 and δ − (x) < 0 such that Φ(x, [δ − (x), 0)) and Φ(x, (0, δ + (x)]) are contained in different connected components of U \S and Φ(x, [δ − (x), δ + (x)]))∩ S = {x}. Here, δ + and δ − can be taken locally uniformly, that is, there exist a neighborhood V ⊂ U of x and δ > 0 such that δ + (y) > δ and δ − (y) < −δ for all y ∈ V ∩ S.  Proof. Let U be a neighborhood of x satisfying the condition of (2) in Definition 3.1. By the continuity of Φ, there exist a neighborhood V 0 of x and The next lemma excludes the possibility of sequences that return to the section very frequently. Lemma 3.3. Let (Φ, X) be a flow and S ⊂ X be topologically transversal to Φ. If x ∈ S, there exist no sequences x n ∈ S and t n > 0 such that x n → x and t n → 0 as n → ∞ and Φ(x n , t n ) ∈ S.
Proof. Let U be a neighborhood of x satisfying condition (2) in Definition 3.1. Let V ⊂ U be a neighborhood of x such that there exists δ > 0 with δ + (y) > δ for all y ∈ V ∩ S. If x n ∈ S and t n > 0 are sequences such that x n → x and t n → 0 as n → ∞ and Φ(x n , t n ) ∈ S, then x n ∈ V ∩ S and consequently t n > δ for a sufficiently large n. This is a contradiction.
Remark 3.5. If the phase space is compact, condition (2) can be weakened to the condition that each x ∈ X has t ∈ R such that Φ(x, t) ∈ S. Indeed, let x ∈ S and consider the ω-limit set of x. Then, ω(x) ∩ S is nonempty by the invariance of the limit set. By Lemma 3.2, we observe that there exists t + > 0 such that Φ(x, t + ) ∈ S. The existence of t − is proved similarly.
Remark 3.6. By definition, a flow with a global Poincaré section has no equilibrium points. By using the argument in [1], we can show that a smooth flow without equilibrium points has a global Poincaré section if the phase space is compact.
According to these definitions, we have the following generalization of well-known results.
Theorem 3.7. Let (Φ, X) be a flow and S ⊂ X be topologically transversal to Φ. If x 0 ∈ S and there exists t for each x ∈ U ∩S and Φ(x, t) ∈ S for 0 < t < T Φ (x). Further, if S is a global Poincaré section, P Φ is defined on the entire S, and it is a homeomorphism.
Proof. First, we show the existence of T Φ (x) and P Φ(x) for each x ∈ U ∩ S, where U is a neighborhood of x 0 in X. Let 0 < r < t + and take a neighborhood V of Φ(x 0 , t + ) by applying Lemma 3.2 with ǫ = r and x = Φ(x 0 , t + ). Let U := Φ(V, −t + ). Then, we have Let us now show that T Φ : U ∩ S → (0, ∞) is continuous. Let x ∈ U ∩ S and ǫ be an arbitrary positive number less than T Φ (x). By We show that there exists an open neighborhood for all y ∈ U 2 ∩ S. If this is not the case, we may take sequences x n ∈ S ∩ U and s n ∈ (0, T Φ (x)−ǫ) so that Φ(x n , s n ) ∈ S and x n → x as n → ∞. As s n ∈ [0, T Φ (x)− ǫ], we may take a convergent subsequence Using the continuity of Φ, we observe that s = 0. Thus, we obtain sequences y n ∈ S ∩ U and t n ∈ (0, T Φ (x) − ǫ) so that y n → x and t n → 0 as n → ∞. However, this is impossible by Lemma 3.3. Therefore, there exists an open neighborhood If S is a global Poincaré section, it is clear that T Φ and P Φ are defined on the entire S. By the definition of a global Poincaré section, the same constructions can be carried out for Ψ(x, t) := Φ(x, −t). Then, we have These are established as follows. For x ∈ S, we have Φ(P Ψ(x), T Ψ (x)) = x ∈ S by definition. Therefore, T Ψ (x) ≥ T Φ (P Ψ(x)). On the other hand, The other relation is obtained by symmetry. Now, we have ( for all x ∈ S.
it is sufficient to prove the first formula. Suppose the series is convergent and let the sum T ∞ and for each N ≥ 1, x n := (P Φ) n (x) converges to x ∞ . For all n, x n is contained in Φ(x, [0, T ∞ ]) ∩ S, which is compact by the definition of topological transversality. Therefore, x ∞ ∈ S. If we set t n := T Φ • (P Φ) n (x), the convergence of the sum implies t n → 0 and Φ(x n , t n ) ∈ S for all n by definition. Thus, we have a pair of sequences (x n , t n ), which does not exist by Lemma 3.3. This is a contradiction.
A flow may admit many different global Poincaré sections, and consequently, a pair of a flow and a section may not necessarily be preserved under a weak morphism. If the sections are preserved by a weak morphism as sets, we have the following correspondence of the first return times between two flows.
which is the desired property.
As a consequence of this lemma, we have the following result. Proof. As P Φ : S → S and P Ψ : S ′ → S ′ are homeomorphisms, they define map dynamical systems. For each x ∈ S, we have Therefore, h| S : (P Φ, S) → (P Ψ, S ′ ) is a morphism of map dynamical systems.
Thus, we may define the following: The category of flows with global Poincaré sections FlowGS is the category whose objects are flows with global Poincaré sections and whose morphisms are morphisms in Flow, which preserves the global Poincaré sections.
Similarly, we may define a category WFlowGS whose objects are flows with global Poincaré sections and whose morphisms are morphisms in WFlow, which preserves the global Poincaré sections.
Objects in WFlowGS or FlowGS are denoted by a triple of the form (Φ, X, S), where (Φ, X) is a flow with a global Poincaré section S.
From Lemma 3.10, we immediately obtain the following:  Proof. Take the composition of P : WFlowGS → Map with the inclusion functor FlowGS ֒→ WFlowGS.

Poincaré maps and suspension flows
In this section, we consider the categorical relationship between a Poincaré map and a suspension.
To establish the notation, we recall the definition of a suspension flow.
Definition 4.1. Let f : X → X be a homeomorphism on a topological manifold X. The mapping torus X f of f is the manifold defined by where ∼ is the smallest equivalence relation with (x, 1) ∼ (f (x), 0) for each x ∈ X. There is a natural surjection π f : X × [0, 1] → X f , which sends each point to the corresponding equivalence class. We denote a point in where x ∈ X, 0 ≤ t < 1 and n ∈ Z is a unique integer satisfying s + t − 1 < n ≤ s + t.
Proof. Let h : (f, X) → (g, Y ) be a morphism in Map. First, we show that h : X f → Y g is well-defined and continuous. Well-definedness is verified by a direct calculation using g • h = h • f. The continuity follows from the commutativity of the following diagram and the universal property of the quotient topology: We show thath commutes with suspension flows. This is verified by a direct calculation: We show thath preserves the sections, that is, Finally, we show that Σ is a functor. It is clear that

Now, we have three categories and three functors between them:
(1) Inclusion functor I : FlowGS → WFlowGS.
(3) Suspension functor Σ : Map → FlowGS. From the existence of these functors, we immediately recover some known results on the preservation of isomorphisms. At this point, we must consider the degree of difference between the original flow and the suspension flow of the Poincaré map. First, we note that there is a pair of flows that are topologically equivalent but not topologically conjugate. The following is a modification of an example in [11].

Example 1. We define two flows on
Then, they are topologically equivalent but not topologically conjugate.
Proof. Topological equivalence is obvious. Suppose there is a homeomorphism h : A → A such that Φ 1 and Φ 2 are topologically conjugate, that is, for all z ∈ A and t ∈ R. By considering t = 1, we obtain h(z) = h(−z) for all z ∈ A, which contradicts the condition that h is injective.
Note that we may take A 0 = {x | 0 < x < 1} as a global Poincaré section for these flows. With this choice, the Poincaré map is the identity id A 0 in either case. Further, the suspension flow for id A 0 coincides with Φ 1 . Thus, the suspension flow of a Poincaré map is not necessarily topologically conjugate with the original flow. On the other hand, topological equivalence can be established.
We check that k (Φ,X,S) commutes with the flows by a direct calculation. When n ≥ 0, where n ∈ Z is a unique integer satisfying s + t − 1 < n ≤ s + t, , we calculate the following for n ≤ −1: The condition that k −1 (Φ,X,S) S = (S P Φ ) 0 can be verified by a direct calculation.
Using the invariance of domain theorem, we observe that (k, τ ) : IΣP I → I is a natural isomorphism. In ordinary terms, this observation can be phrased as follows. Another natural transformation can be constructed. for each x ∈ X.
Proof. First, we show that l (f,X) : (f, X) → P IΣ(f, X) is well-defined as a morphism in Map. As l (f,X) is a composition of continuous maps, it is well-defined and continuous. Additionally, we have for all x ∈ X.
We show that l is natural. Let h : (f, X) → (g, Y ) be a morphism in Map. Then, for all x ∈ X.
These results suggest that there is another category larger than FlowGS and smaller than WFlowGS for which the constructions of Poincaré maps and suspensions become adjoint.
for all x ∈ S 1 , 0 ≤ t < 1 and s ∈ R, where R Φ 1 and R Φ 2 are the same as in Lemma 4.5.
Proof. We show this by a direct calculation. Let x ∈ S 1 and 0 ≤ t < 1. Then, we have Here, we used the result of Lemma 3.9.
Lemma 4.11. The identity morphism in WFlowGS is rate-preserving. The composition of two rate-preserving morphisms is again rate-preserving.
Therefore, we can define a category RWFlowGS, whose objects are flows with global Poincaré sections and whose morphisms are rate-preserving morphisms. If we denote the inclusion functors by J − : FlowGS → RWFlowGS and J + : RWFlowGS → WFlowGS, it is clear that I = J + J − . for all [x, 0] ∈ (S P Φ ) 0 , 0 ≤ t < 1 and s ∈ R because T ΣP Φ ([x, 0]) = 1 for all [x, 0] ∈ (S P Φ ) 0 . We calculate the following: Thus, we have the following result: Proof. It is sufficient to verify the naturality conditions. Let (h, σ) : Further, for all [x, t] ∈ (S 1 ) P Φ 1 with 0 ≤ t < 1 and s ∈ R, where id is the time part of J − (h).
Combining the results above, we obtain the desired result, which gives us the exact relation between the constructions of Poincaré maps and suspension flows.
Proof. We verify that the triangle identities are satisfied by l : 1 Map → P IΣ = (P J + )(J − Σ) and (k, τ ) : In what follows, we omit J + or J − for ease of notation. Let (f, X) be an object in Map. Then, we have for all x ∈ X and 0 ≤ t < 1. Thus, we conclude that J − Σ ⊣ P J + .
The next corollary is an immediate consequence of Remark 4.6 and the injectivity of l (f,X) . We remark that the rate-preserving condition can always be assumed for topologically equivalent flows.

Concluding remarks
The categorical equivalence of Corollary 4.15 enables us to obtain correspondences between various concepts of flows and map dynamical systems. For example, Theorem 4.16 implies that the topological conjugacy of map dynamical systems is categorically equivalent to the topological equivalence of flows. This provides further justification for the use of topological equivalence in the study of flows, in addition to the usual argument that topological conjugacy is too strict.
We also observe a lack of correspondence for some notions. As flows with global Poincaré sections do not have equilibria, it follows that map dynamical systems do not have a concept corresponding to them under the equivalence obtained here. It would be interesting to consider whether there exists another pair of functors under which fixed points correspond to equilibria. A candidate will be the time-one map because it corresponds to the discretization functor, which has been considered in [5]. However, it is known that this construction is not very expressive, and it is unclear whether an interesting equivalence can be found [3].