A categorical view of Poincaré maps and suspension flows

Poincaré 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 Poincaré 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 Poincaré maps is functorial, if a category of flows with global Poincaré sections is adequately defined. Next, we consider the construction of suspension and its functoriality. Finally, we consider the adjointness of the constructions of Poincaré 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 Poincaré sections. We define another category of flows with global Poincaré sections and show that the suspension functor and the Poincaré map functor form an adjoint equivalence if these categories are considered. Hence, a categorical correspondence 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 continuoustime 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: • 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 main result of this paper is a theorem on the adjointness of constructions of Poincaré maps and suspension flows. To formulate these constructions in the categorical setting, we define categories of map dynamical systems and flows. A category Map of map dynamical systems can be defined by setting map dynamical systems as objects and continuous maps satisfying the condition of conjugacy as morphisms. For a category of flow dynamical systems, we consider flows with global Poincaré sections as objects so that the construction of Poincaré maps is possible. However, in the case of flows, morphisms can be defined more than one way because we may change the requirement on the preservation of parametrization of orbits. Here we consider three cases: (i) the parametrization is completely preserved, (ii) morphisms are rate-preserving in the sense of Definition 5.10, or (iii) the parametrization is ignored except for orientation. Correspondingly, we define three categories of flows with global Poincaré sections FlowGS, RWFlowGS and WFlowGS. Using these categories, it is shown that there are Poincaré map functor P : WFlowGS → Map and suspension functor : Map → FlowGS, each representing the corresponding construction. If we define inclusion functors J − : FlowGS → RWFlowGS and J + : RWFlowGS → WFlowGS, we have the following result, which describes the exact relationship between two constructions.

Main Theorem A:
The functor J − is left adjoint to PJ + . Further, the functors J − and PJ + form an adjoint equivalence. Therefore, the category RWFlowGS is equivalent to the category Map. Thus, the constructions of Poincaré maps and suspension flows give a categorical correspondence between flow dynamical systems and map dynamical systems.
To develop the results outlined above, we generalize the construction of Poincaré maps to the flows without smoothness assumptions.
Main Theorem B: Let ( , X) be a flow and S ⊂ X be topologically transversal to . If x 0 ∈ S and there exists t + > 0 such that (x 0 , t + ) ∈ S, there exist a neighbourhood U of x 0 in X and continuous maps P : U ∩ S → S and T : U ∩ S → (0, ∞) such that for each x ∈ U ∩ S and (x, t) ∈ S for 0 < t < T (x). Further, if S is a global Poincaré section, P is defined on the entire S, and it is a homeomorphism.
The rest of this paper is organized as follows. In Section 2, we introduce some basic concepts of the category theory to fix notation. In Section 3, we define several categories of dynamical systems. In Section 4, 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 5, 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 6, we present some concluding remarks.

Preliminaries
In this section, we introduce the concepts of category theory to fix notation. For details, we refer to [4,9,12].

Definition 2.1 (Category):
A category C consists of the following data.
• The class of objects Ob(C).
The composition of morphisms. That is, for each triple (X, Y, Z) of objects and pair of morphisms f : X → Y and g : Y → Z, a morphism g • f : X → Z is specified.
These data are required to satisfy the following axioms.
(1) The composition is associative. That is, we have For every object X, there exists an identity morphism id X : X → X such that f • id X = f and id X • g = g for all f : X → Y and g : Y → X.
We denote a category by bold fonts. For example, the category of sets and maps is denoted as Set.
A functor is a mapping between categories which preserves the structure.

Definition 2.2 (Functor):
Let C and D be categories. A functor F from C to D consists of the following data.
• For each object X in C, an object F(X) in D is specified.
• For each morphism f : These data are required to satisfy the following axioms.
(1) For each object X in C, we have F(id X ) = id F(X) .
(2) For each pair of morphisms f : We denote a functor from C to D by F : C → D. For each category C, there exists an identity functor 1 C : C → C defined by 1 C (X) = X for each object X and 1 C (f ) = f for each morphism f. The composition of two functors can be defined in an obvious manner.
The notion of natural transformation describes the correspondence between functors.

Definition 2.3 (Natural transformation):
Let F : C → D and G : C → D be functors. A natural transformation η from F to G is a family of morphisms in D indexed by the objects in C such that the following diagram is commutative for each morphism f : X → Y in C.
The notion of adjointness describes the situation where two functors are inverse in a weak sense.
for each X in C and Y in D.
The equivalence of categories can be described in terms of adjoint functors. If an adjoint equivalence exists, the categories C and D are equivalent.

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. Definition 3.1: A map dynamical system is a pair (f , X) of a topological manifold (without boundary) X and a homeomorphism f :

Definition 3.2:
A flow is a pair ( , X) of a topological manifold (without boundary) X and a continuous map : A weak morphism (h, τ ) : ( , X) → ( , Y) between flows is a pair of a continuous map h : X → Y and a map τ :

Lemma 3.3: 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.

Proof:
The proof is obvious for (1) and (2). For (3), we need to verify that the 'time-part' composition of the morphism satisfies the conditions of weak morphism. Let (h 1 , τ 1 ) : 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 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 3.4:
The categories in [5] correspond to Map or Flow in this paper.

Remark 3.5:
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 [2,14] with a certain modification.
(1) S is codimension one and locally flat.
(2) For each x ∈ S, there exists a neighbourhood U of x in X and a homeomorphism φ : are contained in different connected components of U\S and Here, δ + and δ − can be taken locally uniformly, that is, there exist a neighbourhood 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 neighbourhood of x satisfying the condition of (2) in Definition 4.1.
By the continuity of , there exist a neighbourhood V 0 of x and δ > 0 such that The next lemma excludes the possibility of sequences that return to the section very frequently.

Lemma 4.3:
Let ( , X) be a flow and S ⊂ X be topologically transversal to . 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 (x n , t n ) ∈ S.
Proof: Let U be a neighbourhood of x satisfying condition (2) in Definition 4.1. Let V ⊂ U be a neighbourhood 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 (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 4.5:
If the phase space is compact, condition (2) can be weakened to the condition that each x ∈ X has t ∈ R such that (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 4.2, we observe that there exists t + > 0 such that (x, t + ) ∈ S. The existence of t − is proved similarly.
Remark 4.6: 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 4.7 (Main Theorem B):
Let ( , X) be a flow and S ⊂ X be topologically transversal to . If x 0 ∈ S and there exists t + > 0 such that (x 0 , t + ) ∈ S, there exist a neighbourhood U of x 0 in X and continuous maps P : U ∩ S → S and T : U ∩ S → (0, ∞) such that for each x ∈ U ∩ S and (x, t) ∈ S for 0 < t < T (x). Further, if S is a global Poincaré section, P is defined on the entire S, and it is a homeomorphism.
Proof: First, we show the existence of T (x) and P (x) for each x ∈ U ∩ S, where U is a neighbourhood of x 0 in X. Let 0 < r < t + and take a neighbourhood V of (x 0 , t + ) by applying Lemma 4.2 with = r and x = (x 0 , t + ). Let U := (V, −t + ). Then, we have 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 (x) − ) so that (x n , s n ) ∈ S and x n → x as n → ∞. As s n ∈ [0, T (x) − ], we may take a convergent subsequence s n i → s ∈ [0, T (x) − ] as i → ∞. Using the continuity of , we observe that s = 0. Thus, we obtain sequences y n ∈ S ∩ U and t n ∈ (0, T (x) − ) so that y n → x and t n → 0 as n → ∞. However, this is impossible by Lemma 4.3. Therefore, there exists an open neighbourhood U 0 : for all y ∈ U 0 ∩ S. Therefore, T (x) is continuous and consequently P (x) is also continuous.
If S is a global Poincaré section, it is clear that T and P are defined on the entire S. By the definition of a global Poincaré section, the same constructions can be carried out for (x, t) := (x, −t). Then, we have These are established as follows. For x ∈ S, we have (P (x), T (x)) = x ∈ S by definition. Therefore, T (x) ≥ T (P (x)). On the other hand, we have ( T (P (x)). The other relation is obtained by symmetry. Now, we have for each N ≥ 1, x n := (P ) n (x) converges to x ∞ . For all n, x n is contained in (x, [0, T ∞ ]) ∩ S, which is compact by the definition of topological transversality. Therefore, x ∞ ∈ S. If we set t n := T • (P ) n (x), the convergence of the sum implies t n → 0 and (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 4.3. 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. respectively, and (h, τ ) : ( , X) → ( , Y) be a weak morphism. Then,

Lemma 4.9: Let ( , X) and ( , Y) be flows with global Poincaré sections S and S ,
which is the desired property.
As a consequence of this lemma, we have the following result.
Therefore, h| S : (P , S) → (P , 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 ( , X, S), where ( , X) is a flow with a global Poincaré section S.
From Lemma 4.10, we immediately obtain the following: • For each morphism h :

Corollary 4.13: The construction of a Poincaré map is functorial for FlowGS.
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 5.1: Let f : X → X be a homeomorphism on a topological manifold X. The mapping torus X f of f is the manifold defined by 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 X f by [x, t], where x ∈ X, 0 ≤ t < 1.

Definition 5.2:
Let (f , X) be a map dynamical system. The suspension flow f : where x ∈ X, 0 ≤ t < 1 and n ∈ Z is a unique integer satisfying s + t − 1 < n ≤ s + t.  Y) be a morphism in Map. First, we show thath : 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: where x ∈ X, 0 ≤ t < 1 and n ∈ Z is a unique integer satisfying s + t − 1 < n ≤ s + t. We show thath preserves the sections, that is, Finally, we show that is a functor. It is clear that Now, we have three categories and three functors between them: (1) Inclusion functor I : FlowGS → WFlowGS. 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 5.5: 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 1 and 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 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.

Lemma 5.6:
There is a natural transformation (k, τ ) : I PI → I defined by the following for each ( , X, S) in FlowGS : Proof: First, we show that (k, τ ) ( ,X,S) : I PI( , X, S) → ( , X, S) is well-defined as a weak morphism in WFlowGS. Well-definedness and continuity of k ( ,X,S) follow from the commutativity of the following diagram: We check that k ( ,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 have Noting that ((P ) −1 (x), t) = (x, t − T • (P ) −1 (x)), we calculate the following for n ≤ −1: The condition that k −1 ( ,X,S) S = (S P ) 0 can be verified by a direct calculation.
Finally, we show that (k, τ ) is natural. Let h : ( 1 , X 1 , S 1 ) → ( 2 , X 2 , S 2 ) be a morphism in FlowGS. Then, we have where x ∈ S 1 and 0 ≤ t < 1. We also have this is a contradiction. Therefore, tT (x) − t T (x ) ≤ 0. By interchanging t and x with t and x , we also have t T (x ) − tT (x) ≤ 0. Therefore, we conclude that x = x and consequently t = t .
Using the invariance of domain theorem, we observe that (k, τ ) : I PI → I is a natural isomorphism. In ordinary terms, this observation can be phrased as follows. Another natural transformation can be constructed.

Lemma 5.9:
There is a natural transformation l : 1 Map → PI defined by for each x ∈ X.
Proof: First, we show that l (f ,X) : (f , X) → PI (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 1 and R 2 are the same as in Lemma 5.6.

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 4.9.

Lemma 5.12:
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 − .
If [x, t] ∈ (S 1 ) P 1 with 0 ≤ t < 1, we have Further, for all [x, t] ∈ (S 1 ) P 1 with 0 ≤ t < 1 and s ∈ R, where p 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. In what follows, we omit J + or J − for ease of notation. Let for all x ∈ X, 0 ≤ t < 1 and s ∈ R. These results show that the following diagram commutes in RWFlowGS.
The next corollary is an immediate consequence of Remark 5.7 and the injectivity of l (f ,X) .

Corollary 5.16: The categories Map and RWFlowGS are equivalent.
We remark that the rate-preserving condition can always be assumed for topologically equivalent flows. Theorem 5.17: Let ( 1 , X 1 , S 1 ) and ( 2 , X 2 , S 2 ) be isomorphic in WFlowGS. Then, they are isomorphic in RWFlowGS.

Concluding remarks
The categorical equivalence of Corollary 5.16 enables us to obtain correspondences between various concepts of flows and map dynamical systems. For example, Theorem 5.17 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].