arXiv:2112.13992v3 [math.DS] 29 May 2022

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MORSE HYPER-GRAPHS OF TOPOLOGICAL SPACES AND

DECOMPOSITIONS

TOMOO YOKOYAMA

Abstract. The cell complex structure is one of the most fundamental struc-tures in topology and combinatorics, the Morse decomposition of a dynamicalsystem analyzes the global gradient behavior, and the Reeb graph of a functionis an elementary tool in Morse theory to represent the global connection and isalso used to analyze continuous and discrete data in topological data analysis.In this paper, we unify three concepts of cell complexes, Morse decompositions,and Reeb graphs into a concept. In fact, we introduce topological invariantsfor topological spaces and decompositions, which are analogous to abstract(weak) orbit spaces and Morse graphs for flows. To achieve these, we defineanalogous concepts of recurrence and “chain-recurrence” for topological spacesand decompositions such that the original and new recurrences correspond toeach other for a flow orbits on a locally compact Hausdorff space and the orbitspace. We show that the Morse hyper-graphs exist for any topological spacesand any invariant decompositions (e.g. compact foliated spaces). Moreover,the abstract weak element spaces generalize the Reeb graphs of Morse func-

tions and the Morse (hyper-)graphs and refine abstract cell complexes.

1. Introduction

The cell complex structure is one of the most fundamental tools in algebraictopology and combinatorics. For instance, the CW complex structure, which isan example of the cell complex structure, was introduced by Whitehead [43] andis widely developed. Moreover, the simplicial complex structure, which is also anexample of the cell complex structure and is led from the barycentric subdivisionof Poincare, is the foundation of the simplicial homology. In addition, the existenceof triangulations (i.e. the simplicial complex structures) of topological spaces is aclassical problem in topology. Thus the cell complex structure is a fundamentalstructure in topology and combinatorics. In this paper, we generalize such anabstract cell structure by using concepts of dynamical systems.

From a dynamical system’s point of view, Birkhoff introduced the concepts ofrecurrent points and described the limit behavior of orbits [6]. Conley defined aweak form of recurrence, called chain recurrence, for a flow on a compact metricspace [12]. The Conley theory says that dynamical systems on compact metricspaces can be decomposed into blocks, each of which is a chain recurrent one ora gradient one. Then this decomposition is called the Morse decomposition, and

Date: May 31, 2022.2020 Mathematics Subject Classification. Primary 37B20; Secondary 54B15,05E45,57-

08,57Q70.Key words and phrases. Cell complex, Morse decomposition, Reeb graph, recurrence, topolog-

ical space, decomposition, quotient space.The author was partially supported by JSPS Grant Number 20K03583.

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http://arxiv.org/abs/2112.13992v3

MORSE HYPER-GRAPHS 2

implies a directed graph, called a Morse graph, which can capture the gradient be-haviors. The Morse decompositions for mappings and semiflows are developed forarbitrary metric spaces [21–24]. Moreover, the Morse decomposition for set-valueddynamical systems is essentially extend by Mcgehee [32] and was developed in sev-eral ways [10,15,26,40]. In addition, the Morse decompositions for semigroups andset-valued semiflows [3,4,7,8,16,34–36], random dynamical systems [9,14,25,27–31],nonautonomous set-valued dynamical systems [1,41,42], and combinatorial dynam-ical systems on simplicial complexes [5, 17, 33], are also introduced and studied invarious ways. In [45], the Morse graph of dynamical systems is refined into abstractorbit spaces, which is also a refinement of Reeb graphs of Hamiltonian flows withfinitely many singular points on surfaces, and the CW decompositions which consistof the unstable manifolds of singular points for Morse flows on closed manifolds.Moreover, using the abstract orbit spaces, the author reconstructed some classes offlows from the time-one mappings.

In this paper, interpreting concepts of recurrence, chain recurrence, and abstractorbits, we introduce abstract (weak) element spaces and Morse hyper-graphs oftopological spaces and decompositions, show the existence of such hyper-graphsfor topological spaces and invariant decompositions (e.g. compact foliated spaces),and reduce abstract (weak) element spaces into Morse hyper-graphs. Moreover, theabstract weak element space generalizes both the Reeb graph of a Morse functionand the Morse graph for a flow, and refines an abstract cell complex.

The present paper consists of eight sections. In the next section, recall someconcepts of dynamical system to interpret such concepts in topological spaces anddecompositions. In §3, we recall notions of combinatorics and topology, and theconcepts of dynamical system are interpreted in topological spaces. In addition,the existence of Morse hyper-graphs is shown. In §4, we demonstrate that abstractelement spaces can be reduced into Morse hyper-graphs. In §5, the correspondencewith abstract element spaces and abstract cell complexes is constructed. Moreprecisely, the abstract element space of a cell complex is an abstract cell complexwith the specialization pre-order and its height. In §6, the recurrence of a flowon a locally compact Hausdorff space coincides with the recurrence for the orbitspace. In §7, to apply results as above to decompositions (and especially compactfoliated spaces), we also introduce concepts for decompositions on topological spacesas above, and show the existence of Morse hyper-graphs and the reducibility ofabstract (weak) element spaces into such hyper-graphs. In addition, we demonstratethat the Reeb graph of a Morse function on a closed manifold is the abstract weakelement space of the set of connected components of level sets as abstract multi-graphs. In the final section, some examples are illustrated to describe recurrentproperties and the necessity of conditions.

2. Preliminaries from dynamical system’s point of view

We recall some concepts of dynamical systems to interpret such concepts intopological spaces and decompositions later.

2.1. Notion of dynamical systems. A flow is a continuous R-action on a topo-logical space. Let v : R ×X → X be a flow on a topological space X . For t ∈ R,define vt : X → X by vt := v(t, ·). For a point x of S, we denote by O(x) the orbitvt(x) | t ∈ R of x.


Definition 1. A subset is invariant (or saturated) if it is a union of orbits.

The saturation of a subset is the union of orbits intersecting it. A nonemptyclosed invariant subset is minimal if it contains no proper nonempty closed invari-ant subsets.

2.1.1. Orbit spaces and orbit class spaces. The orbit space T/v of an invariantsubset T of X is a quotient space T/ ∼ defined by x ∼ y if O(x) = O(y). Notice

that an orbit space T/v is the set O(x) | x ∈ T as a set. The (orbit) class Oof an orbit O is the union of orbits each of whose orbit closure corresponds to O(i.e. O := y ∈ S | O(y) = O), where O is the closure of O. The orbit classspace T/v of an invariant subset T of X is a quotient space T/ ≈ defined by x ≈ y

if O(x) = O(y). Moreover, the orbit class space T/v is the set O(x) | x ∈ T with the quotient topology. Denote by τv (resp. τv) the topology of the orbit spaceX/v (resp. orbit class space X/v). Note that the orbit class space X/v is the T0-tification (see the definition in §3.1.2) of the orbit space X/v, and that T/v (resp.T/v) is a subset of the orbit space X/v (resp. orbit class space X/v).

2.1.2. Properties of points and orbits. A point x of S is singular if x = vt(x) for anyt ∈ R and is periodic if there is a positive number T > 0 such that x = vT (x) andx 6= vt(x) for any t ∈ (0, T ). A point is closed if it is singular or periodic. Denoteby Sing(v) (resp. Per(v), Cl(v)) the set of singular (resp. periodic, closed) points.

The ω-limit (resp. α-limit) set of a point x is ω(x) :=⋂

n∈Rvt(x) | t > n (resp.

α(x) :=⋂

n∈Rvt(x) | t < n).

Definition 2. A point x of S is recurrent if x ∈ ω(x) ∪ α(x).

Denote by R(v) (resp. R(v)) the set of recurrent (reps. non-closed recurrent)points and by P(v) the set of non-recurrent points. An orbit is closed (resp.recurrent) if it contains a closed (resp. recurrent) point.

Definition 3. An orbit is proper if it is embedded.

A point is proper if its orbit is proper.

Definition 4. The derived set of a subset A is the set difference A − A, whereB − C is used instead of the set difference B \ C when B ⊆ C.

2.1.3. Characterizations of properness and recurrence. We have the following ob-servation.

Lemma 2.1. An orbit O of a flow on a topological space is proper if and only if

the derived set O −O is closed.

Proof. Let v be a flow on a topological space X . Suppose that O ⊆ X is proper.Since O is embedded, for any point x ∈ O, there is an open neighborhood Ux of xsuch that O∩Ux = O∩Ux. Then the union U :=

⋃

x∈O Ux is an open neighborhood

of O such that O ∩ U = O. Therefore the derived set O − O = O \ U is closed.Conversely, suppose that O−O is closed. The complement X− (O−O) is an openneighborhood of O such that O ∩ U = O. This means that O is embedded.

We have the following observation.


Lemma 2.2. The following statements hold for a flow v on a paracompact manifold:

(1) The set P(v) of non-recurrent points is the set of non-closed proper points.

(2) The set R(v) of non-closed recurrent points is the set of non-proper points.

(3) The set R(v) of recurrent points is the set of closed or non-proper points.

Proof. Notice that a closed orbit is proper and recurrent, and that M = Cl(v) ⊔P(v) ⊔ R(v), where ⊔ denotes a disjoint union. We claim that any non-closedpoint is proper if and only if it is non-recurrent. Indeed, fix a non-closed pointx ∈ M . By [44, Corollary 3.4], the point x is proper if and only if O(x) = O(x).From invariance of orbit and by definition of recurrence, if x is non-recurrent thenO(x) = O(x) and so proper. By [11, Theorem VI], the closure of a non-closedrecurrent orbit O of v contains uncountably many non-closed recurrent orbits whoseclosures are O. Then if x is proper then x is not recurrent. Therefore, the point xis proper if and only if x is non-recurrent.

Since non-recurrent points are non-closed, the set P(v) of non-recurrent points isthe set of non-closed proper points. Moreover, any non-closed point is non-properif and only if it is recurrent. Since non-proper points are non-closed, the set ofnon-proper points is the set R(v) of non-closed recurrent points.

2.1.4. Abstract weak orbits and abstract orbits. Define an invariant subset [x] for aflow on a topological space X as follows:

[x] =

the connected component of Sing(v) containing x if x ∈ Sing(v)

the connected component of Per(v) containing x if x ∈ Per(v)

the connected component of

y ∈ P(v) | α(x) = α(y), ω(x) = ω(y) containing x if x ∈ P(v)

y ∈ R(v) | α(x) = α(y), ω(x) = ω(y) if x ∈ R(v)

We call that [x] is the abstract weak orbit of x (see [45] for details of propertiesof abstract orbits). Similarly, define an invariant subset 〈x〉 of X as follows:

〈x〉 =

the connected component of Sing(v) containing x if x ∈ Sing(v)

the connected component of Per(v) containing x if x ∈ Per(v)


y ∈ P(v) | α(x) = α(y), ω(x) = ω(y) containing x if x ∈ P(v)

y ∈ R(v) | O(x) = O(y) if x ∈ R(v)

We call that 〈x〉 is the abstract orbit of x (see [45] for details of properties ofabstract weak orbits).

2.1.5. Chain recurrence. Let w : R ×M → M be a flow on a metric space (M,d).

For any ε > 0 and T > 0, a pair (xi)k+1i=0 , (ti)

ki=0 is an (ε, T )-chain from a point

x ∈ M to a point y ∈ M if x0 = x, xp+1 = y, ti > T and d(wti (xi), xi+1) < ε forany i = 0, . . . , k. Define a binary relation ∼CR on M by x ∼CR y if for any ε > 0and T > 0 there is an (ε, T )-chain from x to y.

Definition 5. A point x ∈ M is chain recurrent [13] if x ∼CR x.

Denote by CR(w) the set of chain recurrent points, called the chain recurrentset. It is known that the chain recurrent set CR(w) of a flow on a compactmetric space is closed and invariant and contains the non-wandering set Ω(w) [12,Theorem 3.3B], and that connected components of CR(w) are equivalence classes


of the relation ≈CR on CR(w) [12, Theorem 3.3C], where x ≈CR y if x ∼CR y andy ∼CR x.

2.1.6. Morse graphs. For a flow v on a compact metric space M with a set M =Miλ∈Λ of disjoint compact invariant subsets, a finite directed graph (V,D) withthe vertex set V := Mi | 1 ≤ i ≤ n, and with the directed edge set D :=(Mj,Mk) | Dj,k 6= ∅ is a Morse graph of M if M −

⊔

i Mi =⊔

Dj,k, where

Dj,k :=

x ∈ M −⊔

i

Mi | α(x) ⊆ Mj , ω(x) ⊆ Mk

and⊔

denotes a disjoint union. Then such a graph is denoted by GM and calledby the Morse graph of M.

Definition 6. The graph GM is the Morse graph of v ifM is the set of connectedcomponents of the chain recurrent set CR(v).

Then denoted by Gv the Morse graph of v. Similarly, if M is the set of connectedcomponents of the recurrent set R(v), then the graph GM is denoted by GR(v).Then a Morse graph MM of a flow v is a quotient space of the orbit space witha directed structure. From [45, Theorem 4.5], any connected components of thechain recurrent point set are unions of abstract orbits.

3. Preliminaries for the description of topological and

combinatorial concepts

In this section, we introduce topological invariants for topological spaces. Toconstruct such invariants, we define some concepts (e.g. Morse graph, recurrence,“chain recurrence”, “abstract orbit”) for topological spaces from a dynamical sys-tem’s point of view.

3.1. Fundamental notions of combinatorics and topology.

3.1.1. Notion of combinatorics. An abstract multi-graph is a triple of sets V,Eand a mapping A : E → x, y | x, y ∈ V . By a graph, we mean a cell complexwhose dimension is at most one and which is a geometric realization of an abstractmulti-graph. A (abstract) hyper-multi-graph is a triple of sets V,E and amapping A : E → V ∗, where V ∗ is the family of non-empty finite subsets of V . A(abstract) hyper-graph is a pair of a set V and a family H ⊆ V ∗.

By a decomposition, we mean a family F of pairwise disjoint nonempty subsetsof a set X such that X =

⊔

F . Since connectivity is not required, the sets of orbitsof homeomorphisms are also decompositions.

A binary relation ≤ on a set X is a pre-order if it is reflexive (i.e. a ≤ a forany a ∈ X) and transitive (i.e. a ≤ c for any a, b, c ∈ X with a ≤ b and b ≤ c).For a pre-order ≤, the inequality a < b means both a ≤ b and a 6= b. A pre-orderorder ≤ is a total order (or linear order) if either a < b or b < a for any pointsa 6= b. A chain is a totally ordered subset of a pre-ordered set with respect to theinduced order. Let (X,≤) be a pre-ordered set. Define the height ht(x) of x ∈ Xby ht(x) := sup|C|− 1 | C : chain containing x as the maximal point. Define theheight of the empty set as −1. The height ht(A) of a nonempty subset A ⊆ X isdefined by ht(A) := supx∈A ht(x). A subset A ⊂ X is a downset if b ∈ A for anypoint a ∈ A and any point b ∈ X with b ≤ a.


3.1.2. Notion of topology. Let (X, τ) be a topological space. A point x ∈ X is T0

(or Kolmogorov) if for any point y ∈ X − x, there is an open subset U of suchthat x, y ∩ U is a singleton, T1 (or closed) if the singleton x is closed, and

TD [2] if the derived set x − x is a closed subset. Here a singleton is a setconsisting of a point. A topological space is T0 (resp. T1, TD, etc.) if each point isT0 (resp. T1, TD, etc.).

Definition 7. The class x of a point x ∈ X is defined by a subset y ∈ X | x =

y.

Then the set X := x | x ∈ X of classes is a decomposition of X and is aT0 space as a quotient space, which is called the T0-tification (or Kolmogorovquotient) of X .

Definition 8. A subset is invariant if it is a union of classes.

Notice that any closed subset is invariant. The specialization pre-order ≤τ

on a topological space (X, τ) is defined as follows: x ≤τ y if x ∈ y. The heightsof a point, a subset, and a topological space are defined as the heights with respectto the specialization pre-order. For any k ∈ Z≥0 and a topological space X , denoteby Xk the set of height k points of X and by X≤k the set of points of X whoseheights are less than or equal to k.

3.2. Topological concepts for topological spaces from Dynamical systems.Let (X, τ) be a topological space. By Lemma 2.1, we define properness as follows.

Definition 9. A point of X is proper if it is TD.

Denote by Cl(τ) (resp. P(τ), R(τ)) the set of closed points (resp. non-closedproper points, non-proper points). By definition, we have X = Cl(τ)⊔P(τ)⊔R(τ).From Lemma 2.2, we call that a point x ∈ X is (τ -)recurrent if it is either T0 ornon-TD. Denote by R(τ) the set of recurrent points. Then R(τ) = Cl(τ) ⊔ R(τ)and X = P(τ) ⊔R(τ). We will show that an orbit of a flow v on a locally compactHausdorff space X is recurrent if and only if it is τv-recurrent, where τv is thequotient topology of the orbit space X/v (see Theorem 6.2).

Definition 10. Define an abstract element 〈x〉 for an element x ∈ X :

〈x〉 :=


x′ ∈ X | x − x = x′ − x′ containing x if x ∈ Cl(τ) ⊔ P(τ)


x′ ∈ X | x = x′ containing x if x ∈ R(τ)

We have the following observation.

Lemma 3.1. The abstract elements form a decomposition and satisfy the following

property:

〈x〉 :=

the connected component of Cl(τ) containing x if x ∈ Cl(τ)


x′ ∈ P(τ) | x − x = x′ − x′ containing x if x ∈ P(τ)


x′ ∈ R(τ) | x = x′ containing x if x ∈ R(τ)


Proof. Let (X, τ) be a topological space. Fix a point x ∈ X . If x ∈ Cl(τ), then 〈x〉

is the connected component of x′ ∈ X | x − x = x′ − x′ = ∅ = Cl(τ).

Suppose that x ∈ R(τ). If there is a point y ∈ 〈x〉∩Cl(τ), then x ∈ y = y ⊆Cl(τ), which contradicts x ∈ R(τ) = X − (Cl(τ) ⊔ P(τ)). Thus 〈x〉 ∩ Cl(τ) = ∅.We claim that 〈x〉 ∩P(τ) = ∅. Indeed, assume that there is a point y ∈ 〈x〉 ∩P(τ).

Since x ∈ R(τ), we have x ∈ y − y. Since y − y is closed, we have

y = x ⊆ y − y, which is a contradiction. Thus 〈x〉 is the connected

component of x′ ∈ R(τ) | x = x′.

Suppose that x ∈ P(τ). Then the derived set x − x 6= ∅ is closed, and

the abstract element 〈x〉 is the connected component of x′ ∈ X | x − x =

x′ − x′ 6= ∅. Therefore 〈x〉 ∩ Cl(τ) = ∅. We claim that 〈x〉 ∩ R(τ) = ∅.

Indeed, assume that there is a point y ∈ 〈x〉 ∩ R(τ). Then y ∈ x − x. Since

x − x is closed, the set difference y − y = x − x is closed and soy ∈ P(F), which contradicts y ∈ R(τ). Thus 〈x〉 is the connected component of

x′ ∈ P(τ) | x − x = x′ − x′.

An abstract element is closed (resp. recurrent, proper, etc.) if it is contained inCl(τ) (resp. R, Cl(τ) ⊔P(τ), etc.). Define the abstract element space X/〈τ〉 asa quotient space X/ ∼〈τ〉 defined by x ∼〈τ〉 y if 〈x〉 = 〈y〉.

3.2.1. Quasi-recurrence of points. We will define an analogous concept of chainrecurrence. However, the absence of metrics obstructs a direct interpretation ofchain recurrence. To construct “chain recurrence”, recall facts that any points in theorbit closure for a flow are chain recurrent, and that any connected components ofthe chain recurrent point set are unions of abstract orbits. To achieve the analogousconcept, we define non-maximal points as follows.

Definition 11. A point x ∈ X is non-maximal if there is a point y ∈ X suchthat x ( y.

Note that a point is non-maximal if and only if it is not maximal with respectto the specialization pre-order of τ . A point is maximal if it is not non-maximal.Denote by max τ the set of maximal points. Then we define (τ -)quasi-recurrence fora topological space, which is the analogous concept of chain recurrence, as follows.

Definition 12. A point x ∈ X is (τ -)quasi-recurrent if 〈x〉 contains a recurrentor non-maximal point.

Denote by Q(τ) the set of quasi-recurrent points. Notice that Q(τ) is a unionof abstract elements but is not closed in general (see the example in § 8.1.2).

3.2.2. Morse hyper-graph for a topological space. We define the Morse hyper-graph(V,H) for a topological space as follows: Let (X, τ) be a topological space with aset X = Xλλ∈Λ of disjoint invariant non-empty subsets Xλ (λ ∈ Λ). For anyI ⊆ λ, define a hyper-edge HI as follows: For any point x ∈ X−

⊔

λ∈ΛXλ, we saythat x ∈ HI if there are disjoint invariant non-empty subsets (Ci)i∈I of the subset

x − x such that Ci ⊆ Xi and x − x =⊔

i∈I Ci. Put V := Xλ | λ ∈ Λand H := Xii∈I | HI 6= ∅, I ⊆ Λ.

Definition 13. A hyper-graph GX := (V,H) is a Morse hyper-graph of X ifX −

⊔

λ∈ΛXλ =⊔

I⊆Λ HI .


Definition 14. The hyper-graph GX is the Morse hyper-graph of τ if X is theset of connected components of the set Q(τ) of quasi-recurrent points.

Notice that we can similarly define the Morse hyper-multi-graph. We have thefollowing statement.

Lemma 3.2. The following statements hold for a topological space (X, τ):

(1) R(τ) ⊔ (⋃

x∈X(x − x) ∩ P(τ)) ⊆ Q(τ).(2) X −Q(τ) ⊆ P(τ) ∩max τ .(3) The set Q(τ) of quasi-recurrent points is invariant.

(4) For any element x ∈ P(τ) ∩max τ , each connected component of x − x is

a closed subset contained in a connected component of Q(τ).

Proof. The non-minimal property implies that the set of non-maximal points isinvariant. By definition of quasi-recurrence, the set Q(τ) of quasi-recurrent points

is invariant such that⋃

x∈X x−x ⊆ Q(τ), X−Q(τ) ⊆ max τ , and X−P(τ) =

R(τ) ⊆ Q(τ). Then R(τ) ⊔ (⋃

x∈X(x − x) ∩ P(τ)) ⊆ Q(τ) and X − Q(τ) ⊆P(τ) ∩max τ .

For any point x ∈ P(τ) ∩max τ , the derived subset x − x is a closed subset

contained in X −max τ ⊆ Q(τ) and so each connected component of x − x iscontained in a connected component of Q(τ).

The equality in Lemma 3.2 (1) does not hold in general (see Example in §8.2.3).Lemma 3.2 implies the following existence of Morse hyper-graph for a topologicalspace.

Lemma 3.3. Let be (X, τ) a topological space, Q′ ⊆ X an invariant subset con-

taining Q(τ), and X = Xλλ∈Λ the set of connected components of Q′. Then the

Morse hyper-graph GX exists.

Proof. Since any connected component of a subset is closed in the subset, anyconnected component Xλ is invariant. Therefore X is a set of disjoint invariantsubsets. Lemma 3.2 implies thatX−

⊔

λ∈Λ Xλ = X−Q′ ⊆ X−Q(τ) ⊆ P(τ)∩max τ .Fix any point x ∈ X −

⊔

λ∈ΛXλ ⊆ X − Q(τ) ⊆ P(τ) ∩ max τ . By Lemma 3.2,

the derived subset x − x ⊆ Q(τ) =⊔

λ∈ΛXλ is closed and so the intersection

Cλ := (x − x) ∩ Xλ is invariant. Set I := λ ∈ Λ | Cλ 6= ∅. Then x ∈ HI .This means that X −

⋃

λ∈Λ Xλ =⊔

I⊆ΛHI .

The previous lemmas imply the following statement.

Corollary 3.4. The Morse hyper-graph for a topological space exists.

4. Reductions to Morse hyper-graphs for topological spaces

The Morse hyper-graph of a topological space can be obtained as a quotientspace of the abstract element space as follows.

Theorem 4.1. The Morse hyper-graph Gτ for a topological space (X, τ) is a quo-

tient space of the abstract element space X/〈τ〉.

Proof. Let Gτ = (V,H) be the Morse hyper-graph for a topology τ with V =Xλλ∈Λ and H = Xii∈I | HI 6= ∅, I ⊆ Λ. Then X = Xλλ∈Λ is the setof connected components Xλ of Q(τ). By R(τ) ⊆ Q(τ) =

⊔

λ∈ΛXλ, since any


abstract elements are connected, any recurrent abstract elements are contained insome Xλ.

Fix any point x ∈ P(τ). Then the abstract element 〈x〉 is a connected component

of x′ ∈ P(τ) | x−x = x′−x′ 6= ∅ and the derived set x−x is closed.Suppose that x is quasi-recurrent. Then 〈x〉 is connected and contained in Q(τ).Therefore there is an index i ∈ Λ such that 〈x〉 ⊆ Xi. Suppose that x is not quasi-recurrent. Then 〈x〉 ⊆ X − Q(τ) = X −

⊔

λ∈ΛXλ =⊔

I⊆ΛHI . Fix any x′ ∈ 〈x〉.

Then x − x = x′ − x′ ⊆ Q(τ) =⊔

λ∈Λ Xλ. Put Ci := Xi ∩ (x − x) =

Xi ∩ (x′ − x′). Define I := i ∈ Λ | Ci 6= ∅. Since any connected componentsof a subset are closed in the subset, the intersections Ci for any i ∈ I are invariantsuch that Ci ⊆ Xi and x′−x′ = x−x = (x−x)∩

⊔

λ∈Λ Xλ =⊔

i∈I Ci.Then x′, x ∈ HI and so 〈x〉 ⊆ HI .

In general, the Morse hyper-graph GX even for X as in Lemma 3.3 is not aquotient space of the abstract element space (see the example in 8.1.1).

5. Generalization of abstract cell complex structures

This section shows that the abstract element space is a generalization of anabstract cell complex structures.

5.1. Generalization of abstract cell complex structure. We recall conceptsof cell complexes.

Definition 15. A decomposition eλλ∈Λ on a Hausdorff space X is a cell com-plex if it satisfies the following two conditions:(1) Any element eλ is a k-dimensional ball for some k ∈ Z≥0. In this case, theelement eλ is called a k-cell, and the integer k is called the dimension of eλ.(2) For any k-cell eλ, we have eλ − eλ ⊆ X≤k−1, where X≤k−1 is the union of cellswhose dimension is less than k.

Definition 16. For a set S with a transitive relation ≺ and a function dim: S →Z≥0, the triple (S,≺, dim) is an abstract cell complex if x ≺ y implies dimx <dim y.

Then dimx is called the dimension of x, and x is called a cell. A k-cell is acell whose dimension is k.

5.1.1. Correspondence with abstract element spaces and abstract cell complexes. Wehave the following correspondence.

Proposition 5.1. The abstract element space of a cell complex is an abstract cell

complex with the specialization pre-order and its height.

Proof. Let F be a cell complex on a topological space X . By definition of cellcomplex, the abstract element space X/〈F〉 is F as a set, and the height ht(L)of any element L of the quotient space X/〈F〉 with respect to the specializationpre-order ≤ corresponds to the dimension of the cell of F . Then the tuple (S,≺, dim) := (X/〈F〉,≤, ht) is an abstract cell complex.


6. Correspondence of recurrences

In this section, we demonstrate the correspondence of recurrences for flows andtopological spaces. By definitions, the α-limit set and the ω-limit set of x are closedand invariant. We have the following observation.

Lemma 6.1. Let v be a flow on a Hausdorff space X and x ∈ X a point. Then

O(x) = α(x) ∪O(x) ∪ ω(x).

Proof. By definition of α-limit set and ω-limit set, the orbit closure O(x) containsα(x)∪O(x)∪ω(x). Define a continuous mapping vx : R → X by vx := v(·, x). Sincea closed interval I ⊂ R is compact, the image vx(I) is compact. The Hausdorffseparation axiom of X implies that the image of a closed interval by vx is closed.

Assume that there is a point y ∈ O(x)−(α(x)∪O(x)∪ω(x)). By y /∈ α(x)∪ω(x),

there is a number T > 0, such that y /∈ vx(R<−T ) and y /∈ vx(R>T ). Then

y /∈ vx(R− [−T, T ]) and so there is an open neighborhood U of y such that U ∩vx(R− [−T, T ]) = ∅. Since the image vx([−T, T ]) ⊆ O(x) of the interval [−T, T ] isclosed and y /∈ O(x), the difference V := U \ vx([−T, T ]) is an open neighborhoodof y. Then V ∩ O(x) = (U \ vx([−T, T ])) ∩ O(x) = U ∩ (O(x) − vx([−T, T ])) =

U ∩ vx(R − [−T, T ]) = ∅ and so y /∈ O(x), which contradicts y ∈ O(x). Thus

O(x) = α(x) ∪O(x) ∪ ω(x).

We have the following relation between recurrences.

Theorem 6.2. Let v be a flow on a Hausdorff space X and τX/v the quotient

topology of the orbit space X/v. A τX/v-recurrent orbit is v-recurrent. If X is

locally compact, then the converse holds.

Proof. Suppose that O(x) is an τX/v-recurrent orbit. Then either O(x) is closed or

O(x) − O(x) is not closed. If O(x) is closed, then x ∈ α(x) ∪ ω(x) trivially. Thus

we may assume that O(x) − O(x) is not closed. By Lemma 6.1, this implies that

x ∈ O(x) −O(x) ⊆ α(x) ∪ ω(x) = α(x) ∪ ω(x).Suppose that X is locally compact and O(x) is a v-recurrent orbit. We may

assume that O(x) is not closed. We claim that O(x) − O(x) is not closed. Indeed,

assume O(x)−O(x) is closed. Applying the Baire category theorem, since O(x) islocally compact Hausdorff and so Baire, and since each open subset of a Baire spaceis a Baire space, the orbit O(x) is Baire. Let Un := v(R − [n, n + 1], x) ⊂ O(x)for n ∈ Z. Because O(x) is v-recurrent, each Un is open dense in O(x). SinceO(x) is Baire, we have

⋂

n Un is dense, which contradicts to the definition of Un.

By O(x) − O(x) ⊆ α(x) ∪ ω(x), the non-closedness of O(x) − O(x) and the closedinvariance of α(x) ∪ ω(x) imply that O(x) ⊂ α(x) ∪ ω(x). Therefore O(x) is τX/v-recurrent.

There is a transitive flow on a metrizable space such that each non-singular pointis not τX/v-recurrent but v-recurrent (see an example in §8.2.2 for details).

7. Application to decompositions and foliated spaces

To apply previous results to decompositions and foliated spaces, we recall con-cepts of decompositions and introduce topological invariants for decompositions ontopological spaces.


7.1. Notion of decompositions. By a decomposition, we mean a family F ofpairwise disjoint nonempty subsets of a set X such that X =

⊔

F , where⊔

denotesa disjoint union. Let F be a decomposition on a topological space X . For x ∈ X ,denote by F(x) the element containing x. An element L ∈ F is closed if L = L.

Definition 17. An element L ∈ F is proper if the derived set L− L is closed.

Denote by Cl(F) (resp. P(F), R(F)) the union of closed (resp. non-closedproper, non-closed non-proper) elements. Set R(F) := Cl(F)⊔R(F) = X −P(F).

Definition 18. A subset A of X is F-saturated (or F-invariant) if F(A) = A,where F(A) =

⋃

x∈A F(x).

The subset F(A) is called the saturation of A.

Definition 19. F is said to be Tσ (resp. recurrent, etc.) if the quotient spaceX/F of F is Tσ (resp. recurrent, etc.).

Define the class element L of an element L ∈ F by L := x ∈ X | F(x) = L.

Then the family F := L | L ∈ F is a decomposition, called the class decompo-sition of F . The quotient space of the topological space X by the decompositionF is called the class (element) space of F and is denoted by X/F . Denote byτF := F(U) | U ∈ τ the set of F -saturation of open subsets.

Definition 20. A decomposition F is invariant if τF ⊆ τ .

We will show that if F is either a foliated space or a continuous action of atopological group then F is invariant (see Proposition 7.11). For an invariantdecomposition F , the set τF of saturations of open subsets becomes a topology andcalled the saturated topology on X . In general, the set τF of F -invariant subsetsis not a topology on the quotient space X/F even if L is F -invariant for any L ∈ F(see an example in §8.2.1 for details).

7.1.1. Properties of decompositions. We observe the following statements.

Lemma 7.1. The F-invariant subsets Cl(F), P(F), and R(F) of an invariant

decomposition F on a topological space are F-invariant.

Proof. Let F be an invariant decomposition on a topological space X . For any

point x ∈ Cl(F), we have F(x) = y ∈ X | F(y) = F(x) = F(x) ⊆ Cl(F).

This means that Cl(F) is F-invariant. Fix a point x ∈ P(F). Since the derive set

F(x) − F(x) is closed and F -invariant, if there is a point y ∈ F(x) − F(x), then

F(x) = F(y) ⊆ F(x)−F(x), which is a contradiction. Thus F(x) = F(x) ⊆ P(F).

Since Cl(F) ⊔ R(F) = X − P(F) is F -invariant, so is the complement P(F).

Lemma 7.2. The following statements are equivalent for a decomposition F on a

topological space:

(1) The decomposition F is invariant.

(2) The closure of any F-invariant subset is F-invariant.

Proof. Let F be a decomposition on a topological space (X, τ). Suppose thatτF ⊆ τ . Assume that there is an F -invariant subset A whose closure is not F -invariant. Then there is a point x ∈ F(A) − A. Put U := X − A. This U is anopen neighborhood of x. Since τF ⊆ τ , the saturation V := F(U) is open withV ∩ A = ∅. Then V ∩ A = ∅. Since V is F -invariant, we obtain V ∩ F(A) = ∅.


This contradicts that V is a neighborhood of x ∈ F(A). Thus A is F -invariant forany F -invariant subset A ⊆ X .

Conversely, suppose that the closure of an F -invariant subset is F -invariant.Fix any open subset B. Set F := X − F(B). Since B ∩ F = ∅, we have B ∩ F =∅. The hypothesis implies that F is F -invariant. Then F(B) ∩ F = ∅ and so

F(B)∩X −F(B) = ∅. This implies that X −F(B) is closed and so F(B) is open.Therefore τF ⊆ τ .

We have the following correspondence.

Lemma 7.3. Let F be an invariant decomposition on a topological space X and

p : X → X/F the quotient map. The following statements hold:

(1) ⋃

U | U ∈ τX/F = p−1(U) | U ∈ τX/F = τF .(2) p(U) | U ∈ τF = τX/F .

(3) The quotient map p induces the canonical bijection τF → τX/F by U 7→ p(U).

Proof. The union of any elements of the quotient topology τX/F is invariant open

and so ⋃

U | U ∈ τX/F = p−1(U) | U ∈ τX/F ⊆ τF . From U = p(p−1(U))for any U ∈ τX/F , we have τX/F ⊆ p(U) | U ∈ τF. By U = p−1(p(U)) for anyU ∈ τF , since any elements of τF are invariant open, we obtain p(U) | U ∈ τF ⊆τX/F . Thus p(U) | U ∈ τF = τX/F . Because p−1(p(U)) = U for any U ∈ τF , we

have τF = p−1(U) | U ∈ τX/F. This means that p induces the canonical bijectionτF → τX/F .

From the previous lemma, we can identify the saturated topology of an invariantdecomposition on a topological space as the topology on the decomposition space.

7.2. Topological concepts for decompositions from dynamical systems.We define topological concepts for decompositions from dynamical systems as thecase for topological spaces.

7.2.1. Abstract weak element and abstract element for a decomposition. Let F bea decomposition on a topological space X and q : X → X/F the quotient map.

Definition 21. Define an abstract weak element [L] and abstract element〈L〉 for an element L ∈ F :

[L] :=

the inverse image by q

of the connected component of

L′ ∈ X/F | L ∼= L′, L− L = L′ − L′ containing L if L ⊆ Cl(F) ⊔ P(F)



L′ ∈ X/F | L ∼= L′, L = L′ containing L if L ⊆ R(F)

〈L〉 :=



L′ ∈ X/F | L− L = L′ − L′ containing L if L ⊆ Cl(F) ⊔ P(F)



L′ ∈ X/F | L = L′ containing L if L ⊆ R(F)

Here L ∼= L′ means that L and L′ are homeomorphic.


For a point x ∈ X , define [x] := [F(x)] and 〈x〉 := 〈F(x)〉. As the same argumentof the proof of Lemma 3.1, we have the following observation.

Lemma 7.4. The abstract elements form an invariant decomposition and satisfy

the following property:

〈x〉 =

the inverse image by q of the connected component of

Cl(F)/F containing F(x) if x ∈ Cl(F)


q(x′ ∈ P(F) | F(x)−F(x) = F(x′)−F(x′))

containing F(x) if ∈ P(F)


q(x′ ∈ R(F) | F(x) = F(x′)) containing F(x) if x ∈ R(F)

Proof. Let F be an invariant decomposition on a topological space X . Fix a pointx ∈ X . If x ∈ Cl(F), then 〈x〉 is the inverse image by q of the connected component

of q(x′ ∈ X | F(x)−F(x) = F(x′)−F(x′) = ∅) = q(Cl(F)) = Cl(F)/F .

Suppose that x ∈ R(F). If there is a point y ∈ 〈x〉 ∩ Cl(F), then x ∈ F(y) =F(y) ⊆ Cl(F), which contradicts x ∈ R(F) = X − (Cl(F) ⊔ P(F)). Thus 〈x〉 ∩Cl(F) = ∅. We claim that 〈x〉 ∩ P(F) = ∅. Indeed, assume that there is a point

y ∈ 〈x〉 ∩ P(F). Since x ∈ R(F), we have F(x) ⊆ F(y)−F(y). Since F(y)−F(y)

is closed, we have F(y) = F(x) ⊆ F(y)−F(y), which is a contradiction. Thus 〈x〉

is the inverse image by q of the connected component of q(x′ ∈ R(F) | F(x) =

F(x′)).

Suppose that x ∈ P(F). Then the derived set F(x) − F(x) 6= ∅ is closed, andthe abstract element 〈x〉 is the inverse image by q of the connected component of

q(x′ ∈ X | F(x) − F(x) = F(x′) − F(x′) 6= ∅). Therefore 〈x〉 ∩ Cl(F) = ∅. Weclaim that 〈x〉 ∩ R(F) = ∅. Indeed, assume that there is a point y ∈ 〈x〉 ∩ R(F).

Since F(x)−F(x) is closed, the set difference F(y)−F(y) = F(x)−F(x) is closedand so y ∈ P(F), which contradicts y ∈ R(F). Thus 〈x〉 is the inverse image by q

of the connected component of q(x′ ∈ P(F) | F(x)−F(x) = F(x′)−F(x′)).

The previous lemma implies the following statement.

Lemma 7.5. The abstract weak elements form an invariant decomposition and

satisfy the following property:

[x] =


q(x′ ∈ Cl(F) | F(x) ∼= F(x′)) containing F(x) if x ∈ Cl(F)


q(x′ ∈ P(F) | F(x) ∼= F(x′),F(x)−F(x) = F(x′)− F(x′))

containing F(x) if x ∈ P(F)


q(x′ ∈ R(F) | F(x) ∼= F(x′),F(x) = F(x′))

containing F(x) if x ∈ R(F)

Define the abstract weak element space X/[F ] as a quotient space X/ ∼[F ]

defined by x ∼[F ] y if [x] = [y]. Similarly, define the abstract element spaceX/〈F〉 as a quotient space X/ ∼〈F〉 defined by x ∼〈F〉 y if 〈x〉 = 〈y〉. Since


F(x) ⊆ [x] ⊆ 〈x〉 for any x ∈ X , the abstract weak element space is a quotientspace of the decomposition space, and the abstract element space is a quotientspace of the abstract weak element space.

7.2.2. Quasi-recurrence of elements of a decomposition. A point x ∈ X is non-maximal if there is a point y ∈ X such that F(x) ( F(y). Define quasi-recurrenceas follows.

Definition 22. A point x of a topological space X with a decomposition F isquasi-recurrent if 〈x〉 contains either a point in R(F) or a non-maximal point.

Denote by Q(F) the set of quasi-recurrent points, called the quasi-recurrentset of a decomposition F and by maxF the set of maximal point with respect tothe saturated topology. As the same argument of the proof of Lemma 3.2, we havethe following statement.

Lemma 7.6. The following statements hold for an invariant decomposition F on

a topological space X:

(1) R(F) ⊔ (⋃

L∈F(L− L) ∩ P(F)) ⊆ Q(F).(2) X −Q(F) ⊆ P(F) ∩maxF .

(3) The set Q(F) of quasi-recurrent points and the set of non-maximal points are

F-invariant.

(4) For any element L ⊂ P(F) ∩ maxF , the derived subset L − L is a closed

F-invariant subset contained in Q(F).

Proof. Let F be an invariant decomposition on a topological space X . For any non-maximal point x ∈ X , there is a point y ∈ X such that F(x) ⊆ F(x) ( F(y). This

means that the set of non-maximal points is F -invariant. By definition of quasi-recurrence, from Lemma 7.1, the set Q(F) of quasi-recurrent points is F -invariantsuch that

⋃

L∈F(L − L) ⊆ Q(F), X − Q(F) ⊆ maxF , and X − P(F) = R(F) ⊆

Q(F). Then R(F)⊔(⋃

L∈F(L−L)∩P(F)) ⊆ Q(F) and X−Q(F) ⊆ P(F)∩maxF .

For any element L ⊂ P(F) ∩maxF , since any closed F -invariant subset is F -

invariant, the derived subset L− L is a closed F -invariant subset in X −maxF ⊆Q(F).

The equality in Lemma 7.6 (1) does not hold in general (see Example in §8.2.3).

7.2.3. Morse hyper-graph of a decomposition. As the case of topological spaces, wedefine the Morse hyper-graph of a decomposition on a topological space as follows:Let F be a decomposition on a topological space X with a set M = Mλλ∈Λ

of disjoint F -invariant non-empty subsets Mλ (λ ∈ Λ). For any I ⊆ λ, define ahyper-edge HI as follows: For any point x ∈ X−

⊔

λ∈Λ Mλ, we say that x ∈ HI if

there are disjoint F -invariant non-empty subsets (Ci)i∈I of the subset F(x)−F(x)

such that Ci ⊆ Mi and F(x) − F(x) =⊔

i∈I Ci. Put V := Mλ | λ ∈ Λ andH := Mii∈I | HI 6= ∅, I ⊆ Λ.

Definition 23. A hyper-graph GM := (V,H) is the Morse hyper-graph of M ifM −

⊔

λ∈ΛMλ =⊔

I⊆Λ HI .

For a Morse hyper-graph GM = (V,H), the associated graph GM = (V,E) isdefined by E := Mi,Mj | HI 6= ∅, i, j ∈ I ⊆ Λ. We define the Morse hyper-graph of a decomposition as follows.


Definition 24. The hyper-graph GM is the Morse hyper-graph of F if M isthe set of inverse images by q of connected components of q(Q(F)) ⊂ X/F , whereq : X → X/F is the quotient map.

Then denoted by GF the Morse hyper-graph of F . As the same argument of theproof of Lemma 3.3, the following existence of Morse hyper-graph of the decompo-sition holds.

Lemma 7.7. Let be F an invariant decomposition on a topological space X, Q′ ⊆ Xa F-invariant subset containing Q(F), and M = Mλλ∈Λ the set of inverse images

by q of connected components of Q′/F . Then the Morse hyper-graph GM exists.

Proof. Since any connected component of a subset is closed in the subset, any

connected component Mλ is F -invariant and so the family M is a set of disjointF -invariant non-empty subsets. Lemma 7.6 implies that X−

⊔

λ∈ΛMλ = X−Q′ ⊆X −Q(F) ⊆ P(F) ∩maxF . Fix a point x ∈ X −

⊔

λ∈Λ Mλ ⊆ P(F) ∩maxF . By

Lemma 7.6, the derived subset F(x) − F(x) ⊆ Q(τ) ⊆ Q′ =⊔

λ∈Λ Mλ is closed

F -invariant and so the intersections Cλ := (F(x) − F(x)) ∩ Mλ are F -invariant.Set I := λ ∈ Λ | Cλ 6= ∅. Then x ∈ HI . This means that X −

⋃

λ∈ΛMλ =X −Q(F) =

⊔

I⊆ΛHI .

The previous lemma implies the following existence.

Corollary 7.8. The Morse hyper-graph for an invariant decomposition on a topo-

logical space exists.

In general, the Morse hyper-graph GM even for M as in Lemma 7.7 is not aquotient space of the abstract element space (see the example in 8.1.1).

7.3. Reductions to Morse hyper-graphs for invariant decompositions. The-orem 4.1 implies the following reduction to Morse hyper-graphs for invariant de-compositions on topological spaces.

Theorem 7.9. The Morse hyper-graph for an invariant decomposition on a topo-

logical space is quotient spaces of the abstract element space and the abstract weak

element space.

Proof. Let GF = (V,H) be the Morse hyper-graph for an invariant decompositionF with V = Mλλ∈Λ and H = Mii∈I | HI 6= ∅, I ⊆ Λ. Then M = Mλλ∈Λ

is the set of inverse images by q of connected components of q(Q(F)) ⊂ X/F . ByR(F) ⊆ Q(F) =

⊔

λ∈ΛMλ, since the images by the quotient map q : X → X/F ofany abstract elements are connected, any recurrent abstract elements are containedin some Mλ.

Fix any point x ∈ P(F). Then the abstract element 〈x〉 is the inverse image by

q of a connected component of q(x′ ∈ P(F) | F(x)− F(x) = F(x′)−F(x′) 6= ∅)

and the derived set F(x)−F(x) is closed and F -invariant. Suppose that x is quasi-recurrent. Then 〈x〉 is contained in Q(F) =

⊔

λ∈ΛMλ such that q(〈x〉) is connectedin Q(F)/F . Therefore there is an index i ∈ Λ such that 〈x〉 ⊆ Mi. Suppose thatx is not quasi-recurrent. Then 〈x〉 ⊆ X − Q(F) = X −

⊔

λ∈Λ Mλ =⊔

I⊆ΛHI .

Fix any x′ ∈ 〈x〉. Then F(x) − F(x) = F(x′) − F(x′) ⊆ Q(F) =⊔

λ∈ΛMλ. Put

Ci := Mi ∩ (F(x) − F(x)) = Mi ∩ (F(x′) − F(x′)). Define I := i ∈ Λ | Ci 6= ∅.

Since Mi and F(x)− F(x) are F -invariant, so is the intersection Ci for any i ∈ I.


Moreover, we have that Ci ⊆ Mi and F(x′) − F(x′) = F(x) − F(x) = (F(x) −F(x)) ∩

⊔

λ∈Λ Mλ =⊔

i∈I Ci. Then x′, x ∈ HI and so 〈x〉 ⊆ HI .

7.4. Generalization of Reeb graphs of Morse functions. We recall Reebgraph as follows. For a function f : X → R on a topological space X , the Reebgraph of a function f : X → R on X is a quotient space X/ ∼Reeb defined byx ∼Reeb y if there are a number c ∈ R and a connected component of f−1(c) whichcontains x and y. Then the inverse image of a value of R is called the level set.Notice that the Reeb graph of a Morse function (or more generally a function withfinitely many critical points) on a closed manifold is a finite graph (see [37, The-orem 3.1] for details). Moreover, by the proof of Proposition 7.10, note that anyedge of the Reeb graph of a Morse function on a closed manifold is the leaf spaceof a codimension one product compact foliation, because the leaf space of a con-tinuous codimension two (and so one) compact foliation of a compact manifoldis Hausdorff [18–20, 39]. We show that the abstract element space is a naturalgeneralization of the Reeb graph of a Morse function.

Proposition 7.10. The Reeb graph of a Morse function on a closed manifold is

the abstract weak element space of the set of connected components of level sets as

abstract multi-graphs.

Proof. Let f be a C1 function with finitely many critical points on a closed manifoldM and F the set of connected components of level sets of f . Then any elements of Fare closed. Denote by Crit(f) the set of critical points. We claim that any elementsof F contained in a connected component of the complement X −

⋃

x∈Crit(f)F(x)

are homeomorphic to each other and are submanifolds. Indeed, fix such a connectedcomponent C. Then the restriction f |C is a submersion, and so the restriction F|Cis a codimension one foliation on the connected manifold C. Any element containedin C of F is a closed codimension one submanifold because of the inverse functiontheorem. By the existence of the function f , the codimension one compact foliationF|C is transversely orientable, and so the holonomy group of each leaf of F|C istrivial. The Reeb stability (cf. [38, Theorem 2]) implies that any leaf L ∈ F|C hasits neighborhood on which the restriction of F is a product foliation. Thus anyleaves of F|C are homeomorphic to each other.

Therefore any connected component of the complement X −⋃

x∈Crit(f)F(x) is

contained in the abstract weak element. Suppose that f is Morse. By the Morselemma, any element of F containing a critical point is isolated but is not a manifold,and so such elements are abstract weak elements. This implies that any connectedcomponents of the complement X −

⋃

x∈Crit(f) F(x) correspond to abstract weak

elements. Therefore any vertices and edges of the Reeb graph are abstract weakelements. This means that the Reeb graph of f is the abstract weak element spaceM/F as an abstract multi-graph.

7.5. Foliated spaces. Recall some concepts of foliated spaces. Fix any r ∈ Z≥0 ⊔∞ and any n ∈ Z≥0. Let Z,Z ′ be topological spaces and U an open subsetin Rn × Z with coordinates (x, z). A mapping f : U → Rn is of Cr if its partialderivatives up to order r with respect to x exist and are continuous on U . Amapping h = (h1, h2) : U → Rn ×Z ′ of the form h(x, z) = (h1(x, z), h2(z)) is of C

r

if h1 is of class Cr and h2 is continuous, where .


Definition 25. A collection U = (Ui, φi)i∈Λ if a Cr foliated atlas of dimensionn on a topological space X if it satisfies the following conditions:(1) The subset Uii∈Λ is an open covering of X .(2) For any i ∈ Λ, there are a locally compact separable completely metrizable spaceZi and an open ball Bi in Rn such that φi : Ui → Bi × Zi is a homeomorphism.(3) For any i, j ∈ Λ, the coordinate changes φj φ

−1i | : φi(Ui ∩ Uj) → φj(Ui ∩ Uj)

are locally Cr mappings of the form

φj φ−1i (x, z) = (gij(x, z), hij(z)) .

Then hij is called the local transverse components of the changes of co-ordinates. Each (Ui, φi) is called a Cr foliated chart, and the inverse imagesφ−1i (Bi×z) for any z ∈ Zi are called plaques. The foliated atlas of a topological

space X induces a locally Euclidean topology τU on X . In fact, the basic opensubsets are the plaques of all foliated charts. The connected components of X withrespect to this topology τU are called leaves. Each leaf is a connected n-manifoldwith the Cr structure canonically induced by U . The set of leaves is denoted by Fand called a Cr foliated structure of dimension n on X . The quotient space iscalled the leaf space of F and denoted by X/F . Notice that the leaf space is adecomposition space. For a point x ∈ X , denote by F(x) the leaf containing x.

Definition 26. Two Cr foliated atlases on X define the same Cr foliated structureif their union is a Cr foliated atlas.

Definition 27. A maximal Cr foliated atlas is called a Cr foliated structure.

Definition 28. The pair of a topological space X and its Cr foliated structure iscalled a foliated space.

The Cr foliated space with boundary is defined similarly, and the boundaryof a Cr foliated space is a Cr foliated space without boundary. A subset of afoliated space is saturated (or invariant) if it is a union of leaves. Notice thatany saturated subspace is also a Cr foliated space.

7.5.1. Existence of Morse hyper-graphs of foliated spaces. We have the followingobservations.

Lemma 7.11. If F is either a foliated structure of a foliated space or the set of

orbits of a group-action on a topological space, then F is an invariant decomposition.

Proof. Since the closure of the saturation of any subset is F -invariant, Lemma 7.2and Lemma 7.3 imply the assertion.

We have the following existence of Morse hyper-graphs of foliated spaces.

Theorem 7.12. Let (X,F) be either a compact foliated space or a topological space

equipped with the set of orbits of a group action. The following statements hold:

(1) The Morse hyper-graph GF exists.

(2) The Morse hyper-graph GF is the quotient spaces of the abstract element space

X/〈F〉 and the abstract weak element space X/[F ].

Proof. Lemma 7.11 implies that F is invariant. By Corollary 7.8, the Morse hyper-graph GF exists. Theorem 7.9 implies assertions (2).


A

D

l1

l2ly

Figure 1. A decomposition on a square D = [0, 3]2

8. Examples

We state the following recurrent properties.

8.1. Recurrent properties.

8.1.1. Morse hyper-graphs of M which are not quotient spaces of abstract element

space. There is the Morse hyper-graph GM for M is not a quotient space of theabstract element space. Indeed, letD := [0, 3]2 be a closed square, A := (1, 2)×[1, 2]a square, and F := x | x ∈ D − A ⊔ ly | y ∈ [1, 2] a decomposition as inFigure 1, where ly := (1, 2)×y are open intervals. Then Q(F) = Cl(F) = D−A,

P(F) = A, and Q(F)−Q(F) = A− intA = l1 ⊔ l2. Therefore the abstract (weak)element space is D/[F ] = D/〈F〉 = D−A,A and the Morse hyper-graph GF of Fis (V,H) = (M1 = D−A, H1 = A). On the other hand, the family M := D−

(1, 2)2 is the set of connected components of Q(F) = D− intA = D−(1, 2)2. Thenthe Morse hyper-graph GM forM is (VM, HM) = (M ′

1 = D−intA, H ′1 = intA).

This means that the Morse hyper-graph GM is not a quotient space of the abstractelement space D/[F ] = D −A,A.

Denote by (X, τ) the element space D/F = x | x ∈ D−A⊔ ly | y ∈ [1, 2].Then (X, τ) is a topological space such that Q(τ) = Cl(τ) = (D − A)/F = x |

x ∈ D − A, P(τ) = A/F = ly | y ∈ [1, 2] and Q(τ) = (D − intA)/F = x |x ∈ D−A⊔l1, l2. Moreover, we have D/[τ ] = D/〈τ〉 = (D−A)/F , A/F andthe Morse hyper-graph Gτ of τ is (V ′′, H ′′) = (M ′′

1 = (D −A)/F, H ′′1 = A/F)

and is isomorphic to GF . On the other hand, the family Mτ := Q(τ) the set

of connected components of Q(τ). Then the Morse hyper-graph GMτfor Mτ is

(VMτ, HMτ

) = (M ′′′1 = (D − intA)/F, H ′′′

1 = intA/F) and is isomorphic toGM. This means that the Morse hyper-graph GMτ

is not a quotient space of theabstract element space X/[τ ] = (D −A)/F , A/F.

8.1.2. Non-closedness of quasi-recurrent set. There is a topological space whosequasi-recurrent set is not closed. In fact, define a poset (P = Z × 0, 1,≤) by(n1, n2) < (m1,m2) if n1 = m1, n2 = 0, and m2 = 1. The family τ := ∅ ⊔P − F | F is a finite downset is a topology on P whose specialization pre-orderis the partial order ≤. Then P(τ) = max τ = Z×1 is infinite and so is not open.Therefore Z× 0 = Cl(τ) = Q(τ) is not closed,

8.2. Necessary conditions. We construct some examples to describe the neces-sity of conditions.

8.2.1. Necessity of invariance to become topologies. There is a decomposition whoseset of F -invariant open subsets is not a topology. In fact, for a decompositionF := 0 × [0, 1] ⊔ (x, y) | x 6= 0, y ∈ [0, 1] on a closed disk [0, 1]2, the set τF isnot a topology.


U1 U2

γ1 γ1 T2

p

p γ2

γ1

U2U1

T2/ v

γ2

p

Figure 2. A decomposition on a torus and its abstract element space.

8.2.2. Necessity of local compactness for correspondence of recurrences. There isa transitive flow v on a metrizable space with non-τX/v-recurrent but v-recurrentpoints. Indeed, applying a dump function to an irrational rotation, consider avector field on T2 with one singular point x such that each non-singular orbit isdense. Let X := O ⊔ x be the union of x and a non-singular orbit O and vthe restriction of the flow. Then X is metrizable and v consists of one v-recurrentnon-closed orbit O and one singular point x. Thus O−O = x is closed and so Ois not τX/v-recurrent but v-recurrent.

8.2.3. Necessity for using abstract elements in the definition of quasi-recurrence.

There is a singular codimension one foliation on a torus T2, which can be generatedby a flow, with T2 − P(F) ( R(F) ⊔ (

⋃

L∈F(L − L) ∩ P(F)) ( T2 − (U1 ⊔ U2) =

γ1 ⊔ p ⊔ 〈γ2〉 = Q(F) as in Figure 2. Put X := T2/F . Then X − P(τF ) (

R(τF )⊔(⋃

L∈X(L−L)∩P(τF)) ( X−(U1⊔U2)/F = γ1, p, 〈γ2〉 = Q(τF ).

Note that the abstract element space T2/〈F〉 is weakly homotopic to a circle.

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Applied Mathematics and Physics Division, Gifu University, Yanagido 1-1, Gifu, 501-

1193, Japan,

Email address: [email protected]

arXiv:2112.13992v3 [math.DS] 29 May 2022

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