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General Topology
We generalize and extend the Conley-Morse-Forman theory for combinatorial mul- tivector ﬁelds introduced in Mrozek (Found Comput Math 17(6):1585–1633, 2017). The generalization is threefold. First, we drop the restraining assumption in Mrozek (Found Comput Math 17(6):1585–1633, 2017) that every multivector must have a unique maximal element. Second, we deﬁne the dynamical system induced by the multivector ﬁeld in a less restrictive way. Finally, we also change the setting from Lefschetz complexes to ﬁnite topological spaces. Formally, the new setting is more general, because every Lefschetz complex is a ﬁnite topological space, but the main reason for switching to ﬁnite topologcial spaces is because the latter better explain some peculiarities of combinatorial topological dynamics. We deﬁne isolated invari- ant sets, isolating neighborhoods, Conley index and Morse decompositions. We also establish the additivity property of the Conley index and the Morse inequalities. Research of M.L. was partially supported by the Polish National Science Center under Preludium Grant No. 2018/29/N/ST1/00449. M.M. was partially supported by the Polish National Science Center under Maestro Grant No. 2014/14/A/ST1/00453. T.W. was partially supported by NSF Grant DMS-1407087 and by the Simons Foundation under Award 581334. B Marian Mrozek marian.mrozek@uj.edu.pl Michał Lipinski ´ michal.lipinski@uj.edu.pl Jacek Kubica jacek.kubica@doctoral.uj.edu.pl Thomas Wanner twanner@gmu.edu Division of Computational Mathematics, Faculty of Mathematics and Computer Science, Jagiellonian University, ul. St. Łojasiewicza 6, 30-348 Kraków, Poland Department of Mathematical Sciences, George Mason University, Fairfax, VA 22030, USA 123 M. Lipinski ´ Keywords Combinatorial vector ﬁeld · Finite topological space · Discrete Morse theory · Isolated invariant set · Conley theory Mathematics Subject Classiﬁcation Primary 37B30; Secondary 37E15 · 57M99 · 57Q05 · 57Q15 1 Introduction The combinatorial approach to dynamics has its origins in two papers by Robin Forman (Forman 1998a, b) published in the late 1990s. Central to the work of Forman is the concept of a combinatorial vector ﬁeld. One can think of a combinatorial vector ﬁeld as a partition of the collection of cells of a cellular complex into combinatorial vectors which may be singletons (critical vectors or critical cells) or doubletons such that one element of the doubleton is a face of codimension one of the other (regular vectors). The original motivation of Forman was the presentation of a combinatorial analogue of classical Morse theory. However, soon the potential for applications of such an approach was discovered in data science. Namely, the concept of combinatorial vector ﬁeld enables direct applications of the ideas of topological dynamics to data and eliminates the need of the cumbersome construction of a classical vector ﬁeld from data. Recently, Batko et al. (2020), Kaczynski et al. (2016), Mrozek and Wanner (2021), in an attempt to build formal ties between the classical and combinatorial Morse the- ory, extended the combinatorial theory of Forman to Conley theory (Conley 1978), a generalization of Morse theory. In particular, they deﬁned the concept of an isolated invariant set, the Conley index and Morse decomposition in the case of a combi- natorial vector ﬁeld on the collection of simplices of a simplicial complex. Later, Mrozek (2017) observed that certain dynamical structures, in particular homoclinic connections, cannot have an analogue for combinatorial vector ﬁelds and as a remedy proposed an extension of the concept of combinatorial vector ﬁeld, a combinatorial multivector ﬁeld. We recall that in the collection of cells of a cellular complex there is a natural partial order induced by the face relation. Every combinatorial vector in the sense of Forman is convex with respect to this partial order. A combinatorial mul- tivector in the sense of Mrozek (2017) is deﬁned as a convex collection of cells with a unique maximal element, and a combinatorial multivector ﬁeld is then deﬁned as a partition of cells into multivectors. The results of Mrozek (2017) were presented in the algebraic setting of chain complexes with a distinguished basis (Lefschetz com- plexes), an abstraction of the chain complex of a cellular complex already studied by Lefschetz (1942). The results of Forman were earlier generalized to the setting of Lef- schetz complexes in Jöllenbeck and Welker (2009); Kozlov (2005); Sköldberg (2005), and to the more general setting of ﬁnite topological spaces in Minian (2012). Note that the setting of ﬁnite topological spaces is more general, because every Lefschetz complex is a poset via the face relation, therefore also a ﬁnite topological space via the Alexandrov Theorem (Alexandrov 1937). The aim of this paper is a threefold advancement of the results of Mrozek (2017). We generalize the concept of combinatorial multivector ﬁeld by lifting the assumption 123 Conley-Morse-Forman theory for generalized combinatorial… that a multivector has a unique maximal element. This assumption was introduced in Mrozek (2017) for technical reasons, but turned out to be a barrier for adapting the techniques of continuation in topological dynamics to the combinatorial setting (Dey et al. 2022). We deﬁne the dynamics associated with a combinatorial multivector ﬁeld following the ideas of Dey et al. (2019). This approach is less restrictive, and better adjusted to persistence of Conley index (Dey et al. 2022, 2020, 2022). Finally, we change the setting from Lefschetz complexes to ﬁnite topological spaces. Here, the generalization is not the main motivation. We do so, because the speciﬁc nature of ﬁnite topological spaces helps explain the differences between the combinatorial and the classical theory. For instance, isolated invariant sets are always closed in the classical theory, but this is not true in its combinatorial counterpart, because separation in ﬁnite topological spaces is only T . In this extended and generalized setting we deﬁne the concepts of isolated invariant set and Conley index. We also deﬁne attractors, repellers, attractor-repeller pairs and Morse decompositions, and provide a topological characterization of attractors and repellers. Furthermore, we prove the Morse equation for Morse decompositions, and ﬁnally deduce from it the Morse inequalities. We note that, as in the classical case, attractors of multivector ﬁelds form a bounded, distributive lattice. Therefore, the algebraic characterization of lattices of attractors developed by Kalies, Mischaikow, and Vandervorst in Kalies et al. (2014, 2016, 2021) applies also to the combinatorial case. What unites the two approaches is the combi- natorial multivalued map. The difference is that the approach in Kalies et al. (2014, 2016, 2021) is purely algebraic whereas our approach is purely topological. The rela- tion between the algebraic and topological approaches in the case of gradient-like dynamics is very interesting, but beyond the scope of the present paper. We leave the study of this relation for future investigation. The organization of the paper is as follows. In Sect. 2 we present the main results of the paper. This is an informal section aiming at the presentation of the motiva- tion, intuition, and main ideas of the paper on the basis of an elementary geometric example. The reader interested only in the formal results and their correctness may skip this section. In Sect. 3 we recall basic concepts and facts needed in the paper. Section 4 is devoted to the study of the dynamics of combinatorial multivector ﬁelds and the introduction of isolated invariant sets. In Sect. 5 we deﬁne index pairs and the Conley index. In Sect. 6 we investigate limit sets, attractors and repellers in the combinatorial setting. Finally, Sect. 7 is concerned with Morse decompositions and Morse inequalities for combinatorial multivector ﬁelds. 2 Main results In this section we present the main results of the paper in an informal and intuitive way, for a simple simplicial example. We also indicate the main conceptual differences between our combinatorial approach and the classical theory. Precise deﬁnitions and statements are given in the following sections. 123 M. Lipinski ´ 2.1 Combinatorial phase space In this paper we study dynamics in ﬁnite spaces. We also refer to ﬁnite spaces as com- binatorial spaces. In applications, they typically are collections of cells of a simplicial, a cubical, or a more general cellular complex. All such collections are examples of Lefschetz complexes. We recall that a Lefschetz complex (see Sect. 3.5 for precise def- initions) is a distinguished basis of a ﬁnitely generated chain complex. By Lefschetz homology we mean the homology of this chain complex. In a Lefschetz complex, as in every cellular complex, there is a well deﬁned face relation which makes every Lef- schetz complex a ﬁnite poset and, via the Alexandrov Theorem (Alexandrov 1937), a ﬁnite topological space. The results of this paper apply to any ﬁnite topological space, although from the point of view of applications Lefschetz complexes remain the main object of interest. However, the viewpoint from the perspective of ﬁnite topological spaces is closer to geometric intuition and, as pointed out in the introduction, helps explain similarities and differences between the classical and the combinatorial theory. A Lefschetz complex, as every topological space, has also well deﬁned singular homology. Note that the singular homology and Lefschetz homology of a Lefschetz complex need not be the same in general, although they are the same for many concrete examples of Lefschetz complexes, in particular for cellular complexes. The results of this paper apply to any homology theory for which the excision and Mayer-Vietoris theorems hold. In particular, they apply to singular homology of ﬁnite topological spaces and Lefschetz homology of Lefschetz complexes regardless whether they are the same or not. But, they depend on the speciﬁcally chosen homology theory. For concrete Lefschetz complexes such as simplicial complexes there is also topol- ogy of its geometric realization which is very different from the ﬁnite topology of a Lefschetz complex. Nevertheless, via McCord’s Theorem (McCord 1966), these topologies are weakly homotopic and, consequently, their algebraic invariants such as homotopy and singular homology groups are the same. As an example consider the family X of all simplices of the simplicial complex in Fig. 1 (top). The associated face poset which makes X a ﬁnite topological space is presented in Fig. 1 (bottom). Another topological space associated with the simplicial complex is its polytope, that is the union of all its simplexes with topology induced from the plane. Although the two topological spaces are clearly very different, in particular the poset is only T and the polytope is T , they are related. We can see it by identifying the simplices in the poset with open simplices in the polytope. In this case, the set A ⊂ X is open (respec- tively closed) in the T topology of X if and only if the union of the corresponding open simplices is open (respectively closed) in the Hausdorff topology of the polytope of X. 2.2 Combinatorial dynamical systems Classical dynamical systems, when considered on a ﬁnite space, are very restrictive. On the one hand, as observed in [Chocano et al. 2021, Theorem 2.6], a ﬂow on a ﬁnite T topological space necessarily has only stationary trajectories. On the other hand, it is straightforward to check that in the same setting but for a dynamical system 123 Conley-Morse-Forman theory for generalized combinatorial… Fig. 1 An example of a simplicial complex (top) and the poset (a ﬁnite T topological space, bottom) induced by its face relation with discrete time every trajectory is periodic. Hence, to allow for more interesting dynamics in ﬁnite spaces, we consider combinatorial dynamical systems by which we mean iterates of a multivalued map acting on the ﬁnite space (see Sect. 4.1 for precise deﬁnitions). The question remains whether one can still distinguish a class of multivalued maps whose dynamics is ﬂow-like. One would expect that for a general multivalued map trajectories may arbitrarily jump through space, whereas in the case of ﬂow-like dynamics they should join neighbors in the topological space. We do not attempt to formalize the ﬂow-like concept. Instead, inspired by Forman (Forman 1998a, b), we study the dynamics of a special class of multivalued maps on ﬁnite spaces, generated by a combinatorial analogue of a vector ﬁeld. We introduce it in the next section. Clearly, the dynamics of general multivalued maps in ﬁnite spaces, corresponding to classical discrete time dynamics, is also of interest. However, Conley theory for general multivalued maps in ﬁnite topological spaces requires different ideas and is beyond the scope of the present paper and will be presented in [3] 2.3 Combinatorial multivector fields A combinatorial multivector ﬁeld (see Sect. 4.3 for precise deﬁnitions and more details) is a partition of a ﬁnite topological space X into locally closed sets (or convex sets in terms of posets, see Proposition 3.10), that is, sets A ⊂ X such that their mouth mo A := cl A \ A (the closure of A with A removed) is closed. The elements of the partition are referred to as multivectors. 123 M. Lipinski ´ Fig. 2 A ﬂow on a torus T represented as a green square with bottom and top as well as left and right faces identiﬁed. A cellular structure T is imposed on T . It consists of 9 squares, 18 edges and 9 vertices. The ﬂowlines cross the edges transversally. This leads to a multivector ﬁeld on T . Each multivector consists of cells intersected by the same orange region. For instance, consider the top right orange region. It represents a multivector consisting of three cells: square CADF and edges CA,DF, because these are the only cells in the boundary of CADF crossed by ﬂowlines towards CADF. Similarly, the orange region in bottom right consists only of square AC I G, because none of its faces is crossed inwards There are many ways to obtain multivector ﬁelds in applications. One of the most intuitive methods is via the transversal polygons construction. It consists in the approx- imation of a ﬂow on a manifold by the decomposition of the manifold into convex polygonal cells in such a way that the ﬂow lines cross the faces of every cell transver- sally. The importance of such decompositions was indicated already by Boczko et al. (2007). The top dimensional cells together with their faces (lower dimensional cells) form a cellular decomposition of the manifold. The transversality and the convexity then imply that ﬂowlines originating in the same face enter the same top dimensional cell. By grouping each top dimensional cell with all its faces entering the cell, one obtains a combinatorial multivector ﬁeld (see Mrozek et al. (2022) for details). An example of such a construction is presented in Fig. 2. A special feature of a multi- vector ﬁeld constructed this way is that every multivector contains a cell of maximal dimension and is contained in the closure of the cell. This need not be the case in general as the following example shows. Example 2.1 Consider the partition V := { {A, AC }, {ABC }, {B, AB}, {C , BC }, {CE }, {D, BD, CD, BC D}, {DE , CDE }, {E , EG}, {EF , DE F , EFG}, {F , DF , FG}, {G}} 123 Conley-Morse-Forman theory for generalized combinatorial… Fig. 3 A partition of a poset into multivectors (convex subsets). Nodes as well as corresponding arrows of each multivector are highlighted with a distinct color Fig. 4 A geometric visualization of the combinatorial multivector ﬁeld in Fig. 3. A multivector may be considered as a “black box” whose dynamics is known only via splitting its boundary into the exit and entrance parts of the set of cells X of the simplicial complex in Fig. 1. It is easy to check that this partition is a multivector ﬁeld presented in Fig. 3 with X visualized as a poset, and in Fig. 4 with X visualized as a simplicial complex. Every multivector in Fig. 3 is highlighted with a different color and in Fig. 4 it is indicated by an orange region as in Fig. 2. In terms of the transversal polygons interpretation the dotted part of the boundary of a multivector indicates the outward-directed ﬂow while the solid part of the boundary indicates the inward ﬂow. As this example indicates, a multivector may contain no top dimensional cell but also more than one top dimensional cell. Such multivectors, in particular, are useful in constructing multivector ﬁelds from clouds of vectors. Note that in the case of a multivector ﬁeld constructed from transversal polygons, the transversality implies that the ﬂow may exit the closure of a multivector V only through its mouth. Hence, cl V may be interpreted as an isolating block for the ﬂow with exit set mo V . This allows us to think of a multivector as a black box where the dynamics is known only at its boundary, but not inside. Moreover, the relative homology H (cl V , mo V ) may be interpreted as the Conley index of the invariant set of the ﬂow isolated by cl V (for the deﬁnition of Conley index and isolating block in the classical setting see Conley 1978; Conley and Easton 1971; Stephens and Wanner 2014). 123 M. Lipinski ´ Fig. 5 The combinatorial ﬂow of the multivector ﬁeld in Figs. 3 and 4 represented as the digraph G . V V Downward arrows are induced by the closure components of . Bi-directional edges and self-loops reﬂect dynamics within multivectors. For clarity, we omit edges that can be obtained by between-level transitivity, e.g., the bi-directional connection between node D and BC D. The nodes of critical multivectors are bolded in red 2.4 Combinatorial flow associated with a multivector field With every combinatorial multivector ﬁeld V on a ﬁnite topological space X we asso- ciate a combinatorial dynamical system induced by the multivalued map : X X given by (x ) := cl x ∪[x ] , (1) V V where [x ] denotes the unique multivector in V containing x. Similarly to Forman (1998a) we often refer to the combinatorial dynamical system given by (1)asthe combinatorial ﬂow associated with the multivector ﬁeld V. Formula (1) says that starting from cell x we can either go to the closure of x or we can stay in the multivector of x. In the case of a multivector ﬁeld constructed from transversal polygons as in Fig. 2 the ﬁrst case may be interpreted as the ﬂow-like behaviour, because a ﬂow line cannot leave cell x without crossing the boundary of x. The second case reﬂects the black box nature of a multivector: we only know what happens at the boundary of a multivector, therefore we do not want to exclude any movement inside a multivector. 2.5 Graph interpretation Let F : X X be an arbitrary multivalued map acting on a ﬁnite topological space X. The combinatorial dynamical system induced by F, as in Kalies et al. (2016), may be interpreted as a directed graph G whose vertices are the elements of X and there is a directed arrow from x to y whenever y ∈ F (x ). The graph G := G of the combinatorial ﬂow discussed in Example 2.1 is presented in Fig. 5. A basic concept of multivalued dynamics, a solution, corresponds to a walk in G . We are interested in full solutions, that is, bi-inﬁnite walks, as well as paths by which we mean ﬁnite walks (see Sect. 4.2 for precise deﬁnitions and more details). Translation of problems in combinatorial topological dynamics to the language of directed graphs facilitates their algorithmic study. However, we emphasize that combinatorial topological dynamics cannot be reduced just to graph theory, because 123 Conley-Morse-Forman theory for generalized combinatorial… the topology in the set of vertices of the directed graph matters, as we will see in the following sections, in particular in the concept of essential solution introduced in the next section. In consequence, combinatorial topological dynamics as a ﬁeld is a part of general topological dynamics and not a part of graph theory. The use of the classical terminology of dynamics also in the combinatorial setting helps focusing on this difference. 2.6 Essential solutions As we already explained, formula (1) for the combinatorial dynamical system asso- ciated with a combinatorial multivector ﬁeld has a natural geometric interpretation. However, it also has a drawback, because, as one can easily check, for such a combi- natorial dynamical system there is a stationary (constant) solution through each point. This, in particular, is the consequence of the black box nature of a multivector and the tightness of a ﬁnite topological space. We overcome this problem by distinguishing regular and critical mulltivectors. To deﬁne them we ﬁx a homology theory in the ﬁnite topological space X. For examples based on simplicial complexes we just take sim- plicial homology which in this case coincides with Lefschetz homology and singular homology (see Sect. 3.6). We say that a multivector V is critical if H (cl V , mo V ) = 0. Otherwise we call V regular. There are ﬁve critical multivectors in the multivector ﬁeld presented in Example 2.1: {ABC }, {CE }, {DE F , EF , EFG}, {DF , F , FG}, {G}. In terms of the transversal polygon interpetation a critical multivector may be considered as an isolating block with a non-trivial Conley index. Therefore, accepting the existence of a non-empty invariant set inside may be justiﬁed by the Waze ˙ wski property of the Conley index. In the case of a regular multivector we have no topological justiﬁcation to expect a non-empty invariant set. This motivates the introduction of essential solutions. An essential solution is a full solution γ such that if γ(t ) belongs to a regular multivector V ∈ V then there exist both a k > 0 and an l < 0 satisfying the exclusions γ(t + k), γ (t + l)/ ∈ V (see Sect. 4.4 for precise deﬁnition and more details). An example of an essential solution γ : Z → X for the multivector ﬁeld in Fig. 4 is given by: CE t < 0, Et ∈{0, 2, 3}, γ(t ) = EG t ∈{1, 4}, Gt > 4. 123 M. Lipinski ´ 2.7 Isolated invariant sets and Conley index We say that a set S ⊂ X is invariant if every x ∈ S admits an essential solution through x in S. We say that an invariant set S is an isolated invariant set if there exists a closed set N , called an isolating set such that S ⊂ N , (S) ⊂ N , and every path in N with endpoints in S is a path in S (see Sect. 4.5 for precise deﬁnitions and more details). Note that our concept of isolating set is weaker than the classical concept of isolating neighborhood, because the maximal invariant subset of N may not be contained in the interior of N . The need of a weaker concept is motivated by the tightness in ﬁnite topological spaces. In particular, an isolated invariant set S may intersect the closure of another isolated invariant set S and be disjoint but not disconnected from S . For instance, with respect to Example 2.1 the sets S := {A, AC , C , BC , B, AB} and S := {ABC } are both isolated invariant sets isolated respectively by N := S 2 1 1 and N := cl S = S ∪ S . Observe that S ⊂ N . Thus, the isolating set in the 2 1 1 2 1 2 combinatorial setting of ﬁnite topological spaces is a relative concept. Therefore, one has to specify each time which invariant set is considered as being isolated by a given isolating set. Givenanisolatedinvariant set S of a combinatorial multivector ﬁeld V we deﬁne index pairs similarly to the classical case (see Deﬁnition 5.1), we prove that (cl S, mo S) is one of the possibly many index pairs for S (see Proposition 5.3) and we show that the homology of an index pair depends only on S, but not on the particular index pair (see Theorem 5.16). This enables us to deﬁne the Conley index of an isolated invariant set S (see Deﬁnition 4.8) and the associated Poincaré polynomial (see (4)). In Example 2.1 (see Fig. 4), the Poincaré polynomials of the isolated invariant sets S ={A, AC , C , BC , B, AB} and S ={ABC } are respectively p (t ) = 1 + t and 1 2 S p (t ) = t . 2.8 Morse decompositions The concept of Morse decomposition in combinatorial dynamics is similar in spirit to the classical case although some details are different (see Deﬁnition 7.1). Unlike the classical case, for a combinatorial multivector ﬁeld V we prove that the strongly connected components of the directed graph G which admit an essential solution constitute the minimal Morse decomposition of V (see Theorem 7.3). For Example 2.1 the minimal Morse decomposition consists of six isolated invariant sets: M := {A, AC , C , BC , B, AB}, M := {ABC }, M := {CE }, 1 2 3 M := {DE F , EF , EFG}, M := {DF , F , FG}, M := {G}. 4 5 6 We say that an isolated invariant set S is an attractor (respectively a repeller)if all solutions originating in it stay in S in forward (respectively backward) time (see Sect. 6.1). There are two attractors in our example: M is a periodic attractor, and M 1 6 is an attracting stationary point. Sets M and M are repellers, while M and M are 2 4 3 5 neither attractors nor repellers. 123 Conley-Morse-Forman theory for generalized combinatorial… Fig. 6 The Conley-Morse graph for the multivector ﬁeld in Example 2.1 If there exists a path originating in M and terminating in M , we say that there is a i j connection from M to M . The connection relation induces a partial order on Morse i j sets. The associated poset with nodes labeled with Poincaré polynomials is called the Conley-Morse graph of the Morse decomposition, see also Arai et al. (2009); Bush et al. (2012). The Conley-Morse graph of the minimal Morse decomposition of the combinatorial multivector ﬁeld in Fig. 4 is presented in Fig. 6. The Morse equation (see Theorem 7.9) for this Morse decomposition takes the form: 2t + 3t + 2 = 1 + (1 + t )(1 + 2t ). As this brief overview of the results of this paper indicates, at least to some extent it is possible to construct a combinatorial analogue of classical topological dynamics. Such an analogue may be used to construct algorithmizable models of sampled dynamical systems as well as tools for computer-assisted proofs in dynamics (Mrozek et al. 2022). 3 Preliminaries In this section we recall the background material needed in this paper and we set notation. 3.1 Sets and maps We denote the sets of integers, non-negative integers, non-positive integers, and pos- + − itive integers, respectively, by Z, Z , Z , and N. Given a set A, we write # A for the number of elements in A and we denote by P(A) the family of all subsets of X.We write f : X Y for a partial map from X to Y , that is, a map deﬁned on a subset dom f ⊂ X, called the domain of f , and such that the set of values of f , denoted im f , is contained in Y . 123 M. Lipinski ´ A multivalued map F : X Y is a map F : X → P(Y ) which assigns to every point x ∈ X a subset F (x ) ⊂ Y.Given A ⊂ X,the image of A under F is deﬁned by F (A) := F (a).Bythe preimage of a set B ⊂ Y with respect to F we mean the x ∈A large preimage, that is, −1 F (B) := {x ∈ X | F (x ) ∩ B = ∅} . (2) In particular, if B ={y} is a singleton, we get −1 F ({y}) := {x ∈ X | y ∈ F (x )} . −1 −1 −1 Thus, we have a multivalued map F : Y X given by F (y) := F ({y}).We call it the inverse of F. 3.2 Relations and digraphs Recall that a binary relation or brieﬂy a relation in a space X is a subset E ⊂ X × X. We write xEy as a shorthand for (x , y) ∈ E.The inverse of E is the relation −1 E := {(y, x ) ∈ X × X | xEy} . Given a relation E in X, the pair (X , E ) may be interpreted as a directed graph (digraph) with vertex set X, and edge set E. Relation E may also be considered as a multivalued map E : X X with E (x ) := {y ∈ X | xEy}. Thus, the three concepts: binary relation, multivalued map and directed graph are, in principle, the same and in this paper will be used interchangeably. We recall that a path in a directed graph G = (X , E ) is a sequence x , x ,..., x 0 1 k of vertices such that (x , x ) ∈ E for i = 1, 2,... k. The path is closed if x = x . i −1 i 0 k A closed path consisting of two elements is a loop. Thus, an x ∈ X is a loop if and only if x ∈ E (x ). We note that loops may be present at some vertices of G but at some other vertices they may be absent. Avertexis recurrent if it belongs to a closed path. In particular, if there is a loop at x ∈ X, then x is recurrent. The digraph G is recurrent if all of its vertices are recurrent. We say that two vertices x and y in a recurrent digraph G are equivalent if there is a path from x to y and a path from y to x in G. Equivalence of recurrent vertices in a recurrent digraph is easily seen to be an equivalence relation. The equivalence classes of this relation are called strongly connected components of digraph G. They form a partition of the vertex set of G. We say that a recurrent digraph G is strongly connected if it has exactly one strongly connected component. A non-empty subset A ⊂ X is strongly connected if (A, E ∩ A × A) is strongly connected. In other words, A ⊂ X is strongly connected if and only if for all x , y ∈ A there is a path in A from x to y and from y to x. 123 Conley-Morse-Forman theory for generalized combinatorial… 3.3 Posets Let X be a ﬁnite set. We recall that a reﬂexive and transitive relation ≤ on X is a preorder and the pair (X , ≤) is a preordered set.If ≤ is also antisymmetric, then it is a partial order and (X , ≤) is a poset. A partial order in which any two elements are comparable is a linear (total) order. Given a poset (X , ≤),aset A ⊂ X is convex if x ≤ y ≤ z with x , z ∈ A, y ∈ X implies y ∈ A.Itisan upper set if x ≤ y with x ∈ A and y ∈ X implies y ∈ A. Similarly, A is a down set with respect to ≤ if x ≤ y with y ∈ A and x ∈ X implies x ∈ A.A chain is a totally ordered subset of a poset. Finally, for A ⊂ X we write A := {a ∈ X |∃ a ≤ b}, b∈A < ≤ A := A \ A. One can easily check the following proposition. Proposition 3.1 ([Lipinski ´ 2021, Proposition 1.3.1]) Let (X , ≤) be a poset and let ≤ < A ⊂ X be a convex set. Then the sets A and A are down sets. 3.4 Finite topological spaces Given a topology T on X, we call (X , T ) a topological space. When the topology T is clear from the context we also refer to X as a topological space. We denote the interior of A ⊂ X with respect to T by int A and the closure of A with respect to T by cl A. We deﬁne the mouth of A as the set mo A := cl A \ A. We say that X is T T T a ﬁnite topological space if X is a ﬁnite set. If X is ﬁnite, we also distinguish the minimal open superset (or open hull)of A as the intersection of all the open sets containing A. We denote it by opn A. We note op that when X is ﬁnite then the family T := {X \ U | U ∈ T } of closed sets is also a topology on X, called dual or opposite topology. The following proposition is straightforward. Proposition 3.2 If (X , T ) is a ﬁnite topological space then for every set A ⊂ Xwe have opn A = cl op A. If A ={a} is a singleton, we simplify the notation int {a},cl {a},mo {a} and T T T opn {a} to int a,cl a,mo a and opn a. When the topology T is clear from T T T T T the context, we drop the subscript T in this notation. Given a ﬁnite topological space op op (X , T ) we brieﬂy write X := (X , T ) for the same space X but with the opposite topology. We recall that a subset A of a topological space X is locally closed if every x ∈ A admits a neighborhood U in X such that A ∩ U is closed in U . Locally closed sets are important in the sequel. In particular, we have the following characterization of locally closed sets. Proposition 3.3 ([Engelking 1989, Problem 2.7.1]) Assume A is a subset of a topo- logical space X. Then the following conditions are equivalent. 123 M. Lipinski ´ (i) A is locally closed, (ii) mo A := cl A \ Ais closed in X, T T (iii) A is a difference of two closed subsets of X, (iv) A is an intersection of an open set in X and a closed set in X. As an immediate consequence of Proposition 3.3(iv) we get the following three propositions. Proposition 3.4 The intersection of a ﬁnite family of locally closed sets is locally closed. Proposition 3.5 If A is locally closed and B is closed, then A \ B is locally closed. Proposition 3.6 Let (X , T ) be a ﬁnite topological space. A subset A ⊂ X is locally op closed in the topology T if and only if it is locally closed in the topology T . We recall that the topology T is T or Hausdorff if for any two different points x , y ∈ X, there exist disjoint sets U , V ∈ T such that x ∈ U and y ∈ V.Itis T or Kolmogorov if for any two different points x , y ∈ X there exists a U ∈ T such that U ∩{x , y} is a singleton. Finite topological spaces stand out from general topological spaces by the fact that the only Hausdorff topology on a ﬁnite topological space X is the discrete topology consisting of all subsets of X. Proposition 3.7 ([Lipinski ´ 2021, Proposition 1.4.7]) Let (X , T ) be a ﬁnite topological space and A ⊂ X. Then cl A = cl a. a∈A A remarkable feature of ﬁnite topological spaces is the following theorem. Theorem 3.8 (Alexandrov 1937) For a preorder ≤ on a ﬁnite set X, there is a topology T on X whose open sets are the upper sets with respect to ≤. For a topology T on a ﬁnite set X, there is a preorder ≤ where x ≤ y if and only if x ∈ cl y. T T T The correspondences T →≤ and ≤ → T are mutually inverse. Under these T ≤ correspondences continuous maps are transformed into order-preserving maps and vice versa. Moreover, the topology T is T (Kolmogorov) if and only if the preorder ≤ is a partial order. The correspondence resulting from Theorem 3.8 provides a method to translate concepts and problems between topology and order theory in ﬁnite spaces. In partic- ular, closed sets are translated to down sets in this correspondence and we have the following straightforward proposition. Proposition 3.9 Let (X , T ) be a ﬁnite topological space. Then, for A ⊂ X we have cl A ={x ∈ X |∃ x ≤ a}, a∈A T T opn A ={x ∈ X |∃ x ≥ a}, a∈A T int A ={a ∈ A |∀ x ≥ a ⇒ x ∈ A}. T x ∈X T 123 Conley-Morse-Forman theory for generalized combinatorial… In other words, cl A is the minimal down set with respect to ≤ containing A, T T opn A is the minimal upper set with respect to ≤ containing A and int A is the T T maximal upper set with respect to ≤ contained in A. One can easily verify the following proposition. Proposition 3.10 ([Lipinski ´ 2021, Proposition 1.4.10]) Assume X is a T ﬁnite topo- logical space and A ⊂ X. Then A is locally closed if and only if A is convex with respect to ≤ . 3.5 Lefschetz complexes We say that (X,κ) is a Lefschetz complex (see [Lefschetz 1942, Chapter III, Sect. 1, Deﬁnition 1.1]) if X = (X ) + is a ﬁnite set with gradation, κ : X × X → R is a q q∈Z map with values in a ring with unity such that κ(x , y) = 0 implies both the inclusion x ∈ X and y ∈ X , and for any x , z ∈ X we have q q−1 κ(x , y)κ(y, z) = 0. (3) y∈X One easily veriﬁes that by condition (3) we have a free chain complex (R(X ), ∂ ) κ κ with ∂ : R(X ) → R(X ) deﬁned on generators by ∂ (x ) := κ(x , y)y.The y∈X Lefschetz homology of (X,κ), denoted H (X ), is the homology of this chain complex. Note that the elements of X may be identiﬁed with a basis of R(X ).Viceversa,every ﬁxed basis of a ﬁnitely generated chain complex constitutes a Lefschetz complex. Given x , y ∈ X we say that y is a facet of x if κ(x , y) = 0. It is easily seen that the facet relation extends uniquely to a minimal partial order. Via the Alexandrov Theorem (Alexandrov 1937), this partial order makes every Lefschetz complex a ﬁnite topological space. 3.6 Homology in finite topological spaces In the sequel we need a homology theory in ﬁnite topological spaces. Singular homol- ogy (Munkres 1984) is well deﬁned for any topological space, in particular for a ﬁnite topological space. McCord’s Theorem (McCord 1966) states that every ﬁnite topolog- ical space is weakly homotopy equivalent to the associated order complex K(X ), that is, an abstract simplicial complex consisting of subsets of X linearly ordered by the partial order associated with the topology via the Alexandrov Theorem (Theorem 3.8). This is convenient for computational purposes. For Lefschetz complexes, apart from singular homology we also have the notion of Lefschetz homology. As we already mentioned in Sect. 2.1 the singular homology and Lefschetz homology of a Lefschetz complex need not be the same. Nevertheless, they both satisfy the following theorem which summarizes the features of homology we need in this paper. 123 M. Lipinski ´ Theorem 3.11 Let (X , T ) be a ﬁnite topological space, and for a closed subset A ⊂ X let H (X , A) denote the singular homology of the pair (X , A). Then the following properties hold. (i) If A, B, C , D are closed subsets of X such that B ⊂ A, D ⊂ C and A\B = C \D, then H (A, B) H (C , D). (ii) If B ⊂ A ⊂ X are closed, then the inclusions induce the exact sequence ... → H (A, B) → H (X , B) → H (X , A) → H (A, B) → ... . n n n n−1 (iii) If Y ⊂ X ⊂ X and Y ⊂ X ⊂ X are closed in X then there is an inclusion 0 0 1 1 induced exact sequence ... → H (X ∩ X , Y ∩ Y ) → H (X , Y ) ⊕ H (X , Y ) → n 0 1 0 1 n 0 0 n 1 1 H (X ∪ X , Y ∪ Y ) → H (X ∩ X , Y ∩ Y )... . n 0 1 0 1 n−1 0 1 0 1 Moreover, if X is a Lefschetz complex, then properties (i), (ii), (iii) above also hold with singular homology replaced by Lefschetz homology. Proof In the case of a Lefschetz complex and Lefschetz homology all three properties are easy exercises in homological algebra. Property (ii) for singular homology is stan- dard [Munkres 1984, Theorem 30.2]. Property (i) follows from the excision property of simplicial homology [Munkres 1984, Theorem 9.1] via McCord’s Theorem (McCord 1966) (compare Lipinski ´ 2021). Similarly, property (iii) for singular homology can be established in the same way from the relative simplicial Mayer-Vietoris sequence [Munkres 1984, Chapter 25 Ex.2]. In the sequel we ﬁx a homology theory which satisﬁes properties (i)-(iii) of Theo- rem 3.11. This may be singular homology in the case of an arbitrary ﬁnite topological space or the Lefschetz homology in the case of a Lefschetz complex. Clearly, both Lefschetz homology and singular homology for ﬁnite spaces are ﬁnitely generated. In consequence, for a locally closed A ⊂ X the Poincaré formal power series p (t ) := β (A)t (4) A i i =0 with β (A) := rank H (cl A, mo A) denoting ith Betti number, is well deﬁned and a i i polynomial. 4 Dynamics of combinatorial multivector ﬁelds In this section we introduce and study the main concepts of combinatorial topological dynamics studied in this paper: isolated invariant sets of combinatorial multivector ﬁelds. 123 Conley-Morse-Forman theory for generalized combinatorial… 4.1 Multivalued dynamical systems in finite spaces By a combinatorial dynamical system or brieﬂy, a dynamical system in a ﬁnite space X we mean a multivalued map : X × Z X such that ((x , m), n) = (x , m + n) for all m, n ∈ Z , x ∈ X . (5) Let be a combinatorial dynamical system. Consider the multivalued map : n 1 X X given by (x ) := (x , n). We call the generator of the dynamical system . It follows from (5) that the combinatorial dynamical system is uniquely determined by its generator. Thus, it is natural to identify a combinatorial dynamical system with its generator. In particular, we consider any multivalued map : X X as a combinatorial dynamical system : X × Z X deﬁned recursively by (x , 1) := (x ), (x , n + 1) := ((x , n)), as well as (x , 0) := x. We call it the combinatorial dynamical system induced by a −1 map . In particular, the inverse of also induces a combinatorial dynamical system. We call it the dual dynamical system. 4.2 Solutions and paths By a Z-interval we mean a set of the form Z∩ I where I is an interval in R.A Z-interval is left bounded if it has a minimum; otherwise it is left-inﬁnite.Itis right bounded if it has a maximum; otherwise it is right-inﬁnite.Itis bounded if it has both a minimum and a maximum. It is unbounded if it is not bounded. A solution of a combinatorial dynamical system : X X in A ⊂ X is a partial map ϕ : Z A whose domain, denoted dom ϕ,isa Z-interval and for any i , i + 1 ∈ dom ϕ the inclusion ϕ(i + 1) ∈ (ϕ(i )) holds. The solution passes through x ∈ X if x = ϕ(i ) for some i ∈ dom ϕ. The solution ϕ is full if dom ϕ = Z.It is a backward solution if dom ϕ is left-inﬁnite. It is a forward solution if dom ϕ is right-inﬁnite. It is a partial solution or simply a path if dom ϕ is bounded. A full solution ϕ : Z → X is periodic if there exists a T ∈ N such that ϕ(t + T ) = ϕ(t ) for all t ∈ Z. Note that every closed path may be extended to a periodic solution. If the maximum of dom ϕ exists, we call the value of ϕ at this maximum the right endpoint of ϕ. If the minimum of dom ϕ exists, we call the value of ϕ at this minimum the left endpoint of ϕ. We denote the left and right endpoints of ϕ, respectively, by ϕ and ϕ . By a shift of a solution ϕ we mean the composition ϕ ◦ τ , where the map τ : Z n n m → m + n ∈ Z is translation. Given two solutions ϕ and ψ such that ψ and ϕ exist and ψ ∈ (ϕ ), there is a unique shift τ such that ϕ ∪ (ψ ◦ τ ) is a solution. n n We call this union of paths the concatenation of ϕ and ψ and we denote it by ϕ · ψ.We also identify each x ∈ X with the trivial solution ϕ :{0}→{x }. For a full solution 123 M. Lipinski ´ + − ϕ we denote the restrictions ϕ| + by ϕ and ϕ| − by ϕ . We ﬁnish this section with Z Z the following straightforward proposition. Proposition 4.1 If ϕ : Z → X is a full solution of a dynamical system : X X, −1 then Z t → ϕ(−t ) ∈ X is a solution of the dual dynamical system induced by . op We call it the dual solution and denote it ϕ . 4.3 Combinatorial multivector fields Combinatorial multivector ﬁelds on Lefschetz complexes were introduced in [Mrozek 2017, Deﬁnition 5.10]. In this paper, we generalize this deﬁnition as follows. Let (X , T ) be a ﬁnite topological space. By a combinatorial multivector in X we mean a locally closed and non-empty subset of X. We deﬁne a combinatorial multivector ﬁeld as a partition V of X into multivectors. Therefore, unlike (Mrozek 2017), we do not assume that a multivector has a unique maximal element with respect to ≤ . Such an assumption was introduced in Mrozek (2017) for technical reasons but it is very inconvenient in applications. As an example we mention the following straightforward proposition which is not true in the setting of Mrozek (2017). Proposition 4.2 Assume V is a combinatorial multivector ﬁeld on a ﬁnite topological space X and Y ⊂ X is a locally closed subspace. Then V := { V ∩ Y | V ∈ V, V ∩ Y = ∅ } is a multivector ﬁeld in Y . We call it the multivector ﬁeld induced by V. We say that a multivector V is critical if the relative homology H (cl V , mo V ) is non-zero. A multivector V which is not critical is called regular. For each x ∈ X we denote by [x ] the unique multivector in V which contains x. If the multivector ﬁeld V is clear from the context, we write brieﬂy [x]:=[x ] . We say that x ∈ X is critical (respectively regular) with respect to V if [x ] is critical (respectively regular). We say that a subset A ⊂ X is V-compatible if for each x ∈ X either [x ] ∩ A = ∅ or [x ] ⊂ A. Note that every V-compatible set A ⊂ X V V induces a well-deﬁned multivector ﬁeld V := {V ∈ V | V ⊂ A} on A. The next proposition follows immediately from the deﬁnition of a V-compatible set. Proposition 4.3 The union and the intersection of a family of V-compatible sets is V-compatible. We associate with every multivector ﬁeld V a combinatorial dynamical system on X induced by the multivalued map : X X given by (x ) := [x ] ∪ cl x . (6) V V The following proposition is straightforward. Proposition 4.4 Let V be a multivector ﬁeld on X. Then (x ) =[x ] ∪ mo x . V V 123 Conley-Morse-Forman theory for generalized combinatorial… Fig. 7 An example of a combinatorial multivector ﬁeld V ={{A, C , G}, {D}, {H }, {E , I , J }, {B, F }} on a ﬁnite topological space consisting of ten points. There are two regular multivectors, {A, C , G} and {E , I , J }, the others are critical. Both the nodes and the connecting edges of each multivector are highlighted with a different color One can also easily prove the following proposition. Proposition 4.5 ([Lipinski ´ 2021, Proposition 4.1.5]) Let V be a combinatorial multi- vector ﬁeld on (X , T ).If A ⊂ X, then −1 (A) = [x ] ∪ opn x . x ∈A Note that by Proposition 3.6 a multivector in a ﬁnite topological space X is also op a multivector in X , that is, in the space X with the opposite topology. Thus, a op multivector ﬁeld V in X is also a multivector ﬁeld in X . However, the two multivector ﬁelds cannot be considered the same, because the change in topology implies the change of the location of critical and regular multivectors (see Fig. 8). We indicate op this in notation by writing V for the multivector ﬁeld V considered with the opposite topology. op op The multivector ﬁeld V induces a combinatorial dynamical system : op op op op X X given by (x ) := [x ] ∪ cl x. As an immediate consequence of V V T Proposition 3.2 and Proposition 4.5 we get following result. op Proposition 4.6 The combinatorial dynamical system is dual to the combinatorial op −1 dynamical system , that is, we have = . V V 4.4 Essential solutions Given a multivector ﬁeld V on a ﬁnite topological space X by a solution (full solution, forward solution, backward solution, partial solution or path) of V we mean a corre- sponding solution type of the combinatorial dynamical system . Given a solution 123 M. Lipinski ´ op Fig. 8 An example of a ﬁnite topological space X and X consisting of four points and with the same partition into multivectors V ={{A}, {B}, {C }, {D}}.In X multivectors {B}, {C } and {D} are critical, while op in X only {A} is critical ϕ of V we denote by V(ϕ) the set of multivectors V ∈ V such that V ∩ im ϕ = ∅.We denote the set of all paths of V inaset A by Path (A) and deﬁne Path (x , A) := {ϕ ∈ Path (A) | ϕ(0) = x }, V V Path (x , y, A) := {ϕ ∈ Path (A) | ϕ = x and ϕ = y}. V V We denote the set of full solutions of V in A (respectively backward or forward solu- − + tions in A) by Sol (A) (respectively Sol (A), Sol (A)). We also write Sol (x , A) := V V V V {ϕ ∈ Sol (A) | ϕ(0) = x }. Observe that by (6) x ∈ (x ) for every x ∈ X. Hence, V V a constant map from an interval to a point is always a solution. This means that every solution can easily be extended to a full solution. In consequence, every point is recur- rent which is not typical. To remedy this we introduce the concept of an essential solution. A full solution ϕ : Z → X is left-essential (respectively right-essential)iffor every regular x ∈ im ϕ the set { t ∈ Z | ϕ(t)/∈[x ] } is left-inﬁnite (respectively right-inﬁnite). We say that ϕ is essential if it is both left- and right-essential. We say that a point x ∈ X is essentially recurrent if an essential periodic solution passes through x. Note that a periodic solution ϕ is essential either if #V(ϕ) ≥ 2orif the unique multivector in V(ϕ) is critical. We denote the set of all essential solutions in A ⊂ X (respectively left- or right- + − essential solutions in A) by eSol (A) (respectively eSol (A), eSol (A)) and the set V V of all essential solutions in a set A ⊂ X passing through a point x by eSol (x , A) := {ϕ ∈ eSol(A) | ϕ(0) = x } and we deﬁne the invariant part of A ⊂ X by Inv A := {x ∈ A | eSol(x , A) = ∅} . (7) In particular, if Inv A = A then we say that A is an invariant set for V.Wedropthe subscript V in Sol , eSol and Inv whenever V is clear from the context. V V V 123 Conley-Morse-Forman theory for generalized combinatorial… Proposition 4.7 Let A, B ⊂ X be invariant sets. Then A ∪ B is also an invariant set. Proof Let x ∈ A. By the deﬁnition of an invariant set there exists an essential solution ϕ ∈ eSol(x , A). It is clear that ϕ is also an essential solution in A∪ B. Thus eSol(x , A∪ B) = ∅. The same holds for B. Hence Inv(A ∪ B) = A ∪ B. 4.5 Isolated invariant sets In this subsection we introduce the combinatorial counterpart of the concept of an isolated invariant set. In order to emphasize the difference, we say that an isolated invariant set is isolated by an isolating set, not by an isolating neighborhood. In com- parison to the classical theory of dynamical systems, the crucial difference is that we cannot guarantee the existence of disjoint isolating sets for two disjoint isolated invariant sets. This is caused by the tightness of the ﬁnite topological space. Deﬁnition 4.8 Aclosedset N isolates an invariant set S ⊂ N , if the following two conditions hold: (a) Every path in N with endpoints in S is a path in S, (b) (S) ⊂ N . In this case, we also say that N is an isolating set for S. An invariant set S is isolated if there exists a closed set N meeting the above conditions. An important example is given by the following straightforward proposition. Proposition 4.9 The whole space X isolates its invariant part Inv X. In particular, Inv X is an isolated invariant set. Proposition 4.10 If S ⊂ X is an isolated invariant set, then S is V-compatible. Proof Suppose the contrary. Then there exists an x ∈ S and a y ∈[x ] \ S.Let N be an isolating set for S. It follows from Deﬁnition 4.8(b), that y ∈ (x ) ⊂ N.Itis also clear that x ∈ (y). Thus the path x · y · x is a path in N with endpoints in S, but it is not contained in S, and this in turn contradicts Deﬁnition 4.8(a). The ﬁniteness of the space allows us to construct the smallest possible isolating set. More precisely, we have the following straightforward proposition. Proposition 4.11 Let N be an isolating set for an isolated invariant set S. If M is a closed set such that S ⊂ M ⊂ N , then S is also isolated by M. In particular, cl Sis the smallest isolating set for S. Proposition 4.12 Let S ⊂ X. If S is an isolated invariant set, then S is locally closed. Proof By Proposition 4.11 the set N := cl S is an isolating set for S. Assume that S is not locally closed. By Proposition 3.10 there exist x , z ∈ S and a y ∈ / S such that x ≤ y ≤ z. Hence, it follows from Theorem 3.8 that x ∈ cl y and y ∈ cl z. T T T T In particular, x , y, z ∈ cl S. It follows that ϕ := z · y · x is a solution in cl S with endpoints in S. In consequence, y ∈ S, a contradiction. 123 M. Lipinski ´ In particular, it follows from Proposition 4.12 that if S is an isolated invariant set, then we have the induced multivector ﬁeld V on X. Proposition 4.13 Let S be a locally closed, V-compatible invariant set. Then S is an isolated invariant set. Proof Assume that S is a V-compatible and locally closed invariant set. We will show that N := cl S isolates S.Wehave (S) = cl x ∪ [x ] = cl S ∪ S = cl S ⊂ N . V V x ∈S x ∈S Therefore condition (b) of Deﬁnition 4.8 is satisﬁed. We will now show that every path in N with endpoints in S is a path in S.Let ϕ := x · x · ... · x be a path in N with endpoints in S. Thus, x , x ∈ S. Suppose that 0 1 n 0 n there is an i ∈{0, 1, ..., n} such that x ∈ / S. Without loss of generality we may assume that i is maximal such that x ∈ / S. Then x = x and i < n, because x ∈ S.We i i +1 i n have x ∈ (x ) =[x ] ∪ cl x . Since x ∈ / S, x ∈ S and S is V-compatible, i +1 i i i i i +1 V V we cannot have x ∈[x ] . Therefore, x ∈ cl x . Since ϕ is a path in N = cl S, i +1 i i +1 i we have x ∈ cl S. Hence, x ∈ cl z for a z ∈ S. It follows from Proposition 3.10 that i i x ∈ S, because x , z ∈ S, x ∈ cl x , x ∈ cl z and S is locally closed. Thus, i i +1 i +1 i i we get a contradiction proving that also condition (a) of Deﬁnition 4.8 is satisﬁed. In consequence, N isolates S and S is an isolated invariant set. 4.6 Multivector field as a digraph Let V be a multivector ﬁeld in X. We denote by G the multivalued map interpreted V V as a digraph. Proposition 4.14 Assume A ⊂ X is strongly connected in G . Then the following conditions are pairwise equivalent. (i) There exists an essentially recurrent point x in A, that is, there exists an essential periodic solution in A through x, (ii) A is non-empty and every point in A is essentially recurrent in A, (iii) Inv A = ∅. Proof Assume (i). Then A = ∅.Let x ∈ A be an essentially recurrent point in A and let y ∈ A be arbitrary. Since A is strongly connected, we can ﬁnd a periodic solution ϕ in A passing through x and y.If#V(ϕ) ≥ 2, then ϕ is essential and y is essentially recurrent in A. Otherwise [x ] =[y] and we may easily modify the essential periodic V V solution in A through x to an essential periodic solution in A through y. This proves (ii). Implication (ii)⇒(iii) is straightforward. To prove that (iii) implies (i) assume that ϕ is an essential solution in A,If#V(ϕ) = 1, then the unique multivector V ∈ V(ϕ) is critical and every x ∈ V ⊂ A is essentially recurrent in A. Otherwise we can ﬁnd points x , y ∈ A such that [x ] =[y] . Since A is strongly connected, we can ﬁnd V V paths ψ ∈ Path (x , y, A) and ψ ∈ Path (y, x , A). Then ψ · ψ extends to an 1 V 2 V 1 2 essential periodic solution in A through x proving that x ∈ A is essentially recurrent in A. 123 Conley-Morse-Forman theory for generalized combinatorial… The above result considered the situation of a strongly connected set in G .Ifin addition we assume that this set is maximal, that is, a strongly connected component in G , we obtain the following result. Proposition 4.15 Let V be a multivector ﬁeld on X and let G be the associated digraph. If C ⊂ X is a strongly connected component of G , then C is V-compatible and locally closed. Proof Let x ∈ C and y ∈[x ] . It is clear that x · y ∈ Path (x , y, X ) and y · x ∈ V V Path (y, x , X ). Hence C is V-compatible. Let x , z ∈ C, y ∈ X be such that x ≤ y ≤ z. Since C is strongly connected we T T can ﬁnd a path ρ from x to z. Clearly, by Proposition 3.9 and (6)wehave y ∈ (z) and x ∈ (y). Thus y · ρ ∈ Path (y, z, X ) and z · y ∈ Path (z, y, X ). It follows V V V that y ∈ C. Hence, C is convex and, by Proposition 3.10, C is locally closed. The following theorem shows that some isolated invariant sets of combinatorial multivector ﬁelds may be characterized in purely graph-theoretic terms. However, it is not difﬁcult to give examples of isolated invariant sets which cannot be characterized this way (see [25]). Theorem 4.16 Let V be a multivector ﬁeld on X and let G be the associated digraph. If C ⊂ X is a strongly connected component of G such that eSol(C ) = ∅, then C is an isolated invariant set. Proof According to Proposition 4.13 it sufﬁces to prove that C is a V-compatible, locally closed invariant set. It follows from Proposition 4.15 that C is V-compatible and locally closed. Thus, we only need to show that C is invariant. Since Inv C ⊂ C, we only need to prove that C ⊂ Inv C.Let y ∈ C. Since eSol(C ) = ∅,wemay take an x ∈ C and a ϕ ∈ eSol(x , C ). Since C is strongly connected we can ﬁnd paths ρ and − + ρ in C from x to y and from y to x respectively. Then the solution ϕ · ρ · ρ · ϕ is a well-deﬁned essential solution through y in C. Thus, eSol(y, C ) = ∅, which proves that we have y ∈ Inv C. 5 Index pairs and Conley index In this section we construct the Conley index of an isolated invariant set of a combi- natorial multivector ﬁeld. As in the classical case we deﬁne index pairs, prove their existence and prove that the homology of an index pair depends only on the isolated invariant set and not on the choice of index pair. 5.1 Index pairs and their properties Deﬁnition 5.1 Let S be an isolated invariant set. A pair P = (P , P ) of closed subsets 1 2 of X such that P ⊂ P , is called an index pair for S if 2 1 (IP1) x ∈ P , y ∈ (x ) ∩ P ⇒ y ∈ P (positive invariance), 2 V 1 2 123 M. Lipinski ´ Fig. 9 Digraph G for a multivector ﬁeld from Fig. 7. Black edges are induced by closure relation, while the red bi-directional edges represent connections within a multivector. For clarity, we omit the edges that can be obtained by the between-level transitivity (e.g., from A to G). Nodes that are part of a critical multivector are additionally bolded in red (IP2) x ∈ P , (x ) \ P = ∅ ⇒ x ∈ P (exit set), 1 V 1 2 (IP3) S = Inv(P \ P ) (invariant part). 1 2 An index pair P is said to be saturated if S = P \ P . 1 2 Proposition 5.2 Let P be an index pair for an isolated invariant set S. Then P isolates S. Proof According to our assumptions, the set P is closed, and we clearly have S = Inv(P \ P ) ⊂ P \ P ⊂ P . Thus, it only remains to be shown that conditions (a) 1 2 1 2 1 and (b) in Deﬁnition 4.8 are satisﬁed. Suppose there exists a path ψ := x · x · ... · x in P such that x , x ∈ S and 0 1 n 1 0 n x ∈ P \ S for some i ∈{1, 2,..., n − 1}. First, we will show that im ψ ⊂ P \ P . i 1 1 2 To this end, suppose the contrary. Then, there exists an i ∈{1, 2,..., n − 1} such that x ∈ P and x ∈ P \ P . Since ψ is a path we have x ∈ (x ).But, (IP1) i 2 i +1 1 2 i +1 V i implies x ∈ P , a contradiction. i +1 2 Since S is invariant and x , x ∈ S,wemay take a ϕ ∈ eSol(x , S) and a ϕ ∈ 0 n 0 0 n eSol(x , S). The solution ϕ · ψ · ϕ is an essential solution in P \ P through x . n 1 2 i 0 n Thus, x ∈ Inv(P \ P ) = S, a contradiction. This proves that every path in P with i 1 2 1 endpoints in S is contained in S, and therefore Deﬁnition 4.8(a) is satisﬁed. In order to verify (b), let x ∈ S be arbitrary. We have already seen that then x ∈ P \ P ⊂ P . Now suppose that (x ) \ P = ∅. Then (IP2) implies x ∈ P , 1 2 1 V 1 2 which contradicts x ∈ P \ P . Therefore, we necessarily have (x ) \ P = ∅, that 1 2 V 1 is, (x ) ⊂ P , which immediately implies (b). Hence, P isolates S. V 1 1 One can easily see from the above proof that in Deﬁnition 5.1 one does not have to assume that S is an isolated invariant set. In fact, the proof of Proposition 5.2 implies that any invariant set which admits an index pair is automatically an isolated invariant set. Furthermore, the following result shows that every isolated invariant set S does indeed admit at least one index pair. 123 Conley-Morse-Forman theory for generalized combinatorial… Fig. 10 Schematic depiction of the two cases of a set A(P, Q) Proposition 5.3 Let S be an isolated invariant set. Then (cl S, mo S) is a saturated index pair for S. Proof To prove (IP1) assume that x ∈ mo S and y ∈ (x ) ∩ cl S. Since S is V- compatible we have [x ] ∩ S = ∅. Therefore, [x ] ∩ cl S ⊂ cl S \ S = mo S. Clearly, V V due to Propositions 3.3 and 4.12,cl x ⊂ mo S ⊂ cl S. Hence, y ∈ (x ) ∩ cl S = ([x ] ∪ cl x ) ∩ cl S = ([x ] ∩ cl S) ∪ (cl x ∩ cl S) ⊂ mo S. V V V To see (IP2) note that by Proposition 4.10 the set S is V-compatible and (S) = cl x ∪[x ] = cl x ∪ S = cl S. V V x ∈S x ∈S Thus, if x ∈ S, then (x )\cl S = ∅. Therefore, (x )\cl S = ∅ for x ∈ P = cl S V V 1 implies x ∈ cl S \ S = mo S. Finally, directly from the deﬁnition of mouth we have cl S \ mo S = S, which proves (IP3), as well as the fact that (cl S, mo S) is saturated. We write P ⊂ Q for index pairs P, Q meaning P ⊂ Q for i = 1, 2. We say that i i index pairs P, Q of S are semi-equal if P ⊂ Q and either P = Q or P = Q .For 1 1 2 2 semi-equal pairs P, Q,welet Q \ P if P = Q , 1 1 2 2 A(P, Q) := Q \ P if P = Q . 2 2 1 1 Proposition 5.4 Let P and Q be semi-equal index pairs for S. Then there is no essential solution in the set A(P, Q). Proof First note that the deﬁnition of A(P, Q) implies either A(P, Q) = Q \ P ⊂ Q \ P = Q \ Q and A(P, Q) ∩ (P \ P ) = ∅, 1 1 1 2 1 2 1 2 or A(P, Q) = Q \ P ⊂ Q \ P = P \ P and A(P, Q) ∩ (Q \ Q ) = ∅. 2 2 1 2 1 2 1 2 123 M. Lipinski ´ Therefore, by (IP3) and the ﬁrst inclusions in the above two statements we get Inv A(P, Q) ⊂ S. Yet, the second identities above clearly show that A(P, Q) is disjoint from S. Thus, Inv A(P, Q) = ∅, and by the deﬁnition of the invariant part (see (7)) there is no essential solution in A(P, Q). Lemma 5.5 Assume S is an isolated invariant set. Let P and Q be saturated index pairs for S. Then H (P , P ) H (Q , Q ). 1 2 1 2 Proof By the deﬁnition of a saturated index pair Q \ Q = S = P \ P . Hence, 1 2 1 2 using Theorem 3.11(i) we get H (P , P ) H (Q , Q ). 1 2 1 2 Proposition 5.6 Assume S is an isolated invariant set. Let P be an index pair for S. Then the set P \ P is V-compatible and locally closed. 1 2 Proof Assume that P \ P is not V-compatible. This means that for some x ∈ P \ P 1 2 1 2 there exists a y ∈[x ] \ (P \ P ). Then y ∈ P or y ∈ / P . Consider the case V 1 2 2 1 y ∈ P . Since [x ] =[y] ,wehave x ∈ (y).Itfollows from (IP1) that x ∈ P , 2 V V V 2 a contradiction. Consider now the case y ∈ / P . Then from (IP2) one obtains x ∈ P , 1 2 which is again a contradiction. Together, these cases imply that P \P is V-compatible. 1 2 Finally, the local closedness of P \ P follows immediately from Proposition 1 2 3.3(iii). Proposition 5.7 Assume S is an isolated invariant set. Let P ⊂ Q be semi-equal index pairs for S. Then A(P, Q) is V-compatible and locally closed. Proof First note that our assumptions give P , Q ⊂ P and P , Q ⊂ Q .If P = 2 2 1 2 2 1 2 Q , then A(P, Q) = Q \ P = (Q \ P ) \ (P \ P ) = (Q \ Q ) \ (P \ P ). 1 1 1 2 1 2 1 2 1 2 If P = Q , then 1 1 c c A(P, Q) = Q \ P = Q ∩ P = (P ∩ P ) ∩ Q 2 2 2 1 2 2 2 c c c = (P ∩ P ) ∩ (Q ∩ Q ) = (P \ P ) \ (Q \ Q ), 1 1 1 2 1 2 2 2 where the superscript c denotes the set complement in X. Thus, by Proposition 5.6, in both cases, A(P, Q) may be represented as a difference of V-compatible sets. Therefore, it is also V-compatible. The local closedness of A(P, Q) follows from Proposition 3.3. Lemma 5.8 Let A be a V-compatible, locally closed subset of X such that there is no essential solution in A. Then H (cl A, mo A) = 0. Proof Let A := {V ∈ V | V ⊂ A}. Since A is V-compatible, we have A = A.Let denote the transitive closure of the relation in A given for V , W ∈ A by A A V W ⇔ V ∩ cl W = ∅. (8) 123 Conley-Morse-Forman theory for generalized combinatorial… We claim that is a partial order in A. Clearly, is reﬂective and transitive. A A Hence, we only need to prove that is antisymmetric. To verify this, suppose the contrary. Then there exists a cycle V V ··· V = V with n > 1 and n A n−1 A A 0 n V = V for i = j and i , j ∈{1, 2,... n}. Since V ∩ cl V = ∅ we can choose i j i i −1 v ∈ V ∩ cl V and v ∈ V such that v ∈ cl v . Then v ∈ (v ) and i i i −1 i −1 i i V i −1 i −1 i −1 v ∈ (v ). Thus, we can construct an essential solution V i −1 i −1 ... · v · v · v · v · v · ... · v · v · v · v · ... . 1 2 n 1 n 1 2 n−1 n This contradicts our assumption and proves that is a partial order. Moreover, since a constant solution in a critical multivector is essential, all multi- vectors in A have to be regular. Thus, H (cl V , mo V ) = 0 for every V ∈ A. (9) Since is a partial order, we may assume that A ={V } where the numbering i =1 of V extends the partial order to a linear order ≤ , that is, A A V ≤ V ≤ ··· ≤ V . 1 A 2 A A m We claim that i < j ⇒ cl V \ V = cl V . (10) i j i Indeed, if this is not satisﬁed, then V ∩ cl V = ∅ which, by the deﬁnition (8)of j i gives V V as well as V V , and therefore j ≤ i, a contradiction. For j i j i A A A k ∈{0, 1,... m} deﬁne set W := V . Then W = ∅ and W = A.Now ﬁx a k j 0 m j =1 k ∈{0, 1,... m}. Observe that by (10)wehave k m k k cl W \ A = cl V \ V = cl V \ V = cl W \ W = mo W . k j j j j k k k j =1 j =1 j =1 j =1 Therefore, mo W = cl W \ A ⊂ cl A \ A = mo A. k k It follows that W ∪ mo A = cl W ∪ mo A. Hence, the set Z := W ∪ mo A is closed. k k k k For k > 0we have Z \ Z = W \ W \ mo A = V ∩ A = V = cl V \ mo V . k k−1 k k−1 k k k k Hence, we get from Theorem 3.11(i) and (9) H (Z , Z ) = H (cl V , mo V ) = 0. k k−1 k k 123 M. Lipinski ´ Now it follows from the exact sequence of the triple (Z , Z , cl A) that k−1 k H (cl A, Z ) H (cl A, Z ). k k−1 Note that Z = W ∪ mo A = mo A and Z = W ∪ mo A = A ∪ mo A = cl A. 0 0 m m Therefore, we ﬁnally obtain H (cl A, mo A) = H (cl A, Z ) H (cl A, Z ) = H (cl A, cl A) = 0, 0 m which completes the proof of the lemma. Lemma 5.9 Let P ⊂ Q be semi-equal index pairs of an isolated invariant set S. If P = Q , then H (Q , P ) = 0, and analogously, if P = Q , then H (Q , P ) = 0. 1 1 2 2 2 2 1 1 Proof By Theorem 5.7 the set A(P, Q) is locally closed and V-compatible. Hence, the conclusion follows from Proposition 5.4 and Lemma 5.8. Lemma 5.10 Let P ⊂ Q be semi-equal index pairs of an isolated invariant set S. Then H (P , P ) = H (Q , Q ). 1 2 1 2 Proof Assume P = Q . We get from Lemma 5.9 that H (Q , P ) = 0. Using Theo- 2 2 1 1 rem 3.11(ii) for the triple P ⊂ P ⊂ Q then implies 2 1 1 H (P , P ) = H (Q , P ) = H (Q , Q ). 1 2 1 2 1 2 Similarly, if P = Q we consider the triple P ⊂ Q ⊂ Q and obtain 1 1 2 2 1 H (P , P ) = H (Q , P ) = H (Q , Q ). 1 2 1 2 1 2 In order to show that two arbitrary index pairs carry the same homological informa- tion, we need to construct auxiliary, intermediate index pairs. To this end, we deﬁne the push-forward and the pull-back of a set A in B. π (A, B) := {x ∈ B |∃ ϕ ∈ A,ϕ = x }, (11) ϕ∈Path (B) π (A, B) := {x ∈ B |∃ ϕ = x,ϕ ∈ A} (12) ϕ∈Path (B) + − Proposition 5.11 Let A ⊂ X then π (A, X ) (π (A, X )) is closed (open) and V- V V compatible. Proof Let x ∈ π (A, X ) be arbitrary. Then there exists a point a ∈ A and a ϕ ∈ Path (a, x , X ). For any y ∈[x ] the concatenation ϕ·y is also a path. Thus, π (A, X ) V V is V-compatible. 123 Conley-Morse-Forman theory for generalized combinatorial… To show closedness, take an x ∈ π (A, X ) and y ∈ cl x.By(11) there exists an a ∈ A and a ϕ ∈ Path (a, x , X ). Then the path ϕ · y is a path from A to y, implying that y ∈ π (A, X ). Since X is ﬁnite one obtains + + cl π (A, X ) = cl x = π (A, X ), V V x ∈ (A,X ) + − and therefore π (A, X ) is closed. The proof for π (A, X ) is symmetric. V V Let P be an index pair for S. Deﬁne the set P ⊂ P of all points x ∈ P for which 1 1 there exists no path in P which starts in x and ends in S, that is, P := {x ∈ P | π (x , P ) ∩ S = ∅}. (13) 1 1 Proposition 5.12 If P is an index pair for an isolated invariant set S, then S ∩ P = ∅ and P ⊂ P. Proof The ﬁrst assertion is obvious. In order to see the second take an x ∈ P and suppose that x ∈ / P. This means that there exists a path ϕ in P such that ϕ = x and ϕ ∈ S. The condition (IP1) of Deﬁnition 5.1 implies im ϕ ∈ P . Therefore, ϕ ∈ P 2 2 and P ∩ S = ∅ which contradicts S ⊂ P \ P . 2 1 2 Proposition 5.13 If P is an index pair for an isolated invariant set S, then mo S ⊂ P. Moreover, (S) ⊂ S ∪ P. Proof To prove that mo S ⊂ P assume the contrary. Then there exists an x ∈ mo S, such that π (x , P ) ∩ S = ∅. It follows that there exists a path ϕ in P from x to S. 1 1 Since x ∈ mo S ⊂ cl S, we can take a y ∈ S such that x ∈ cl y ⊂ (y).Itfollowsthat ψ := y · ϕ is a path in P through x with endpoints in S. Since, by Proposition 5.2, P 1 1 isolates S, we get x ∈ S, a contradiction. Finally, by V-compatibility of S guaranteed ˆ ˆ by Proposition 4.10, we have the inclusion (S) = cl S ⊂ S ∪ mo S ∪ P = S ∪ P, which proves the remaining assertion. Proposition 5.14 Let P be an index pair for an isolated invariant set S. Then the sets ˆ ˆ P and P ∪ S are closed. Proof Let x ∈ P and let y ∈ cl x. Then y ∈ (x ). Moreover, y ∈ P , because P is closed. Clearly, if ϕ ∈ Path (y, P ), then x · ϕ ∈ Path (x , P ). Therefore 1 V 1 V 1 + + π (y, P ) ⊂ π (x , P ). Since, by (13), the latter set is disjoint from S,soisthe 1 1 V V ˆ ˆ former one. Therefore, y ∈ P. It follows that P is closed. ˆ ˆ ˆ ˆ Proposition 5.13 implies that cl(S ∪ P) = cl S ∪ P = S ∪ mo S ∪ P = S ∪ P, which proves the closedness of S ∪ P. Lemma 5.15 If P is an index pair for an isolated invariant set S, then P := (S ∪ ∗∗ ˆ ˆ ˆ P, P ) is an index pair for S and P := (S ∪ P, P) is a saturated index pair for S. 123 M. Lipinski ´ ∗ ∗ Proof First consider P . By Proposition 5.14 set P = S ∪ P is closed. By Proposition ˆ ˆ 5.12 we have P ⊂ P ⊂ S ∪ P. ∗ ∗ Let x ∈ P = P and let y ∈ (x ) ∩ P . Then y ∈ (x ) ∩ P .Itfollows from 2 1 V V 2 1 (IP1) for P that y ∈ P . Thus, (IP1) is satisﬁed for P . ∗ ∗ Now, let x ∈ P = S ∪ P and suppose that there is a y ∈ (x ) \ P = ∅.We 1 1 have x ∈ / S, because otherwise (x ) ⊂ mo S ∪ S = cl S ⊂ cl(S ∪ P) and then ∗ ∗ Proposition 5.14 implies (x ) ⊂ S ∪ P ⊂ P which contradicts (x ) \ P = ∅. V V 1 1 ˆ ˆ Hence, x ∈ P.Wehave y ∈ / P because otherwise y ∈ π (x , P ) ⊂ P ⊂ P ,a 1 1 contradiction. Thus (x ) \ P = ∅. Since x ∈ P ⊂ P ,by (IP2) for P we get V 1 1 x ∈ P = P . This proves (IP2) for P . ∗ ∗ ∗ Clearly, P \ P = P \ P ⊂ P \ P , and therefore we have the inclusion 2 1 2 1 2 1 ∗ ∗ Inv P \ P ⊂ Inv (P \ P ) = S. To verify the opposite inclusion, let x ∈ S be 1 2 1 2 arbitrary. Since S is an invariant set, there exists an essential solution ϕ ∈ eSol(x , S). ∗ ∗ We have im ϕ ⊂ S ⊂ (P ∪ S) \ P = P \ P , because P ∩ S = ∅. Consequently, 2 2 1 2 ∗ ∗ ∗ ∗ ∗ x ∈ Inv(P \ P ) and S = Inv(P \ P ). Hence, P also satisﬁes (IP3), which 1 2 1 2 completes the proof that P is an index pair for S. ∗∗ ∗∗ Consider now the second pair P .Let x ∈ P = P be arbitrary and choose y ∈ ∗∗ ˆ ˆ (x ) ∩ P = (x ) ∩ (P ∪ S). Since x ∈ P we get from (13) that (x ) ∩ S = ∅. V V V ∗∗ ∗∗ ˆ ˆ Thus, y ∈ (x ) ∩ P ⊂ P = P . This proves (IP1) for the pair P . ∗∗ ∗∗ To see (IP2) take an x ∈ P = P ∪ S and assume (x ) \ P = ∅.We 1 1 cannot have x ∈ S, because then (x ) ⊂ (S) and Proposition 5.13 implies V V ∗∗ ∗∗ ˆ ˆ (x ) ⊂ S ∪ P = P , a contradiction. Hence, x ∈ P = P which proves (IP2) 1 2 ∗∗ for P . ˆ ˆ ˆ Finally, we clearly have S ∩ P = ∅. Therefore, (S ∪ P) \ P = S and ∗∗ ∗∗ ˆ ˆ Inv(P \ P ) = Inv((S ∪ P) \ P) = Inv S = S. 1 2 ∗∗ This proves that P satisﬁes (IP3) and that it is saturated. Theorem 5.16 Let P and Q be two index pairs for an invariant set S. Then H (P , P ) H (Q , Q ). 1 2 1 2 ∗ ∗ ∗∗ Proof It follows from Lemma 5.15 that P ⊂ P as well as P ⊂ P are semi-equal index pairs. Hence, we get from Lemma 5.10 that ∗ ∗ ∗∗ ∗∗ ∼ ∼ H (P , P ) = H (P , P ) = H (P , P ). 1 2 1 2 1 2 Similarly, one obtains ∗ ∗ ∗∗ ∗∗ ∼ ∼ H (Q , Q ) = H (Q , Q ) = H (Q , Q ). 1 2 1 2 1 2 ∗∗ ∗∗ Since both pairs P and Q are saturated, it follows from Lemma 5.5 that ∗∗ ∗∗ ∗∗ ∗∗ ∼ ∼ H (P , P ) H (Q , Q ). Therefore, H (P , P ) H (Q , Q ). = = 1 2 1 2 1 2 1 2 123 Conley-Morse-Forman theory for generalized combinatorial… 5.2 Conley index We deﬁne the homology Conley index of an isolated invariant set S as H (P , P ) 1 2 where (P , P ) is an index pair for S. We denote the homology Conley index of S by 1 2 Con(S). Proposition 5.3 and Theorem 5.16 guarantee that the homology Conley index is well-deﬁned. Given a locally closed set A ⊂ X we deﬁne its ith Betti number β (A) and Poincaré polynomial p (t ), respectively, as the ith Betti number and the Poincaré polynomial of the pair (cl A, mo A), that is, β (A) := β (cl A, mo A) and p (t ) := p (t ) i i A cl A,mo A (see (4)). The theorem used in the following Proposition originally comes from Rybakowski and Zehnder (1985), but we use its more general version that was stated in Mrozek (2017). Proposition 5.17 If (P , P ) is an index pair for an isolated invariant set S, then 1 2 p (t ) + p (t ) = p (t ) + (1 + t )q(t ), (14) S P P 2 1 where q(t ) is a polynomial with non-negative coefﬁcients. Moreover, if H (P ) = H (P ) ⊕ H (cl S, mo S) 1 2 then q(t ) = 0. Proof An index pair (P , P ) induces a long exact sequence of homology modules 1 2 ... → H (P ) → H (P ) → H (P , P ) → H (P ) → ... . (15) n 2 n 1 n 1 2 n−1 2 By Proposition 5.3 and Theorem 5.16 we have H (P , P ) = H (cl S, mo S). Thus, we 1 2 can replace (15) with ... → H (P ) → H (P ) → H (cl S, mo S) → H (P ) → ... . n 2 n 1 n n−1 2 In view of [Mrozek 2017, Theorem 4.6] we further get p (t ) + p (t ) = p (t ) + (1 + t )q(t ). S P P 2 1 for some polynomial q with non-negative coefﬁcients. The second assertion follows directly from the second part of [Mrozek 2017, Theorem 4.6] (see also Rybakowski and Zehnder 1985). We say that an isolated invariant set S decomposes into the isolated invariant sets S and S if cl S ∩ S = ∅, S ∩ cl S = ∅,aswellas S = S ∪ S . Proposition 5.18 Assume an isolated invariant set S decomposes into the isolated invariant sets S and S . Then Sol(S) = Sol(S ) ∪ Sol(S ). 123 M. Lipinski ´ Proof The inclusion Sol(S ) ∪ Sol(S ) ⊂ Sol(S) is trivial. To see the opposite inclu- sion, let ϕ ∈ Sol(S).Wehavetoprove that im ϕ ⊂ S or im ϕ ⊂ S . If this were not the case, then without loss of generality we can assume that there exists a j ∈ Z such that both ϕ( j ) ∈ S and ϕ( j + 1) ∈ S are satisﬁed. This immediately implies ϕ( j + 1) ∈ cl ϕ( j ) ∪[ϕ( j )] .Wehave ϕ( j + 1)/∈[ϕ( j )] , because otherwise the V V V-compatibility of S (see Propostition 4.10) implies ϕ( j + 1) ∈ S and, in conse- quence, S ∩ S = ∅, a contradiction. Hence, ϕ( j + 1) ∈ cl ϕ( j ) ⊂ cl S which yields cl S ∩ S = ∅, another contradiction, proving that ϕ ∈ Sol(S ) ∪ Sol(S ). Theorem 5.19 Assume an isolated invariant set S decomposes into the isolated invari- ant sets S and S . Then we have Con(S) = Con(S ) ⊕ Con(S ). Proof In view of Proposition 5.3, the two pairs P = (cl S , mo S ) and Q = (cl S , mo S ) are saturated index pairs for S and S , respectively. Consider the fol- lowing exact sequence given by Theorem 3.11(iii): ... →H (P ∩ Q , P ∩ Q ) → H (P , P ) ⊕ H (Q , Q ) n 1 1 2 2 n 1 2 n 1 2 (16) →H (P ∪ Q , P ∪ Q ) → H (P ∩ Q , P ∩ Q ) → ... . n 1 1 2 2 n−1 1 1 2 2 Note that S ∩ Q ⊂ S ∩ cl S = ∅ and similarly S ∩ P = ∅. Since both P and Q 2 2 are saturated and S ∩ S = ∅ we get P ∩ Q = (S ∪ P ) ∩ (S ∪ Q ) 1 1 2 2 = (S ∩ S ) ∪ (S ∩ Q ) ∪ (P ∩ S ) ∪ (P ∩ Q ) = P ∩ Q . 2 2 2 2 2 2 Thus, H (P ∩ Q , P ∩ Q ) = 0, which together with the exact sequence (16) implies 1 1 2 2 H (P ∪ Q , P ∪ Q ) = H (P , P ) ⊕ H (Q , Q ). (17) ∗ 1 1 2 2 ∗ 1 2 ∗ 1 2 Notice further that S ∩ cl S = ∅ implies S \ Q = S . Similarly S \ P = S . 2 2 Therefore, one obtains the identity (P \ P \ Q ) ∪ (Q \ Q \ P ) = (S \ Q ) ∪ (S \ P ) = S ∪ S = S. 1 2 2 1 2 2 2 2 Hence, by Theorem 3.11(i), H (cl S, mo S) = H (P ∪ Q , P ∪ Q ). (18) 1 1 2 2 Finally, from (17) and (18) we get Con(S) = H (cl S, mo S) = H (P ∪ Q , P ∪ Q ) 1 1 2 2 = H (P , P ) ⊕ H (Q , Q ) = Con(S ) ⊕ Con(S ), 1 2 1 2 which completes the proof of the theorem. 123 Conley-Morse-Forman theory for generalized combinatorial… 6 Attractors, repellers and limit sets In the rest of the paper we assume that the ﬁnite topological space X is invariant with respect to a given combinatorial multivector ﬁeld V on X. We need the invariance assumption to guarantee the existence of an essential solution through every point in X. This assumption is not very restrictive, because if X is not invariant, then we can replace the space X by its invariant part Inv X and the multivector ﬁeld V by its restriction V (see Propositions 4.2 and 4.9). Inv X 6.1 Attractors, repellers and indecomposable sets We say that an invariant set A ⊂ X is an attractor if (A) = A. In addition, an −1 invariant set R ⊂ X is a repeller if (R) = R. The following proposition shows that we can also express the concepts of attractor and repeller in terms of push-forward and pull-back. Proposition 6.1 Let A be an invariant set. Then A is an attractor (a repeller) in X if + − and only if π (A, X ) = A(π (A, X ) = A). V V Proof Let A be an attractor. The inclusion S ⊂ π (S, X ) is true for an arbitrary set. Suppose that there exists a y ∈ π (A, X ) \ A. Then by (11) we can ﬁnd an x ∈ A and ϕ ∈ Path (x , y, X ). This implies that there exists a k ∈ Z such that ϕ(k) ∈ A and ϕ(k + 1)/ ∈ A.But ϕ(k + 1) ∈ (ϕ(k)) ⊂ (A) = A, a contradiction. Therefore, V V π (A, X ) = A. + + Now assume that π (A, X ) = A.Again,by(11), we get A = π (A, X ) = V V (π (A, X )) = (A). V V The proof for a repeller is analogous. Theorem 6.2 The following conditions are equivalent: (1) A is an attractor, (2) A is closed, V-compatible, and invariant, (3) A is a closed isolated invariant set. Proof Let A be an attractor. It follows immediately from Propositions 6.1 and 5.11 that condition (1) implies condition (2). Moreover, Proposition 4.13 shows that (2) implies (3). Finally, suppose that (3) holds. By Proposition 4.10 set A is V-compatible. It is also closed. Therefore, we have (A) = cl x ∪[x ] = cl x ∪ [x ] = cl A ∪ A = A, V V V x ∈A x ∈A x ∈A which proves that A is an attractor. Theorem 6.3 The following conditions are equivalent: (1) R is a repeller, 123 M. Lipinski ´ (2) R is open, V-compatible, and invariant, (3) R is an open isolated invariant set. Proof Assume R is a repeller. It follows from Propositions 6.1 and 5.11 that condition (1) implies condition (2), and Proposition 4.13 shows that (2) implies (3). Finally, assume that condition (3) holds. Then R is V-compatible by Proposition 4.10.The openness of R and Proposition 4.5 imply −1 (R) = opn x ∪[x ] = opn x ∪ [x ] = R, V V x ∈R x ∈R x ∈R which proves that R is a repeller. Let ϕ be a full solution in X. We deﬁne the ultimate backward and forward image of ϕ respectively by uim ϕ := ϕ ((−∞, t ]) , t ∈Z uim ϕ := ϕ [t , +∞) . ( ) t ∈Z Note that in a ﬁnite space a descending sequence of sets eventually must become constant. Therefore, we get the following result. Proposition 6.4 There exists a k ∈ N such that uim ϕ = ϕ((−∞, −k]) and + − + uim ϕ = ϕ([k, +∞)). In particular, the sets uim ϕ and uim ϕ are always non- empty. Proposition 6.5 If ϕ is a left-essential (a right-essential) solution, then we can ﬁnd an − + essential solution ψ such that im ψ ⊂ uim ϕ (im ψ ⊂ uim ϕ). Proof We only consider the case of a right-essential solution ϕ. By Proposition 6.4 there exists a k ∈ Z such that uim ϕ = ϕ([k, +∞)). We consider two cases. If uim ϕ passes through a critical multivector, then we can easily build a stationary essential solution. In the second case, we have at least two different multivectors + + V , W ∈ V such that V ∩ uim ϕ = ∅ = W ∩ uim ϕ. Then there exist t , s, u ∈ Z with k < t < s < u and ϕ(t ) ∈ V , ϕ(s) ∈ W , and ϕ(u) ∈ V . But then the concatenation ··· · ϕ([t , u]) · ϕ([t , u]) · ... is clearly essential. Deﬁnition 6.6 We say that a non-empty invariant set A ⊂ X is Morse indecomposable, or brieﬂy indecomposable, if the only non-empty attractor in A is the entire set A. Proposition 6.7 Let A ⊂ X be a non-empty invariant set. Then A is indecomposable if and only if A is a strongly connected set in G . Proof Let A be an indecomposable invariant set. Suppose it is not strongly connected. Then we can ﬁnd points x , y ∈ A such that Path (x , y, A) = ∅. Deﬁne A := Inv π (x , A). Clearly, y ∈ / A . We will show that A is a non-empty attractor in A. 123 Conley-Morse-Forman theory for generalized combinatorial… + + Let z ∈ π (x , A). Since A is invariant there exists ϕ ∈ eSol (z, A). By Proposition V V 6.5 we can construct an essential solution ψ such that im ψ ⊂ uim ϕ ⊂ π (x , A). Thus, A is non-empty. + + Now suppose that π (A , A) = A . Then there exists an a ∈ π (A , A) \ A .It V V follows from (11)that forevery b ∈ A we have Path (a, b, A) = ∅, since otherwise we could construct an essential solution through a which lies in π (x , A).Using exactly the same reasoning as above, one can further show that Inv(π (A , A) \ A ) = + + ∅. But since we clearly have the inclusion Inv(π (A , A) \ A ) ⊂ Inv(π (x , A)) = V V A , this leads to a contradiction. Thus, Proposition 6.1 shows that the set A is indeed an attractor, which is non-empty and a proper subset of A. Since this contradicts the minimality of A, we therefore conclude that A is strongly connected. Now assume conversely that A is strongly connected. It is clear that for any point x ∈ A we get π (x , A) = A. It follows by Proposition 6.1 that the only non-empty attractor in A is the entire set A. The duality allows us to adapt the proof of Proposition 6.7 to get the following proposition. Proposition 6.8 An invariant set R is an indecomposable invariant set if and only if the only non-empty repeller in R is the entire set R. Proposition 6.9 Let S ⊂ X be an indecomposable invariant set and let A ⊂ Xbe an attractor (a repeller). If A ∩ S = ∅ then S ⊂ A. Proof Let x ∈ A ∩ S and let y ∈ S. There exists ϕ ∈ Path (x , y, S). Now deﬁne t = min dom ϕ and s = max dom ϕ. Clearly ϕ(t + 1) ∈ (ϕ(t )) = (x ) ⊂ (A) = A. V V V Now, by induction let k ∈{t , t + 1,..., s − 1} and ϕ(k) ∈ A then ϕ(k + 1) ∈ (ϕ(k)) ⊂ (A) = A. V V Therefore, y = ϕ(k + 1) ∈ A, and this implies S ⊂ A. For a full solution ϕ in X, deﬁne the sets − − V (ϕ) := V ∈ V | V ∩ uim ϕ = ∅ , (19) + + V (ϕ) := V ∈ V | V ∩ uim ϕ = ∅ . (20) − + We refer to a multivector V ∈ V (ϕ) (respectively V (ϕ))asa backward (respectively − + forward) ultimate multivector of ϕ. The families V (ϕ) and V (ϕ) will be used in the sequel, in particular in the proof of the following theorem. Theorem 6.10 Let A ⊂ X be an attractor. Then A := Inv (X \ A) is a repeller in X, which is called the dual repeller of A. Conversely, if R is a repeller, then R := Inv (X \ R) is an attractor in X, called the dual attractor of R. Moreover, the dual repeller (or the dual attractor) is non-empty, unless we have A = X(or R = X). 123 M. Lipinski ´ Proof We will show that A is open. Let x ∈ A and let y ∈ opn x. Then one has x ∈ cl y by Proposition 3.9. Since A is closed as an attractor (Proposition 6.2), we immediately get y ∈ / A. The invariance of X lets us select a ϕ ∈ eSol(y, X ). Then − − im ϕ ∩ A = ∅, because otherwise there exists a t ∈ Z such that ϕ(t ) ∈ A and ϕ(t + 1)/ ∈ A, which gives ϕ(t + 1) ∈ (ϕ(t )) ⊂ (A) = A, V V a contradiction. Now, let ψ ∈ eSol(x , A ). Clearly, x ∈ cl y ⊂ (y). Thus, ϕ · ψ ∈ eSol(y, X \ A).Itfollows that y ∈ Inv(X \ A) = A which proves that opn A ⊂ A . Therefore, the set A is open. Since A is V-compatible, also X \ A is V-compatible. Let x ∈ A and let y ∈ [x ] . Since x ∈ A ⊂ X \ A, V-compatibility of X \ A implies y ∈ / A. Select a − + ϕ ∈ eSol(x , A ) Thus, ϕ · y · ϕ is a well-deﬁned essential solution in X \ A, that is, eSol(y, X \ A) = ∅. It follows that y ∈ Inv(X \ A) = A . Hence, A is V-compatible. Altogether, the set A is invariant, open and V-compatible. Thus, by Theorem 6.3 it is a repeller. Finally, we will show that A = ∅ unless A = X. Suppose that X \ A = ∅, and let x ∈ X \ A. Since X is invariant, there exists a ϕ ∈ eSol(x , X ). As in the ﬁrst part − − of the proof one can show that im ϕ ∩ A = ∅, that is, we have im ϕ ⊂ X \ A. According to Proposition 6.5 there exists an essential solution ψ such that im ψ ⊂ − − uim ϕ ⊂ im ϕ ⊂ X \ A, and this immediately implies A = Inv (X \ A) = ∅. 6.2 Limit sets We deﬁne the V-hull of a set A ⊂ X as the intersection of all V-compatible, locally closed sets containing A, and denote it by A . As an immediate consequence of Proposition 3.4 and Proposition 4.3 we get the following result. Proposition 6.11 For every A ⊂ X its V-hull is V-compatible and locally closed. We deﬁne the α- and ω-limit sets of a full solution ϕ respectively by α(ϕ) := uim ϕ , ω(ϕ) := uim ϕ . The following proposition is an immediate consequence of Proposition 3.6. op Proposition 6.12 Assume ϕ is a full solution of V and ϕ is the associated dual op op op solution of V . Then α(ϕ) = ω(ϕ ) and ω(ϕ) = α(ϕ ). Proposition 6.13 Let ϕ be an essential solution. Then α(ϕ) = V (ϕ) and ω(ϕ) = V (ϕ) . 123 Conley-Morse-Forman theory for generalized combinatorial… Proof Clearly − − − uim ϕ ⊂ V ∈ V | V ∩ uim ϕ = ∅ = V (ϕ) and therefore − − α(ϕ) = uim ϕ ⊂ V (ϕ) . − − Now let x ∈ V (ϕ). Then there exists a y ∈[x ] such that y ∈ uim ϕ. Then y ∈ α(ϕ) and, since α(ϕ) is V-compatible, [y] =[x ] ⊂ α(ϕ).Thus,wehave V V V (ϕ) ⊂ α(ϕ). Since α(ϕ) is locally closed and V-compatible, the set α(ϕ) is a superset of the V-hull of α (ϕ). Hence, V (ϕ) ⊂ α(ϕ). The proof for ω(ϕ) is analogous. Lemma 6.14 Assume ϕ : Z → X is a full solution of V and V (ϕ) (respectively V (ϕ)) contains at least two different multivectors. Then for every V ∈ V such that V ⊂ α(ϕ) (respectively V ⊂ ω(ϕ)) we have ( (V ) \ V ) ∩ α(ϕ) = ∅ (respectively ( (V ) \ V ) ∩ ω(ϕ) = ∅) (21) V V and −1 −1 (V ) \ V ∩ α(ϕ) = ∅ (respectively (V ) \ V ∩ ω(ϕ) = ∅). (22) V V Proof Assume V ∈ V is such that V ⊂ α(ϕ). This happens if V ∈ V (ϕ), but might also happen for some V ∈ / V (ϕ). Assume ﬁrst that V ∈ V (ϕ). Since there are at least two different multivectors − − in the set V (ϕ) there exists a strictly decreasing sequence k : N → Z such that ϕ(k ) ∈ V and ϕ(k + 1)/ ∈ V . Since the set {ϕ(k + 1) | n ∈ N}⊂ X is ﬁnite, n n n after taking a subsequence, if necessary, we may assume that ϕ(k + 1) = y ∈ / V.Let − − W := [y] . Then W = V and y ∈ W ∩ uim ϕ ∩ (V ). This implies W ∈ V (ϕ) V V and (V ) ∩ W = ∅. By Proposition 6.13 we have ∅ = (V ) ∩ W ⊂ ( (V ) \ V ) ∩ W ⊂ ( (V ) \ V ) ∩ α(ϕ). V V V Thus, (21) is satisﬁed. Now assume that V ∈ / V (ϕ). We have (V ) = cl V ∪ V = cl V . Suppose that (21) does not hold. Then ∅ = ( (V ) \ V ) ∩ α(ϕ) = (cl V \ V ) ∩ α(ϕ) = mo V ∩ α(ϕ) 123 M. Lipinski ´ and therefore α(ϕ) \ V = (cl α(ϕ) \ mo α(ϕ)) \ (V ∪ mo V ) = (cl α(ϕ) \ mo α(ϕ)) \ cl V = cl α(ϕ) \ (mo α(ϕ) ∪ cl V ) . By Proposition 3.10 the set α(ϕ) \ V is locally closed as a difference of closed sets. Clearly, α(ϕ) \ V is V-compatible. This shows that α(ϕ) is not a minimal locally closed and V-compatible set containing V (ϕ). This contradicts Proposition 6.13. Hence (21) holds for V ⊂ α(ϕ). The proof of (21)for V ∈ V (ϕ) is a straightforward adaptation of the proof for op op op V ⊂ α(ϕ).Tosee (22) observe that since ϕ is a full solution of V , ω(ϕ ) = + op − α(ϕ) by Proposition 6.12 and, clearly, V (ϕ ) = V (ϕ), we may apply (21)to op op op −1 V , ϕ and ω(ϕ ). Thus, by Proposition 4.6 we get (V ) \ V ∩ α(ϕ) = op op ( (V ) \ V ) ∩ ω(ϕ ) = ∅, and the claim for ω(ϕ) follows similarly. Theorem 6.15 Let ϕ be an essential solution in X. Then both limit sets α(ϕ) and ω(ϕ) are non-empty strongly connected isolated invariant sets. Proof The nonemptiness of α(ϕ) and ω(ϕ) follows from Proposition 6.4. The sets α(ϕ) and ω(ϕ) are V-compatible and locally closed by Proposition 6.11. In order to prove that they are isolated invariant sets it sufﬁces to apply Proposition 4.13 as long as we prove that α(ϕ) and ω(ϕ) are also invariant. We will ﬁrst prove that α(ϕ) is invariant. Let x ∈ α(ϕ). Suppose that V (ϕ) is a singleton. Then by Proposition 6.13, α(ϕ) =[x ] . Since ϕ is essential, this is possible only if [x ] is critical. It follows that the stationary solution ψ(t ) = x is essential. Hence α(ϕ) is an isolated invariant set. Assume now that there are at least two different multivectors in V (ϕ). Then the assumptions of Lemma 6.14 are satisﬁed and, as a consequence of (21), for every x ∈ α(ϕ) there exist a point x ∈[x ] and a y ∈ α(ϕ) such that y ∈ (x ) \[x ] ∩ V V V α(ϕ). Hence, we can construct a right-essential solution x · x · x · x · x · x · ..., 0 1 2 0 1 2 where x = x, x ∈[x ] , and x ∈ (x ) \[x ] ∩α(ϕ). Property (22) provides 0 i V i +1 V i V i i a complementary left-essential solution. Concatenation of both solutions gives an essential solution in α(ϕ). Hence, we proved that α(ϕ) is invariant and consequently an isolated invariant set. Finally, we prove that α(ϕ) is strongly connected. To this end consider points x , y ∈ α(ϕ). We will show that then Path (x , y, α(ϕ)) = ∅. Using the two abbreviations V := [x ] and V := [y] it is clear that x y V V V , V ∈ V (ϕ) ⇒ Path (x , y, α(ϕ)) = ∅. (23) x y V − − Assume now that V ⊂ α(ϕ) \ V (ϕ) and V ∈ V (ϕ).By(23) it is enough to x y show that there exists at least one point z ∈ V (ϕ) such that Path (x , z, α(ϕ)) = ∅. Suppose the contrary. 123 Conley-Morse-Forman theory for generalized combinatorial… The set π (V , α(ϕ)) is closed and V-compatible in α(ϕ) by Proposition 5.11. By Proposition 3.5 set A := α(ϕ) \ π (V , α(ϕ)) is locally closed. Clearly, it is V-compatible and contains V (ϕ). Yet this results in a contradiction, because we have now found a smaller V-hull for V (ϕ). − − Now, consider the case when V ∈ V (ϕ) and V ⊂ α(ϕ) \ V (ϕ).The set x y π (V , α(ϕ)) is V-compatible and locally closed by Proposition 5.11.Inviewof(23) + + we have V (ϕ) ⊂ π (V , α(ϕ)). Thus, one either has V ⊂ π (V , α(ϕ)) or x y x V V + − π (V , α(ϕ)) is a smaller V-compatible, locally closed set containing V (ϕ).In both cases we get a contradiction. − − Finally, let V , V ⊂ α(ϕ) \ V (ϕ) and let z ∈ V (ϕ). Using the previous x y cases we can ﬁnd ψ ∈ Path (x , z, α(ϕ)) and ψ ∈ Path (z, y, α(ϕ)). Then, ψ ·ψ ∈ 1 2 1 2 V V Path (x , y, α(ϕ)). This ﬁnishes the proof that α(ϕ) is strongly connected. The proof for ω(ϕ) is analogous. Let ϕ be an essential solution. We say that an isolated invariant set S absorbs ϕ in positive (respectively negative) time if ϕ(t ) ∈ S for all t ≥ t (for all t ≤ t )for some 0 0 t ∈ Z. We denote by (ϕ) (respectively by A(ϕ)) the family of isolated invariant sets absorbing ϕ in positive (respectively negative) time. Proposition 6.16 For an essential solution ϕ we have α(ϕ) = A(ϕ), (24) ω(ϕ) = (ϕ). (25) Proof Let ϕ be an essential solution. It follows from Proposition 6.4 that there exists − − a k ∈ Z such that ϕ((−∞, k]) = uim ϕ ⊂ α(ϕ). Moreover, by Proposition 6.15 we have α(ϕ) ∈ A(ϕ). Hence A(ϕ) ⊂ α(ϕ). To see the opposite inclusion take an S ∈ A(ϕ). Then, there exists a t ∈ Z such that ϕ((−∞, t ]) ⊂ S. It follows that 0 0 α(ϕ) =uim ϕ ⊂ϕ((−∞, t ]) ⊂S = S. V 0 V V Hence, α(ϕ) ⊂ A(ϕ). This proves (24). The proof of (25) is analogous. Let A, B ⊂ X. We deﬁne the connection set from A to B by: C (A, B) := x ∈ X |∃ α(ϕ) ⊂ A and ω(ϕ) ⊂ B . (26) ϕ∈eSol(x ,X ) Proposition 6.17 Assume A, B ⊂ X. Then the connection set C (A, B) is an isolated invariant set. Proof To prove that C (A, B) is invariant, take an x ∈ C (A, B) and choose a ϕ ∈ eSol(x , X ) as in (26). It is clear that ϕ(t ) ∈ C (A, B) for every t ∈ Z. Thus, ϕ ∈ eSol(x , C (A, B)), and this in turn implies x ∈ Inv C (A, B) and shows that C (A, B) − + is invariant. Now consider a point y ∈[x ] . Then the solution ρ = ϕ · y · ϕ is a well-deﬁned essential solution through y such that α(ρ) ⊂ A and ω(ρ) ⊂ B. Thus, C (A, B) is V-compatible. 123 M. Lipinski ´ In order to prove that C (A, B) is locally closed, consider x , z ∈ C (A, B), and a y ∈ X such that z ≤ y ≤ x. Select essential solutions ϕ ∈ eSol(x , C (A, B)) and T T x − + ϕ ∈ eSol(z, C (A, B)). Then ψ := ϕ · y · ϕ is a well-deﬁned essential solution x z through y such that α(ψ) ⊂ A and ω(ψ) ⊂ B.Itfollows that y ∈ C (A, B). Thus, by Proposition 3.10, C (A, B) is locally closed. Finally, Proposition 4.13 proves that C (A, B) is an isolated invariant set. Proposition 6.18 Assume A is an attractor. Then C (A, A ) = ∅. Similarly, if R is a repeller, then C (R , R) = ∅. Proof Suppose there exists an x ∈ C (A, A ). Then by (26) we can choose a ϕ ∈ eSol(x , X ) and a t ∈ Z such that ϕ(t ) ∈ A and ϕ(t + 1)/ ∈ A. However, since A is an attractor, ϕ(t ) ∈ A implies ϕ(t + 1) ∈ A, a contradiction. The proof for a repeller is analogous. 7 Morse decomposition, Morse equation, Morse inequalities In this ﬁnal section we deﬁne Morse decompositions and prove the Morse inequali- ties for combinatorial multivector ﬁelds. We recall the general assumption that X is invariant with respect to a ﬁxed combinatorial multivector ﬁeld V on X. 7.1 Morse decompositions Deﬁnition 7.1 Assume X is invariant and (P, ≤) is a ﬁnite poset. Then the collection M ={ M | p ∈ P } is called a Morse decomposition of X if the following conditions are satisﬁed: (i) M is a family of mutually disjoint, isolated invariant subsets of X. (ii) For every essential solution ϕ in X either im ϕ ⊂ M for an r ∈ P or there exist p, q ∈ P such that q > p and α(ϕ) ⊂ M and ω(ϕ) ⊂ M . q p We refer to the elements of M as Morse sets. Note that in the classical deﬁnition of Morse decomposition the analogue of con- dition (ii) is formulated in terms of trajectories passing through points x ∈ / M.In our setting we have to consider all possible solutions. There are two reasons for that: the non-uniqueness of a solution passing through a point and the tightness of ﬁnite topological spaces. In particular, in the ﬁnite topological space setting it is possible to have a non-trivial Morse decomposition such that every point is contained in a Morse set. Without our modiﬁcation of the deﬁnition of Morse decomposition, recurrent behavior spreading into several sets is a distinct possibility. Figure 11 illustrates such an example. Proposition 7.2 Let X be an invariant set, let A ⊂ X be an attractor, and let A denote its non-empty dual repeller. Furthermore, deﬁne M := A, M := A , and let 1 2 123 Conley-Morse-Forman theory for generalized combinatorial… Fig. 11 A sample combinatorial multivector ﬁeld V = {{A, D, F , G}, {B, C , E , H }} on the ﬁnite topolog- ical space X = {A, B, C , D, E , F , G, H } with Alexandroff topology induced by the partial order indicated by arrows. If we consider M = V, then one obtains a partition into isolated invariant sets with X \ M = ∅. Note that ... D · H · B · F · D · ... is a periodic trajectory which passes through both “Morse sets” P := {1, 2} be an indexing set with the order induced from N. Then M ={M , M } 1 2 is a Morse decomposition of X. Proof By Theorems 6.2 and 6.3 both A and A are isolated invariant sets which are clearly disjoint. Let x ∈ X and let ϕ ∈ eSol (x , X ). By Theorem 6.15 the set ω(ϕ) is strongly connected and invariant. It is also indecomposable by Proposition 6.7.By Proposition 6.9 it is either a subset of A or a subset of Inv(X \ A) = A .The same holds for α(ϕ). We therefore have four cases. The situation α(ϕ) ⊂ M and ω(ϕ) ⊂ M is consis- 2 1 tent with the deﬁnition. The case α(ϕ) ⊂ M and ω(ϕ) ⊂ M is clearly in conﬂict 1 2 with the deﬁnition of an attractor and a repeller. Now suppose that we have α(ϕ) ⊂ M and ω(ϕ) ⊂ M . It follows that there exists a t ∈ Z such that ϕ((−∞, t ])) ⊂ A. Since A is an attractor we therefore have ϕ(t + 1) ∈ (ϕ(t )) ⊂ A, and induction easily implies im ϕ ⊂ A = M . The same argument holds for M . 1 2 7.2 Strongly connected components as Morse decomposition We recall that G stands for the digraph interpretation of the multivalued map V V associated with the multivector ﬁeld V on X. Theorem 7.3 Assume X is invariant. Consider the family M of all strongly connected components M of G with eSol(M ) = ∅. Then M is a minimal Morse decomposition of X. Proof For convenience, assume that M ={M | i ∈ P} is bijectively indexed by a ﬁnite set P. Any two strongly connected components M , M ∈ M are clearly disjoint i j and by Theorem 4.16 they are isolated invariant sets. Hence, condition (i) of a Morse decomposition is satisﬁed. Deﬁne a relation ≤ on the indexing set P by i ≤ j ⇔∃ ϕ ∈ M and ϕ ∈ M . ϕ∈Path (X ) j i 123 M. Lipinski ´ It is clear that ≤ is reﬂexive. To see that it is transitive consider M , M , M ∈ M i j k such that k ≤ j ≤ i. It follows that there exist paths ϕ and ψ such that ϕ ∈ M , ϕ ,ψ ∈ M and ψ ∈ M . Since M is strongly connected we can ﬁnd ρ ∈ j k j Path (ϕ ,ψ , X ). The path ϕ · ρ · ψ clearly connects M with M proving that V i k k ≤ i. In order to show that ≤ is antisymmetric consider sets M , M with i ≤ j and j ≤ i. i j It follows that there exist paths ϕ and ψ such that ϕ ,ψ ∈ M and ϕ ,ψ ∈ M . i j Since the sets M , M are strongly connected we can ﬁnd paths ρ and ρ from ϕ to i j ψ and from ψ to ϕ respectively. Clearly, ϕ ∈ Path (ϕ ,ϕ , X ) and ρ · ψ · ρ ∈ (ϕ ,ϕ , X ). This proves that M and M are the same strongly connected component. i j Let x ∈ X and let ϕ ∈ eSol(x , X ). We will prove that α(ϕ) ⊂ M and ω(ϕ) ⊂ M q p −1 + for some M , M ∈ M. Note that ϕ (V ) is right-inﬁnite for any V ∈ V (ϕ). p q It follows that for any V , W ∈ V (ϕ) we can ﬁnd 0 < t < t < t such that 0 1 2 ϕ(t ), ϕ(t ) ∈ V and ϕ(t ) ∈ W . Thus, points in V (ϕ) are in the same strongly 0 2 1 connected component and therefore ω (ϕ) ⊂ C for some strongly path connected component of G . Moreover, ... · ϕ| · ϕ| · ... is clearly an essential solution V [t ,t ] [t ,t ] 0 2 0 2 in C. Thus C = M ∈ M for some p ∈ P. By Proposition 4.15 the set M is V- p p compatible and locally closed. Hence, M is a superset of the V-hull of V (ϕ) and from Proposition 6.13 we get ω(ϕ) ⊂ M . A similar argument gives α(ϕ) ⊂ M for p q some q ∈ P. It is clear from the deﬁnition of ≤ that p ≤ q. Next, we show that α(ϕ) ⊂ M ∈ M and ω(ϕ) ⊂ M implies im ϕ ⊂ M. Thus, take a y ∈ im ϕ. Then y = ϕ(t ) for some t ∈ Z. Since α(ϕ) and ω(ϕ) are subsets of 1 1 M, we can ﬁnd t < t and t > t , such that x := ϕ(t ) ∈ M and z := ϕ(t ) ∈ M. 0 1 2 1 0 2 Since M is strongly connected there exists a path ρ from z to x. Then ϕ| · ρ ∈ [t ,t ] 1 2 Path (y, x , X ). Since ϕ| ∈ Path (x , y, X ), we conclude that y belongs to the V [t ,t ] V 0 1 strongly connected component of x, that is, y ∈ M. This completes the proof that M is a Morse decomposition. To show that M is a minimal Morse decomposition assume the contrary. Then we can ﬁnd a Morse decomposition M of an M ∈ M with at least two different Morse sets M and M in M . Since M and M are disjoint and V-compatible we can ﬁnd 1 2 1 2 disjoint multivectors V ⊂ M and V ⊂ M . Since the set M is strongly connected we 1 1 2 2 can ﬁnd paths ϕ ∈ Path (x , y, M ) and ρ ∈ Path (y, x , M ) with x ∈ V and y ∈ V . 1 2 V V The alternating concatenation of these paths ψ := ... · ϕ · ρ · ϕ ·ρ. . . is a well-deﬁned essential solution. Then ∅ = im ψ ⊂ α(ψ) ∩ ω(ψ) which implies im ψ ⊂ M for an M ∈ M . However, im ψ ∩ M = ∅ = im ψ ∩ M , a contradiction. 3 1 2 7.3 Morse sets For a subset I ⊂ P we deﬁne the Morse set of I by M (I ) := C (M , M ). i j i , j ∈I Theorem 7.4 The set M (I ) is an isolated invariant set. 123 Conley-Morse-Forman theory for generalized combinatorial… Proof Observe that M (I ) is invariant, because, by Proposition 6.17, every connection set is invariant, and by Proposition 4.7 the union of invariant sets is invariant. We will prove that M (I ) is locally closed. To see that, suppose the contrary. Then, by Proposition 3.10, we can choose a, c ∈ M (I ) and a point b ∈ / M (I ) such that c ≤ b ≤ a. There exist essential solutions ϕ ∈ eSol(a, X ) and ϕ ∈ eSol(c, X ) such T a c − + that α(ϕ ) ⊂ M and ω(ϕ ) ⊂ M for some p, q ∈ I.Itfollows that ψ := ϕ · b · ϕ a q c p a c is a well-deﬁned essential solution such that α(ψ) ⊂ M and ω(ψ) ⊂ M . Hence, q p b ∈ C (M , M ) ⊂ M (I ) which proves that M (I ) is locally closed. Moreover, M (I ) q p is V-compatible as a union of V-compatible sets. Thus, the conclusion follows from Proposition 4.13. Theorem 7.5 If I is a down set in P, then M (I ) is an attractor in X. Proof We will show that M (I ) is closed. For this, let x ∈ cl(M (I )). By Proposition 3.7 we can choose a y ∈ M (I ) such that x ∈ cl y. Consider essential solutions ϕ ∈ eSol(x , X ) and ϕ ∈ eSol(y, M (I )) with α(ϕ ) ⊂ M for some i ∈ I.The x y y i − + concatenated solution ϕ := ϕ · ϕ is well-deﬁned and satisﬁes α(ϕ) ⊂ M and y x ω(ϕ) ⊂ M for some j ∈ P. Deﬁnition 7.1 implies that i > j. Since I is a down set, we get j ∈ I . It follows that x ∈ C (M , M ) ⊂ M (I ). Thus, cl M (I ) ⊂ M (I ), i j which proves that M (I ) is closed. Finally, Theorem 6.2 implies that the set M (I ) is an attractor. ≤ < Theorem 7.6 If I ⊂ P is convex, then (M (I ), M (I )) is an index pair for the isolated invariant set M (I ). ≤ < Proof By Proposition 3.1 the sets I and I are down sets. Thus, by Theorem 7.5 ≤ < ≤ ≤ both M (I ) and M (I ) are attractors. It follows that (M (I )) ⊂ M (I ) and < < (M (I )) ⊂ M (I ). Therefore, conditions (IP1) and (IP2) of an index pair are satisﬁed. ≤ < Let A := M (I ) \ M (I ).The set A is V-compatible as a difference of V- ≤ < compatible sets. By Proposition 3.3 it is also locally closed, because M (I ) and M (I ) are closed as attractors (see Theorems 6.2 and 6.3). We claim that M (I ) ⊂ A.Tosee this, assume the contrary and select an x ∈ M (I ) \ A. By the deﬁnition of M (I ) we can ﬁnd an essential solution ϕ through x such that ω(ϕ) ⊂ M for some p ∈ I . Since ≤ < < M (I ) ⊂ M (I ) and x ∈ / A we get x ∈ M (I ).But M (I ) is an attractor. Therefore ω(ϕ) ⊂ M (I ), which in turn implies p ∈ / I , a contradiction. ≤ < To prove the opposite inclusion take an x ∈ Inv(M (I ) \ M (I )). Then we can ≤ < ﬁnd an essential solution ϕ ∈ eSol(x , M (I ) \ M (I )), and clearly one has im ϕ ⊂ ≤ < M (I ) \ M (I ). In particular, − < + < uim ϕ ∩ M (I ) = ∅ and uim ϕ ∩ M (I ) = ∅. (27) ≤ ≤ We also have ϕ ∈ eSol(x , M (I )), which means that there exist p, q ∈ I such that p ≥ q, α(ϕ) ⊂ M , ω(ϕ) ⊂ M . We cannot have p ∈ I , because then we get ∅ = p q − < ≤ < uim ϕ ⊂ α(ϕ) ⊂ M ⊂ M (I ) which contradicts (27). Therefore, p ∈ I \I = I . By an analogous argument we get q ∈ I . It follows that x ∈ C (M , M ) ⊂ M (I ). p q 123 M. Lipinski ´ ≤ < Since for a down set I ⊂ P we have I = I , I = ∅, as an immediate consequence of Theorem 7.6 we get the following corollary. ≤ < Corollary 7.7 If I is a down set in P, then I = I, I = ∅, (M (I ), ∅) is an index pair for M (I ). Theorem 7.8 Assume X is invariant, A ⊂ X is an attractor and A is its dual repeller. Then we have p (t ) + p (t ) = p (t ) + (1 + t )q(t ) (28) A A X for a polynomial q(t ) with non-negative coefﬁcients. Moreover, if q = 0, then C (A , A) = ∅. Proof Let P := {1, 2} with order induced from N, M := A and M := A . Then 1 2 M := {M , M } is a Morse decomposition of X by Proposition 7.2.For I := {2} one 1 2 ≤ < ≤ obtains I ={1, 2} and I ={1}. Yet, this immediately implies both M (I ) = X and M (I ) = M ({1}) = A. We have ≤ < p (t ) = p (t ) and p (t ) = p (t ). (29) X A M (I ) M (I ) ≤ < By Theorem 7.6 the pair (M (I ), M (I )) is an index pair for M (I ) = A . Thus, by ≤ < substituting P := M (I ), P := M (I ), S := A into (14) in Corollary 5.17 we get 1 2 (28)from(29). By Proposition 6.18 we have the identity C (A, A ) = ∅. Therefore, if in addition C (A , A) = ∅, then X decomposes into A and A , and Theorem 5.19 implies H (P ) = Con(X ) = Con(A) ⊕ Con(A ) = H (P ) ⊕ H (A ), 1 2 as well as q = 0 in view of Proposition 5.17. This ﬁnally shows that q = 0 implies C (A , A) = ∅. 7.4 Morse equation and Morse inequalities The following two theorems follow from the results of the preceding section by adapt- ing the proofs of the corresponding results in Mrozek (2017). Theorem 7.9 Assume X is invariant and P ={1, 2, ..., n} is ordered by the linear order of the natural numbers. Let M := {M | p ∈ P} be a Morse decomposition of X and set A := M ({i } ),A := ∅. Then (A , M ) is an attractor-repeller pair in i 0 i −1 i A . Moreover, n n p (t ) = p (t ) + (1 + t ) q (t ) M X i i =1 i =1 for some polynomials q (t ) with non-negative coefﬁcients and such that q (t ) = 0 i i implies C (M , A ) = ∅ for i = 2, 3, ..., n. i i −1 123 Conley-Morse-Forman theory for generalized combinatorial… As before, for a locally closed set A ⊂ X we deﬁne its kth Betti number by β (A) := rank H (cl A, mo A). k k Theorem 7.10 Assume X is invariant. For a Morse decomposition M of X deﬁne m (M) := β (M ). k k r r ∈P Then for any k ∈ Z we have the following inequalities. (i) The strong Morse inequalities: m (M) − m (M) + ... ± m (M) ≥ β (X ) − β (X ) + ... ± β (X ), k k−1 0 k k−1 0 (ii) The weak Morse inequalities: m (M) ≥ β (X ). k k Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. 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Syst. 13(4), 1847–1878 (2014) Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional afﬁliations.
Journal of Applied and Computational Topology – Springer Journals
Published: Jun 1, 2023
Keywords: Combinatorial vector field; Finite topological space; Discrete Morse theory; Isolated invariant set; Conley theory; Primary 37B30; Secondary 37E15; 57M99; 57Q05; 57Q15
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