Continuation Sheaves in Dynamics: Sheaf Cohomology and Bifurcation
aa r X i v : . [ m a t h . D S ] F e b PreprintFebruary 4, 2021 pp. 1-60 , Appendix: 48-60.
CONTINUATION SHEAVES IN DYNAMICS:SHEAF COHOMOLOGY AND BIFURCATION
K. A
LEX D OWLINGRutgers University110 Frelinghusen RoadPiscataway, NJ 08854, USA W ILLIAM
D. K
ALIESFlorida Atlantic University777 Glades RoadBoca Raton, FL 33431, USA R OBERT
C.A.M. V
ANDERVORSTVU UniversityDe Boelelaan 1081a1081 HV, Amsterdam, The NetherlandsA
BSTRACT . Continuation of algebraic structures in families of dynamical systems is de-scribed using category theory, sheaves, and lattice algebras. Well-known concepts in dy-namics, such as attractors or invariant sets, are formulated as functors on appropriate cate-gories of dynamical systems mapping to categories of lattices, posets, rings or abelian groups.Sheaves are constructed from such functors, which encode data about the continuation ofstructure as system parameters vary. Similarly, morphisms for the sheaves in question arisefrom natural transformations. This framework is applied to a variety of lattice algebras andring structures associated to dynamical systems, whose algebraic properties carry over totheir respective sheaves. Furthermore, the cohomology of these sheaves are algebraic invari-ants which contain information about bifurcations of the parametrized systems. Introduction.
The global dynamics of a system can be describes by the structure of itsattractors. Indeed, in his fundamental decomposition theorem, Conley [5] uses the set ofall attractors in a system to establish a global decomposition into minimal (chain)-recurrentcomponents and connecting orbits between them [32]. The algebraic structure that under-lies this decomposition is codified in the fact that the set of all attractors naturally forms a
Mathematics Subject Classification.
Primary: 37Bxx, 55N30; Secondary: 06Fxx.
Key words and phrases.
Continuation, attractor sheaves, distributive lattice, sheaf cohomology, parametrizeddynamical systems. bounded, distributive lattice [18]. In a series of papers [17, 18, 19, 20, 16, 13] the theoreti-cal framework for such dynamically meaningful algebraic structures has been developed.This framework has been used to design algorithms to compute global dynamical informa-tion rigorously for explicit maps, cf. [1]. In applications, systems depend on parameters,and the variation of the dynamics over the parameter space is often of fundamental impor-tance. This paper describes the natural mathematical framework to capture this variationwith respect to parameters.Recall that a set U Ă X is an attracting neighborhood if ω p U, φ q Ă int p U q . A set A Ă X is an attractor if there exists an attracting neighborhood U such that A “ ω p U, φ q . The bounded,distributive lattices of attractors and attracting neighborhoods are denoted by Att p φ q and ANbhd p φ q respectively, cf. Sect. 3 and [18]. With these lattice structures, the ω -limit setoperator ω : ANbhd ! Att is a surjective lattice homomorphism. For a parametrized familyof dynamical systems, structures like the lattice of attractors at a fixed parameter value mayvary dramatically under small perturbations. Indeed, bifurcations can occur on a Cantorset with positive measure in the parameter space. Continuation relates these structures forvarying parameter values.Continuation for some algebraic structures in dynamics has been well described. For aflow φ on a compact topological space X , cf. Defn. 2.1, a set S Ă X is an isolated invariantset if there exists a closed neighborhood N of S such that S is the largest invariant set in N .The set of isolated invariant sets for φ is denoted by Isol p φ q . As Montgomery shows in [28],the Conley index of isolated invariant sets is invariant under perturbation. This propertyis extremely valuable for applications of the Conley index. Montgomery uses the conceptof invariant set space to fully represent the continuation data of isolated invariant sets. Theinvariant set space is defined as follows: Π r Isol s : “ ! p φ, S q ˇˇ φ a flow on X and S an isolated invariant set for φ ) . By choosing an appropriate topology on Π r Isol s the map p φ, S q π φ, where the flow φ is an element of a topological space of flows on X , is a local homeo-morphism. The pair ` Π r Isol s , π ˘ is called an ´etal´e space . Such a space is in general a non-Hausdorff topological space. Montgomery uses this concept to define continuation of iso-lated invariant sets : two pairs p S, φ q and p S , φ q are related by continuation if they are con-tained in the same quasi-component of Π r Isol s . For example in a 1-parameter family offlows this implies that p S, φ q and p S , φ q are connected by a continuous path in Π r Isol s .Two pairs p S, φ q and p S , φ q that are related by continuation have isomorphic Conley in-dices, cf. [5], [33]. The concept of ´etal´e spaces was later on used by Kurland and Franzosa ONTINUATION SHEAVES IN DYNAMICS 3 to describe continuation of attractor-repeller pairs and Morse representations respectively,cf. [22] and [12].The set of isolated invariant sets can be given the structure of a meet-semilattice, i.e.the intersection of two isolated invariant sets is again an isolated invariant set. This alge-braic structure is not exploited in [28]. This paper explores the algebraic structures andfurther investigates the continuation of a variety of algebraic structures in dynamics, witha flexible, general theory. Inspired by how topological invariants are treated in algebraictopology Sections 2 and 3 frame dynamical invariants as functors from a category of dy-namical systems to a category of lattices, posets, rings, abelian groups, etc. The ´etal´e spaceconstruction in [28, 12, 22] is then generalized to this categorical setting. Of importance inthis paper is the utilization of sheaf cohomology, not to be confused with the continuationof the cohomological Conley index [28]. Sheaf cohomology is used here to define algebraicinvariants that are capable of detecting bifurcations.1.1. ´Etal´e spaces and sheaves.
In Sections 2 and 3 we regard the concept of attractors, re-pellers, etc. as (contravariant) functors on suitable categories of dynamical systems, and thehomomorphisms between the various lattices as natural transformations, cf. Fig. 1[middle].In Section 4 we use the functorial formulation ω : ANbhd ñ Att as the building block forconstructing a canonical ´etal´e space by essentially locally fixing an attracting neighbor-hood U and considering the pairs p φ, A q such that A “ ω φ p U q to comprise the ´etal´e space.This yields the projection p φ, A q π φ and the ´etal´e space Π r Att s . Furthermore, naturaltransformations of dynamical invariants are shown to correspond to continuous maps of´etal´e spaces in Fig. 1[right]. Summarizing the algebraic framework in [17, 18, 19, 20] imme-diately manifests ´etal´e spaces for the various dynamical entities, and thus in continuation,cf. Sections 4. In Section 6 we show that the binary operations on attractors and attractingneighborhoods yields continuous functions which induces an additional algebraic struc-ture for the associated sheaves. ANbhd p φ q RNbhd p φ q Att p φ q Rep p φ q ω φ c α φ ˚ ANbhd RNbhdAtt Rep ω c α ˚ Π r ANbhd s Π r RNbhd s Π r Att s Π r Rep s Π r ω s Π r c s Π r α s Π r ˚ s F IGURE
1. The diagram from [18] [left] appears categorically in the seconddiagram [middle], and then in the associated ´etal´e spaces [right].While the ´etal´e space constructions provide a full picture of continuation, they are obtuseboth topologically and algebraically, even for simple parametrized systems. For instance,these spaces are mostly not Hausdorff. This problem is encountered in algebraic topol-ogy and algebraic geometry. The well-established remedy is a fundamental observation in
K. DOWLING, W.D. KALIES AND R.C.A.M. VANDERVORST sheaf theory: ´etal´e spaces and sheaves are categorically equivalent . However, sheaves enjoy amore accessible and comprehensive theory [4, 14, 6].In an effort to utilize ideas from sheaf theory, we consider the sheaves of sections as-sociated to the ´etal´e spaces, cf. Section 7 and [39]. In the case of attractors we obtain the attractor lattice sheaf : S Att : O ! BDLat sheaf of sections , where O is the posetal category of open sets of dynamical systems. A continuous section in Π r Att s over an open set Ω P O is a continuous map σ : Ω ! Π r Att s such that π ˝ σ “ id . The set of all continuous sections over Ω is denoted by S Att p Ω q and comprises the attractorlattice sheaf. In Section 6 we study algebraic constructions to obtain abelian sheaves. Thecategorical equivalence between sheaves of sections and ´etal´e spaces guarantees that weretain the results in Sections 4 and 5. Via Booleanization functor and monoid ring functorwe obtain the abelian sheaves: A : “ ` R S Att ˘ : O ! Ring , and Att : “ ` Z S Att ˘ : O ! Ring , where denotes the sheafification functor. We refer to these sheaves as the attractor sheaf and the free attractor sheaf respectively. For the abelian sheaves A and Att sheaf cohomol-ogy is defined, cf. Appendix D.1.2.
Parametrized systems, bifurcations, and sheaf cohomology.
For a topological space Λ we define a parametrized dynamical system on X as a continuous map φ : T ˆ X ˆ Λ ! X such that φ λ : “ φ p¨ , ¨ , λ q is dynamical system on X for all λ P Λ . The map λ φ ˚˚˚ φ λ ,called the transpose , is a continuous map without additional assumptions on the topologicalspaces Λ and X , cf. Appendix B. This puts a sheaf on the parameter space Λ whose sectionsprecisely detail the continuation of their germs. In the case of attractors we obtain theattractor lattice sheaf over Λ : π : φ ´ ˚˚˚ Π r Att s ! Λ ´etal´e space ùùùùùùñ φ ´ ˚˚˚ S Att : O p Λ q ! BDLat sheaf of sections , where O p Λ q is the posetal category of open sets in Λ . The pull-backs for the attractor sheafand the free attractor sheaf are denoted by A φ ˚˚˚ : “ p R φ ´ ˚˚˚ S Att q , and Att φ ˚˚˚ : “ p Z φ ´ ˚˚˚ S Att q . After fully adapting the continuation theory to parametrized systems in Sections 8.1 and8.2, we will use sheaf cohomology to study bifurcations. Sheaf cohomology is a powerful
ONTINUATION SHEAVES IN DYNAMICS 5 algebraic invariant from sheaf theory. The continuation of some structure, say an attractor,can always been solved locally using an attracting neighborhood, but this problem is some-times impossible globally . Sheaf cohomology quantifies when and how this obstructionoccurs. In Section 8.2 we show that the continuation of attractors is conjugacy invariant.
Theorem (Conjugacy Invariance Theorem, cf. Thm. 8.7) . Let X , Y be compact metric spaces.Suppose φ ˚˚˚ : Λ ! DS p T , X q and ψ ˚˚˚ : Λ ! DS p T , Y q are conjugate parametrized dynamicalsystems. Then, the ´etal´e spaces φ ´ ˚˚˚ Π r Att s and ψ ´ ˚˚˚ Π r Att s are homeomorphic. Conjugate systems have isomorphic attractor lattice sheaves and thus isomorphic at-tractor sheaves and free attractor sheaves. This implies isomorphic cohomologies. Hence,the (free) attractor sheaf’ cohomology is an invariant to study parametrized dynamicalsystems. It can detect bifurcations.
Theorem (cf. Thm. 9.10) . Let Λ be a contractible manifold, X a compact metric space, and φ ˚˚˚ : Λ ! DS p T , X q a parametrized dynamical system on X . Suppose that H k p Λ; A φ ˚˚˚ q ‰ , for some k ą . Then, there exist a bifurcation point in λ P Λ . In some cases we can use sheaf cohomology as a relative invariant for detecting bifurca-tions which may not be noticed by regular sheaf cohomology. A parametrized dynamicalsystem φ ˚˚˚ : Λ ! DS p T , X q is stable at a point λ P Λ if there exists an open neighborhoodof λ such that the restriction of φ ˚˚˚ is conjugate to the constant parametrization. Theorem (cf. Thm. 9.13) . Let Λ be a contractible manifold and Λ Ă Λ be a submanifold which isa deformation retract of Λ with φ ˚˚˚ stable on Λ . Suppose that H k p Λ , Λ ; A φ ˚˚˚ q ‰ , for some k ě . Then, there exist a bifurcation point in λ P Λ r Λ . Section 10 explores examples of these cohomology groups for different types of bifurca-tions. We describe a number of elementary bifurcation in 1-dimensional systems.
Theorem (cf. Thm. 10.5) . Let φ ˚˚˚ be a parametrized dynamical system over R with a pitchforkbifurcation at λ . Then, A φ ˚˚˚ is acyclic and H p Λ; A φ ˚˚˚ q – Z . Moreover, there exists a value λ P R such that H k ` Λ , Λ ; A φ ˚˚˚ ˘ – $&% Z if k “ and a ą λ ;0 if k “ or a ď λ , K. DOWLING, W.D. KALIES AND R.C.A.M. VANDERVORST where Λ “ r a, , Furthermore, for Λ : “ p´8 , a s , then H k ` Λ , Λ ; A φ ˚˚˚ ˘ – for all k and for all a P R . Different types of bifurcations can have different cohomology in their attractor sheaves,but if two systems experience the same type of bifurcation, the cohomology is isomor-phic. We propose this as a tool for classifying bifurcations, in much the same way singularhomology classifies topological spaces. We believe this invariant to be computable by uti-lizing the existing theory for combinatorial dynamics [19, 1] and cellular sheaf cohomology[7].1.3.
Outline of the Paper.
In Section 2 we review the topological and categorical structuresassociated to dynamical systems. Section 3 demonstrates how existing dynamical invari-ants fit into the functorial language. Then, Section 4 builds the ´etal´e space constructionfrom continuation frames, and investigates induced maps from natural transformations.Section 6 utilizes Section 4 for algebraic constructions in dynamics. Section 7 translates tosheaves of sections, and Section 8 constructs abelian sheaves, discusses the implications ofparametrized systems, and establishes conjugacy invariance. Section 9 begins explorationof sheaf cohomology’s connection to bifurcations, and Section 10 computes examples ofthis cohomology for several one-parameter bifurcations.2.
Categories of dynamical systems.
We give spaces of dynamical systems a categoricalstructure as well as a topology. The following definition of dynamical systems is used:
Definition 2.1.
Let p X, T q be a topological space and let T be the (additive) topologicalmonoid (group) with topology T T . A dynamical system is a continuous map φ : T ˆ X ! X that satisfies(i) φ p , x q “ x for all x P X ;(ii) φ p t, φ p s, x qq “ φ p t ` s, x q for all s, t P T and all x P X .The set of all dynamical systems on the phase space X with time space T is denoted by DS p T , X q .In this paper T is either Z , Z ` , R , or R ` . In case of Z ` and R ` we extend φ for negativetime by φ ´ t “ φ ´ t , the inverse image. If the topology T T on T “ R , or R ` is the discretetopology, then φ is a one-parameter family of continuous maps φ t : “ φ p t, ¨q on X satisfyingthe (semi)group property in (ii), and the continuity in time does not place any restrictionon the system. In applications, for example those arising from differential equations, it iscommon for the topology on T “ R to be the standard topology, but certain results in thispaper do not require the topology T T to have specific properties. Therefore we typicallyuse the notation φ t p x q . However, certain properties of the phase space topology T do playa crucial role. In particular, for clarity of the presentation of the main ideas of this paper, we ONTINUATION SHEAVES IN DYNAMICS 7 always consider the phase space X to be compact . For some results, such as Theorem 8.7,we additionally assume a metric on X . Such restrictions are explicitly stated and explainedwhen needed.First, we endow DS p T , X q with a suitable topology. One natural choice arises by view-ing DS p T , X q as a function space with the compact-open topology, i.e. the topology gener-ated by the subbasis of sets of the form φ | φ p K q Ă U for K compact in T ˆ X and U open in X ( , varying of pairs p K, U q , cf. Appendix B.Next, we endow DS p T , X q with a categorical structure and as to refer to DS p T , X q asthe category of dynamical systems on X over T . An object φ P ob p DS p T , X qq is a dynamicalsystem φ : T ˆ X ! X . A morphism τ ˆ f P hom p φ, ψ q is defined by(i) a continuous map f : X ! X ;(ii) a continuous reparametrization τ : T ˆ X ! T which is strictly monotone and bijec-tive for each x and satisfies τ p , x q “ , such that the following diagram commutes: T ˆ X X T ˆ X X τ ˆ f φ fψ cf. [9]. We refer to such a morphism τ ˆ f as a (topological) quasiconjugacy. Note that hom p φ, ψ q can also be endowed with the compact-open topology, so that both the objectsand the hom-set of DS p T , X q are topological spaces. We abuse notation so that an opensubset Ω Ă ob p DS p T , X qq is referred to as an open set Ω in DS p T , X q . Remark 2.2.
Note that the conditions on reparametrizations imply that τ “ id in the casethat T “ Z , cf. [8, II(7.2)]. This in part motivates the terminology of quasiconjugacy. When T “ R , the case f “ id yields a reparametrization τ p t, x q of time. Remark 2.3.
For special subsets of dynamical systems, such as smooth flows on a man-ifold, topologies other than the compact-open topology may be more appropriate.
Forclarity of presentation, we use the notation DS p T , X q to mean that the objects and mor-phisms of this category are given the compact-open topology. However, in other cases,similar results to those obtained for DS p T , X q follow from the abstract theory presented inSection 4. Remark 2.4.
A more restrictive choice of the set of morphisms leads to a subcategory.For example, one could consider from least restrictive to most restrictive: topological(semi)equivalence with reparametrization of time, topological (semi)conjugacy, or no struc-ture on the morphism set, ie. hom p φ, φ q “ t id ˆ id u and hom p φ, ψ q “ ∅ when φ ‰ ψ ,cf. [9, 32]. K. DOWLING, W.D. KALIES AND R.C.A.M. VANDERVORST
Considering dynamical systems without specifying the time space T or the topologi-cal space X yields the category DS of all dynamical systems. In this case we extend themorphisms to be maps τ ˆ f P hom p φ, ψ q consisting of continuous maps f : X ! Y andcontinuous time reparametrizations τ : T ˆ X ! S (strictly monotone, bijective for each x and τ p , x q “ ) such that the following diagram commutes: T ˆ X X S ˆ Y Y τ ˆ f φ fψ Morphisms will again be referred to as quasiconjugacies. Morphisms with f surjectiveare called semiconjugacies , and when f : X ! Y is a homeomorphism, they are called con-jugacies . If the time space is fixed, the associated subcategory is denoted by DS p T q . Wedo typically not equip DS or DS p T q with a topological structure on the objects and mor-phisms.3. Functoriality of dynamics.
The study of a dynamical system often focuses on the prop-erties of the invariant sets of a system. A subset S Ă X is invariant if it is the union ofcomplete orbits, or equivalently φ t p S q “ S for all t P T . One of the most important classesof invariant sets are the attractors of a system. In [18], it is shown that the attractors havethe algebraic structure of a bounded, distributive lattice. In the next section, we character-ize such algebraic structures in terms of functors on the category of dynamical systems.For a given dynamical system φ : T ˆ X ! X the omega limit set of U Ă X is definedby ω φ p U q : “ č t ě cl ď s ě t φ s p U q . Recall from [18] some properties of ω φ p U q .(i) ω φ p U q is compact, closed, and nonempty whenever U “ ∅ ,(ii) ω φ p U q is an invariant set for the dynamics,(iii) ω φ ` ω φ p U q ˘ “ ω φ p U q ,(iv) ω φ p cl U q “ ω φ p U q ,(v) ω φ p U Y V q “ ω φ p U q Y ω φ p V q .A subset U Ă X is called an attracting neighborhood if ω φ p U q Ă int U . Attracting neighbor-hoods form a bounded, distributive lattice denoted by ANbhd p φ q . The binary operationsare X and Y , see [18]. A subset A Ă X is called an attractor if there exists an attractingneighborhood U Ă X such that A “ ω φ p U q , which is a neighborhood of A by definition.Attractors are compact, closed invariant sets, and the set of all attractors is a bounded, dis-tributive lattice Att p φ q with binary operations: A _ A “ A Y A and A ^ A : “ ω φ p A X A q ,cf. [18]. ONTINUATION SHEAVES IN DYNAMICS 9
Remark 3.1.
In the above listed properties of omega limit sets and attractors, the compact-ness of X is crucial. If we drop the compactness assumption on X , some of the properties,such as invariance and idempotency, do not hold in general.The categorical structure of DS can now be used to reformulate the above lattices interms of functors. For notational convenience we write ψ : t : “ ψ p τ p t, ¨q , ¨q Lemma 3.2.
Let φ, ψ P ob p DS q and let τ ˆ f P hom p φ, ψ q . Then, for all U Ă Y we have φ t p f ´ p U qq Ă f ´ p ψ : t p U qq @ t ě . In particular, ω φ p f ´ p U qq “ ω φ p f ´ p ω ψ p U qqq Ă f ´ p ω ψ p U qq . (1) Proof.
See Appendix A.Now suppose we have τ ˆ f P hom p φ, ψ q and U P ANbhd p ψ q . Then, by Lemma 3.2, ω φ p f ´ p U qq Ă f ´ p ω ψ p U qq Ă f ´ p int p U qq Ă int p f ´ p U qq , where the latter inclusion follows from the continuity of f . Therefore f ´ p U q P ANbhd p φ q , and the inverse image operator induces a well-defined map f ´ : ANbhd p ψ q ! ANbhd p φ q .This map is in fact a homomorphism by the properties of inverse images, since the lat-tice operations on ANbhd p φ q and ANbhd p ψ q are union and intersection, so using functornotaton, ANbhd p τ ˆ f q “ f ´ . Thus, by assigning to each dynamical system its attractingneighborhood lattice and to each morphism its inverse image operator, by the properties ofinverse images and Lemma 3.2, we have a contravariant functor, ANbhd : DS ! BDLat ,from the category of dynamical systems to the category of bounded, distributive lattices.Recall that U P ANbhd p φ q is an attracting block if φ t p cl U q Ă int p U q for all t ą . Nowsuppose τ ˆ f P hom p φ, ψ q and U P ABlock p ψ q . Then for all t ą φ t ` cl p f ´ p U qq ˘ Ă φ t ` f ´ p cl U q ˘ Ă f ´ ` ψ : t p cl U q ˘ Ă f ´ p int p U qq Ă int p f ´ p U qq , which implies that f ´ p U q P ABlock p φ q so that we can restrict f ´ to f ´ : ABlock p ψ q ! ABlock p φ q . As before, ABlock p τ ˆ f q “ f ´ . This makes ABlock : DS ! BDLat a con-travariant functor.
Remark 3.3.
Similar statements as in Lemma 3.2 also hold for α -limit sets, as defined in[17], see Appendix A. Therefore we can define functors RNbhd , RBlock : DS ! BDLat forrepelling neighborhoods and repelling blocks analogously.A similar construction can be used to define a contravariant functor
Att : DS ! BDLat ,but its action on morphisms must be modified, since the inverse image of an attractor neednot be an attractor.
Proposition 3.4.
Suppose τ ˆ f P hom p φ, ψ q and A P Att p ψ q . Then ω φ p f ´ p A qq P Att p φ q .Moreover, for τ ˆ f P hom p φ, ψ q , the map ω φ ˝ f ´ : Att p ψ q ! Att p φ q is a lattice homomorphism.Proof. See Appendix A.Thus, by assigning each dynamical system its attractor lattice and each morphism τ ˆ f P hom p φ, ψ q the operator Att p τ ˆ f q “ ω φ ˝ f ´ , we have a contravariant functor Att : DS ! BDLat . Remark 3.5. If τ ˆ f P hom p φ, ψ q is a conjugacy, i.e. f : X ! Y is a homeomorphism, thenalso τ ´ ˆ f ´ P hom p ψ, φ q is a conjugacy, where τ ´ p s, y q is defined by s “ τ p t, f ´ p y qq .As a consequence, A P Att p φ q if and only if f p A q P Att p ψ q . Remark 3.6.
The same process yields
Rep : DS ! BDLat by replacing ω with α . Note theproofs are simpler with Rep , since the lattice operations are simply union and intersection.Summarizing, for an attracting neighborhood U P ANbhd p φ q the application U ω φ p U q defines a surjective lattice homomorphism and by the functoriality of ANbhd and
Att wehave
ANbhd p φ q ANbhd p ψ q Att p φ q Att p ψ q (cid:15) (cid:15) (cid:15) (cid:15) ω φ (cid:15) (cid:15) (cid:15) (cid:15) ω ψ o o ANbhd p τ ˆ f q o o Att p τ ˆ f q These properties make ω φ the φ -component of a natural transformation ω : ANbhd ñ Att .A second commutative diagram can be constructed by considering Boolean algebras ofsubsets of X and Y . For a dynamical system φ the assignment φ Set p X q , the Booleanalgebra of subsets of X , is a contravariant functor from DS to BDLat which we denoteby P . Indeed, for a morphism τ ˆ f P hom p φ, ψ q , P p τ ˆ f q : “ f ´ : Set p Y q ! Set p X q isa Boolean homomorphism. The map ι φ : ANbhd p φ q Set p X q , given by set-inclusion, is alattice homomorphism which displays ι φ as the φ -component of the natural transformation ι between the functors ANbhd and P . In terms of commutative diagrams we have: ANbhd p φ q ANbhd p ψ q Set p X q Set p Y q (cid:15) (cid:15) (cid:15) (cid:15) ι φ (cid:15) (cid:15) (cid:15) (cid:15) ι ψ o o ANbhd p τ ˆ f q o o P p τ ˆ f q ONTINUATION SHEAVES IN DYNAMICS 11
The construction of the functors
Att , ANbhd , and P is best summarized in the followingdiagram of categories, functors, and natural transformations: DS BDLat
ANbhdP ι DS BDLat
ANbhdAtt ω (2) Remark 3.7.
In some situations it is useful to consider attracting neighborhoods in thealgebra of regular closed sets using analogous constructions, cf. [19].The objective in this paper is to regard the continuation of dynamical features in terms ofsheaves over DS p T , X q . Formulating continuation in terms of ´etal´e spaces was carried outby Montgomery [28] for isolated invariant sets. There are a variety of dynamical featuresof interest: attractors and repellers as well as their neighborhoods, and also Morse sets andMorse representations, to name a few. The categories of algebraic structures correspondingto these features are also varies. To keep the underlying theory flexible, and so as notto repeat theoretical arguments, we first introduce the underlying concepts and theoremsabstractly, and then apply this general theory in specific contexts.4. Abstract continuation.
Recall from the introduction that a fundamental feature of Con-ley theory is that an isolated invariant set continues under perturbation of a dynamicalsystem. In this section, we provide an abstract framework to expand continuation to alge-braic structures of certain isolated invariant sets such as attractors and repellers.4.1. C -structures and categories of elements. Let D be a category such that ob p D q forma topological space and let C be a concrete category, i.e. there exists a faithful functor, F : C ! Set , into the category of sets. In applications D is a category of dynamical systemsequipped with a topology on the ob p D q , such as DS p T , X q , and C is the category char-acterizing the algebraic structure of the dynamical feature to be continued, for examplebounded, distributive lattices.A C -valued contravariant functor on D is referred to as a C -structure on D . Let E , G : D ! C be C -structures and let w : E ñ G be a natural transformation. For objects φ P ob p D q the functors E and G yield objects E p φ q and G p φ q in C and the component w φ ofthe natural transformation yields a morphism w φ : E p φ q ! G p φ q .In applications, typically the functor G represents a dynamical feature such as attrac-tors, and the functor E denotes a corresponding neighborhood feature such as attractingneighborhoods or attracting blocks. Furthermore, we assume the existence of a locally constant, contravariant functor P : D ! C , referred to as the universe functor , which defines an injective natural trans-formation ι : E ñ P . In dynamics applications when D “ DS p T , X q , the universe functorassigns to each φ a fixed subalgebra of the Boolean algebra Set p X q , the power set of thephase space, or a fixed subalgebra of the Boolean algebra R p X q , the regular closed subsetsof X . For example, if E “ ANbhd , the lattice of attracting neighborhoods, E p φ q , is a sublat-tice of Set p X q . For larger subcategories the functor P is locally constant. For the remainderof this section we assume P p φ q “ P to be constant.Now we have a span of functors and natural transformations P ι ðù E w ùñ G which aresummarized in the following diagrams (compare to (2)): D C EP ι D C
EGw (3)Since C is a concrete category, we may regard a functor E into C as a Set -valued functor,and thus consider Π r E s , its category of elements , cf. [24, 25]. The category of elements con-struction is used in the next section to generate an ´etal´e space. To define the category of ele-ments Π r E s , let ob p Π r E sq be the set of all pairs p φ, U q such that φ P ob p D q and U P E p φ q . Themorphisms of Π r E s are maps p φ, U q ! p φ , U q for which there is a D -morphism h : φ ! φ with E p h qp U q “ U . The projection p φ, U q φ defines a canonical projection functor π : Π r E s ! D . Moreover, given a natural transformation between functors, w : E ñ G , we have the functorbetween the associated categories of elements Π r w s : Π r E s ! Π r G sp φ, U q p φ, w φ p U qq . From the span of functors P ι ðù E w ùñ G we obtain a span of functors on the associatedcategories of elements: Π r P s Π r ι s −−− Π r E s Π r w s −−−! Π r G sp φ, U q −− ß p φ, U q p φ, w φ p U qq . (4)Note that in (4), the set U P E p φ q . To localize Π r E s , for a fixed element U P P we define thesubcategory Π r E ; U s via ob ` Π r E ; U s ˘ : “ ! p φ, U q P ob p Π r E sq | U P E p φ q ) with morphisms p φ, U q ! p φ , U q for which there is a D -morphism h : φ ! φ with E p h qp U q “ U . ONTINUATION SHEAVES IN DYNAMICS 13
Applying the projection functor π yields a corresponding subcategory Φ r E ; U s of D .The objects of Φ r E ; U s are given by ob ` Φ r E ; U s ˘ “ φ P ob p D q | U P E p φ q ( with morphisms h : φ ! φ with E p h qp U q “ U . This yields the following commutative diagrams: Π r E s D Π r E ; U s Φ r E ; U s π Ă π Ă Π r G s Φ r E ; U s D (cid:15) (cid:15) ✤✤✤✤✤✤✤✤ π ? ? ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ Θ r w ; U s / / Ă (5)where Θ r w ; U sp φ q : “ ` φ, w φ p U q ˘ is called the partial section functor , which satisfies: Θ r w ; U s ˝ π “ Π r w s , π ˝ Θ r w ; U s “ id . (6) Remark 4.1.
In settings where Φ r E ; U s is a used as a subset of D we abuse notation andwrite φ P Φ r E ; U s . The same applies to open subsets Ω Ă D . Remark 4.2.
The partial section functor is indeed a functor which is useful when topolo-gizing the categories of elements, cf. Appendix A.4.2.
Continuation frames and ´etal´e spaces. A C -structure E : D ! C is called stable if Φ r E ; U s is open in D for all elements U P P . Otherwise a C -structure is said to be unstable . In the remainder of the paper we will always assume that E : D ! C admits a universe P for which it is stable.Definition 4.3. A C -continuation frame on D is a triple p G , E , w q consisting of C -structures E , G : D ! C and a natural transformation w : E ñ G such that(i) w φ : E p φ q ։ G p φ q is surjective for all φ P ob p D q ;(ii) E is a stable C -structure.Condition (i) can be paraphrased by saying that w is componentwise surjective. The C -structure E in a continuation frame is is called a stable extension of G .When the natural transformation is not surjective componentwise, we restrict thecodomain to the image of w to define a continuation frame. In the application of C -structures in dynamics, G is typically unstable as the examples in the next section show.The next step is to topologize Π r G s with the topology generated by the subbasis B p G q : “ ! Θ r w ; U sp Ω q | U P P , Ω Ă Φ r E ; U s open ) , where Θ r w ; U sp Ω q is the image under Θ r w ; U s of objects φ P Ω . This is the finest topologysuch that all maps Θ r w ; U s : Φ r E ; U s ! Π r G s are continuous. The functor π : Π r G s ! D may be regarded as a projection π : Π r G s ։ ob p D q , and with the above defined topology on Π r G s , it is also a continuous map between topological spaces. We now show that p Π r G s , π q is an ´etal´e space in the category Set by establishing that π is a local homeomorphism. Theorem 4.4.
Let p G , E , w q be a C -continuation frame on D . Then, the pair p Π r G s , π q is an´etal´e space on D .Proof. To establish p Π r G s , π q as an ´etal´e space with the above defined topology on Π r G s weshow that π is a local homeomorphism.Let p φ, S q be a point in Π r G s . Then, since w φ : E p φ q ։ G p φ q is surjective for all φ , thereexists U P E p φ q such that w φ p U q “ S . Consequently, the point p φ, S q is contained in theimage of the map Θ r w ; U s : Φ r E ; U s ! Π r G s , which is open by the definition of the topology.The image under π of the set Im ` Θ r w ; U s ˘ is the set Φ r E ; U s which is open by assumption.It remains to show that π : Im ` Θ r w ; U s ˘ ! Φ r E ; U s is a homeomorphism.First we show bijectivity. By definition π : Im ` Θ r w ; U s ˘ ! Φ r E ; U s is onto and since φ p φ, w φ p U qq for φ P Φ r E ; U s is a section, we establish bijectivity.Second we show that π : Im ` Θ r w ; U s ˘ ! Φ r E ; U s is continuous and open. Let Ω Ă Φ r E ; U s be open. Then, π ´ p Ω q “ Θ r w ; U sp Ω q is open by the definition of the topologywhich proves the continuity of π . Let Θ r w ; U i sp Ω i q , with U i P P and Ω i Ă Φ r E ; U i s , be afinite collection of basic open sets. Then, π ˆč i Θ r w ; U i sp Ω i q X Im ` Θ r w ; U s ˘˙ “ č i Ω i X Φ r E ; U s , is open and thus π is a homeomorphism. This proves that Π p G q is an ´etal´e space in Set .In the spirit of [28] two points p φ, S q and p φ , S q are related by continuation if they arecontained in the same quasi-component of Π r G s , or equivalently p φ , S q is contained in thequasi-component of p φ, S q . Recall that a quasi-component of p φ, S q of Π r G s is the intersec-tion of all clopen subsets of Π r G s containing p φ, S q . We will study continuation in terms ofsheaves as we will see in forthcoming sections.The stable C -structure in a C -continuation frame yields a second ´etal´e space. Topologize Π r E s as follows. Define the embedding Θ r id ; U s : Φ r E ; U s ! Π r E s as the trivial section φ p φ, U q and define a subbasis for the topology on Π r E s as follows: B p E q : “ ! Θ r id ; U sp Ω q | U P P , Ω Ă Φ r E ; U s open ) . Corollary 4.5.
Let p G , E , w q be a C -continuation frame on D . Then, the pair p Π r E s , π q is an´etal´e space on D .Proof. The projection π : Π r E s ! ob p D q given by p φ, U q φ is a local homeomorphismwith the above defined topology, i.e. π : Π r E ; U s ! Φ r E ; U s is a homeomorphism.The category of elements Π r P s trivially gives an ´etal´e space and makes the span of func-tors in (4) into a span of ´etal´e spaces. Corollary 4.6.
The map Π r w s : Π r E s ! Π r G s is a morphism of ´etal´e spaces and a local homeomor-phism. ONTINUATION SHEAVES IN DYNAMICS 15
Proof.
We start with proving that Π r w s is continuous. By definition of the topology on Π r E s and Π r G s the inverse image under Π r w s of a subbase element Θ r w ; U sp Ω q , Ω Ă D open, isopen in Π r E s which proves that Π r w s is open. This gives the commutative triangle Π r E s Π r G s D / / Π r w s (cid:31) (cid:31) ❄❄❄❄❄❄ π (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧ π (7)which proves that Π r w s is a morphism of ´etal´e spaces and a local homeomorphism, cf. [3,Proposition 2.4.8], [39, Lemma 3.5]. Remark 4.7.
A continuous map Π r w s : Π r E s ! Π r G s is called an ´etal´e morphism if the dia-gram in (7) commutes, cf. [39, Definition 3.3]. Such a map is then a local homeomorphism[3, Proposition 2.4.8], [39, Lemma 3.5].4.3. Induced ´etal´e morphisms.
The following lemma provides a criterion to construct an´etal´e space morphism from a natural transformation of structures.
Lemma 4.8.
Let E , E be stable C -structures on D and let n : E ñ E be a natural transformation.Then, the induced functor Π r n s : Π r E s ! Π r E s , defined by p φ, U q p φ, n φ p U qq , defines a morphism of ´etal´e spaces if and only if the sets t φ P Φ r E ; U s : n φ p U q “ U u , are open for every pair U, U P P .Proof. Suppose Π r n s is continuous, and U, U P P . Then, φ P Φ r E ; U s | n φ p U q “ U ( “ π ´ Π r n s ´ ` Im p Θ r id ; U sq ˘ X Im p Θ r id ; U sq ¯ , which is open. Now for the converse. Let Θ r id ; U sp Ω q be a subbasis element of Π r E s . Then, ď U P P ´ Ω X φ P Φ r E ; U s | n φ p U q “ U (¯ “ Π r n s ´ ` Θ r id ; U sp Ω q ˘ , which is a union of open sets, and therefore open.When the action of n φ is independent of φ P ob p D q , the openness condition is triviallysatisfied. This condition for stable structures translates to unstable structures in the follow-ing proposition. Proposition 4.9.
Let p G , E , w q and p G , E , w q be continuation frames on D and let v : G ñ G bea natural transformation. Suppose there exists a natural transformation r v : E ñ E — the lift of v — such that Π r r v s is continuous, and the following diagram commutes: E E G G r vw w v Then, Π r v s is a morphism of ´etal´e spaces. The lift r v is called a stable extension of v .Proof. We have the following maps on ´etal´e spaces: Π r E s Π r E s Π r G s Π r G s D Π r r v s Π r w s Π r w s Π r v s π π By Corollary 4.6 Π r w s and Π r w s are ´etal´e morphisms and by [3, Proposition 2.4.8], [39,Lemma 3.5] the map Π r ˜ v s is an ´etal´e morphism. Diagram chasing then shows that also Π r v s is a ´etal´e morphism by using the same results. Corollary 4.10.
Let p G , E , w q , p G , E , w q be two continuation frames sharing G . If w factorsthrough some natural transformation r ι : E ñ E , i.e the following diagram commutes: E E G r ι w w and Π r r ι s is continuous. Then, p G , E , w q and p G , E , w q induce the same ´etal´e space p Π r G s , π q on D .Proof. Let Π r G s and Π r G s denote the ´etal´e spaces induced by p G , E , w q and p G , E , w q respec-tively. The natural transformation r ι lifts the identity transformation ι : G ñ G . By applyingProposition 4.9 we achieve a continuous, open bijection Π r ι s : Π r G s ! Π r G s which provesthat the ´etal´e topologies are homeomorphic. Corollary 4.11.
Let p G , E , w q be a C -continuation frame and suppose C has concrete pullbacks, i.e.pullbacks exist and are preserved by the forgetful functor into Set . Then, the ´etal´e space induced byany stable extension w : E ! G is unique.Proof. Let p G , E , w q , p G , E , w q be two continuation frames sharing G . We will show theyinduced the same ´etal´e space. Consider the functor E ˆ G E : D ! C , along with canonicalprojections ONTINUATION SHEAVES IN DYNAMICS 17 E ˆ G E E E G p E p E w w Call the composite w ˝ p E “ w ˝ p E “ p . First, we check that p G , E ˆ G E , p q is a continuationframe. p φ is surjective component-wise, since for each S P G p φ q , there exists an U P E p φ q and U P E p φ q such that w φ p U q “ w φ p U q “ S . Then p φ p U, U q “ S . To see this constructionis stable, notice that Φ r E ˆ G E ; p U, U qs “ t φ P Φ r E ; U s X Φ r E ; U s : w φ p U q “ w φ p U qu“ π ´ p Θ r w ; U sp Φ r E ; U sq X Θ r w ; U sp Φ r E ; U sqq“ π ´ p Θ r w ; U sp Φ r E ; U sqq X π ´ p Θ r w ; U sp Φ r E ; U sqq , with π the induced ´etal´e maps from the continuation frames. Applying continuity of π foreach ´etal´e space gives that Φ r E ˆ G E ; p U, U qs is open. Finally, p E and p E are independentof the dynamical system, trivially satisfying the openness condition in Proposition . .Thus, applying Corollary 4.10 to both p E and p E , we have that p G , E , w q , p G , E ˆ G E , p q , and p G , E , w q all induce the same ´etal´e space.5. Continuation of attractors.
In Section 3 we establised the functors
ANbhd and
Att actingbetween the categories of dynamical systems and the category of bounded, distributivelattices. The topologies introduced in Section 2 yields the following result.
Lemma 5.1.
ANbhd : DS p T , X q ! BDLat is a stable structure.Proof.
As subset of DS p T , X q we define Φ r ANbhd ; U s : “ t φ P DS p T , X q | U P ANbhd p φ qu for any subset U Ă X . The condition U P ANbhd p φ q is equivalent to ω φ p U q Ă int U . By[13, 18] we have the equivalent characterization: U P ANbhd p φ q if and only if there exists atime τ ą such that φ t p cl U q Ă int U, @ t ě τ, (8)which is equivalent to ď t Pr τ, τ s φ t p cl U q “ φ ` r τ, τ s ˆ cl U ˘ Ă int U. (9)Indeed, if (8) is satisfied then (9) follows. On the other hand if (9) is satisfied then φ ` r τ, τ s ˆ cl U ˘ “ ď t Pr τ, τ s φ t ` τ p cl U q “ φ τ ´ ď t Pr τ, τ s φ t p cl U q ¯ “ φ τ p int U q Ă φ τ p cl U q Ă int U. By induction φ ` r nτ, p n ` q τ s ˆ cl U ˘ Ă int U for all n ě , which establishes (8). Summa-rizing, φ P Φ r ANbhd ; U s if and only φ ` r τ, τ s ˆ cl U ˘ Ă int U, for some τ ą . (10)For any τ ą define K τ “ r τ, τ s ˆ cl U Ă T ˆ X which is a compact set. Consider thebasic open sets B ` K τ , int U ˘ : “ ! φ | φ p K τ q “ φ ` r τ, τ s ˆ cl U ˘ Ă int U ) , which are contained in the subbasis for the compact-open topology on DS p T , X q . By(10) an element φ P Φ r ANbhd ; U s is contained in B ` K τ , int U ˘ for some τ ą and thus Φ r ANbhd ; U s Ă Ť τ ą B ` K τ , int U ˘ . On the other hand if φ P B ` K τ , int U ˘ for some τ ą , then (10) implies that φ P Φ r ANbhd ; U s which shows that Ť τ ą B ` K τ , int U ˘ Ă Φ r ANbhd ; U s and thus Φ r ANbhd ; U s “ Ť τ ą B ` K τ , int U ˘ which is a union of basic openset and thus open.Consequently, ` Att , ANbhd , ω ˘ is a BDLat -continuation frame on DS p T , X q and ANbhd is a stable extension for
Att . By Theorem 4.4 we have that p Π r Att s , π q is an ´etal´e space in Set . Stable extensions in a continuation frame are not unique. For example for attractorswe can also consider other continuation frames that define the same ´etal´e space. A subset U Ă X is called an attracting block if φ t p cl U q Ă int U for all t ą . This defines the lattice ABlock p φ q of attracting blocks for φ with ω φ : ABlock p φ q ։ Att p φ q . By the same token asbefore we may regard ABlock : DS p T , X q ! BDLat as a contravariant functor which isa stable extension of
Att : DS p T , X q ! BDLat . The continuation frame p Att , ABlock , ω q defines the same ´etal´e space Π r Att s , cf. Corollary 4.11.The functor Att is the structure we wish to continue, with stable extension ω : ANbhd ñ Att . This gives DS p T , X q BDLat
ANbhdAtt ω Π r Att s Φ r ANbhd ; U s DS p T , X q (cid:15) (cid:15) π : : tttttttttttttt Θ r w ; U s / / Ă We the partial section map Θ r ω ; U s : Φ r ANbhd ; U s ! Π r Att s , which maps a dynamical system φ with attracting neighborhood U to the pair p φ, A q withits associated attractor A “ ω φ p U q . Since ω φ is surjective, given a pair p φ, A q P Π r Att s , thereexists an attracting neighborhood U such that Θ r ω ; U sp φ q “ p φ, A q . In Remarks 3.3 and ONTINUATION SHEAVES IN DYNAMICS 19 p Rep , RNbhd , α q based re-pelling neighborhoods and repellers which yields the ´etal´e space Π r Rep s . For a dynamicalsystem φ : T ` ˆ X ! X we define φ ´ t : “ φ ´ t as the inverse image. The map φ p´ t, x q alsosatisfies the semi-group property. This allows us to define the notion alpha-limit set: α φ p U q : “ č t ě cl ď s ě t φ ´ s p U q . Some properties of α φ p U q are: (i) α φ p U q is compact, closed, (ii) α φ p U q is a forward-backward invariant set for the dynamics, (iii) α φ ` α φ p U q ˘ Ą α φ p U q , (iv) α φ p U Y V q “ α φ p U q Y α φ p V q . A neighborhood U Ă X is called an repelling neighborhood if α φ p U q Ă int U . Attracting neighborhoods form a bounded, distributive lattice denoted by RNbhd p φ q .The binary operations are X and Y . A subset A Ă X is called an repeller if there existsan repelling neighborhood U Ă X such that R “ α φ p U q , which is a neighborhood of R by definition. Repellers are compact, closed forward-backward invariant sets and com-pose a bounded, distributive lattice Rep p φ q with binary operations Y and X . As before φ ANbhd p φ q and φ Rep p φ q define the contravariant functors RNbhd and
Rep from DS p T , X q ! BDLat . The functor
RNbhd : DS p T , X q ! BDLat is a stable structure and p Rep , RNbhd , α q forms a continuation frame in a similar way. From the continuation frame p Rep , RNbhd , α q we obtain the ´etal´e space p Π p Rep q , π q .For a dynamical system φ consider the duality isomorphism A A ˚ , A P Att p φ q . Sincefor U P ANbhd p φ q the maps U U c and ω φ p U q α φ p U c q define lattice isomorphismswe also have the natural transformations c : ANbhd ðñ RNbhd and ˚ : Att ðñ Rep . Thisyields the following commutative diagram:
ANbhd p ψ q RNbhd p ψ q ANbhd p φ q RNbhd p φ q Att p ψ q Rep p ψ q Att p φ q Rep p φ q cf ´ ω ψ α ψ f ´ cω φ α φ ˚ ψ Att p f q Rep p f q˚ φ where Att p f q “ ω φ ˝ f ´ and Rep p f q : “ ˚ φ ˝ ω φ ˝ f ´ ˝ ˚ ψ . This implies the middlecommutative diagram in Fig. 1,where we lift ˚ via r ˚ : “ c . This verifies commutativity in Proposition 4.9, and since r ˚ φ is independent of φ , continuity of Π r r ˚s is automatically satisfied. Applying Proposi-tion 4.9 to the dual repeller transformation yields the following important isomorphism of ´etal´e spaces: Π r˚s : Π r Att s ! Π r Rep s p φ, A q p φ, A ˚ q , which yields the right diagram in Fig. 1. Remark 5.2.
The dual repeller operator ˚ is dependent on the underlying system φ : A A ˚ “ t x P X : ω φ p x q X A “u . But for convenience of notation, we will omit the subscript when the underlying system isunderstood.6.
Algebraic constructions.
In this section we incorporate the binary operations of lattices,groups, rings, etc. and augment the ´etal´e spaces with these operations.6.1.
Binary operations and lattices.
Given two ´etal´e spaces p Π , π q , p Π , π q over a topolog-ical space. Define Π ‚ Π : “ p σ, σ q P Π ˆ Π : π p σ q “ π p σ q ( , which is also an ´etal´e space with the same projection map and the product topology, cf. [39,Sect. 2.5]. Proposition 6.1.
Suppose C has concrete binary products. Let p G , E , w q , p G , E , w q be C -continuation frames on D . Then, p G ˆ G , E ˆ E , w ˆ w q is a continuation frame and the map g : Π r G ˆ G s ! Π r G s ‚ Π r G s , ` φ, p S, S q ˘ ` p φ, S q , p φ, S q ˘ is a homeomorphism.Proof. Since both w and w are surjective componentwise, so is w ˆ w . For the opennesscondition: Φ r G ˆ G ; p U, U qs “ φ P ob p D q | p U, U q P p G ˆ G qp φ q ( “ Φ r G ; U s X Φ r G ; U s , which is open. Bijectivity of g is immediate; it remains to be shown that g is continuousand open on subbasis elements. Let U, U P P and Ω , Ω open in Φ r E ; U s and Φ r E ; U s respectively. Then, g ´ ´ Θ r w ; U sp Ω q ˆ Θ r w ; U sp Ω q X Π r G s ˝ Π r G s ¯ “ Θ “ w ˆ w ; p U, U q ‰ p Ω X Ω q , which is open. Similarly, letting U, U P P and Ω Ă Φ r E ˆ E ; p U, U qs open we have: g ´ Θ r w ˆ w ; p U, U qs ¯ “ Θ r w ; U sp Ω q ˆ Θ r w , U sp Ω q X Π r G s ˝ Π r G s which proves that g is an open map and therefore a homeomorphism. Remark 6.2.
The universe functor in the product continuation frame is the product of itsfactors’ universe functors: p U, U q P P ˆ P . ONTINUATION SHEAVES IN DYNAMICS 21
We can apply Propositions 4.9 and 6.1 to the the
BDLat -continuation frames p Att , ANbhd , ω q and p Rep , RNbhd , α q to interpret lattice operations as morphisms of´etal´e spaces. This permits us to regard Π r Att s and Π r Rep s as BDLat -valued.For example ^ φ : Att p φ q ˆ Att p φ q ! Att p φ q given by p A, A q A ^ A forms a naturaltransformation ^ : Att ˆ Att ñ Att . From Proposition 4.9 r ^ : ANbhd ˆ ANbhd ñ ANbhd , given by p U, U q U X U with ω φ p U q “ A and ω φ p U q “ A , acts as a lift for ^ which yields an ´etal´e space morphism from Π r Att ˆ Att s to Π r Att s . Combining the latter with Proposition 6.1 yields an ´etal´e space morphism: Π r^s : Π r Att s ‚ Π r Att s ! Π r Att s , ` p φ, A q , p φ, A q ˘ p φ, A ^ A q , which establishes ^ as a continuous binary operation on Π r Att s . The same can be achievedfor _ . Absorption, distributivity and associativity follows immediately from the propertiesof ^ and _ . It remains to show that the assignments of the neutral elements φ p φ, ∅ q P Π r Att s , φ ` φ, ω φ p X q ˘ P Π r Att s , are continuous. The first one is trivial and for the second one we choose the constantsection p φ, X q P Π r ANbhd s . For Rep the latter is constant, i.e. φ X which triviallyproves continuity. Consequently, we have established Π r Att s and Π r Rep s as BDLat -valued´etal´e spaces. We will later explore abelian structures and ring structures which are used inthe treatment of sheaf cohomology.6.2.
The Conley form.
Recall that the Conley form assigns to two attractors
A, A P Att p φ q an associated invariant set p A, A q C Att p A, A q : “ A X A , where A P Rep p φ q is dual to A in the sense that A “ α φ p U c q where U P ANbhd p φ q with ω φ p U q “ A . The repeller A is called the dual repeller to A . The Conley form has a universal property in the sense thatit is a unique extension of set-difference for bounded, distributive lattices, cf. [20].For a dynamical system φ and subset U Ă X define the invariant set Inv φ p U q : “ Ş t S Ă U | S is invariant u . A Morse neighborhood is a subset T Ă X given by T “ U X V with U P ANbhd p φ q and V P RNbhd p φ q . It holds that Inv φ p T q “ ω φ p U q X α φ p V q : “ M and which is called a Morse set . By construction M Ă int T , cf. [20]. The Morse setsand denoted by Morse p φ q which is a bounded, meet-semilattice with binary operation M ^ M : “ Inv φ p M X M q . The Morse neighborhoods are denoted by MNbhd p φ q and whichform a bounded, meet-semilattice with intersection as binary operation. Both ∅ and ω φ p X q are neutral elements. As before Inv : MNbhd ñ Morse is a stable
MLat -structure, where C “ MLat is the category of bounded, meet-semilattices. The triple p Morse , MNbhd , Inv q is a MLat -continuation frame and by the general theory in Section 4 we obtain the
MLat -´etal´e space Π r Morse s of Morse sets. By the same token we can treat the Conley form as natural transformation C Att : Att ˆ Att ñ Morse , where the functor Morse assigns the bounded, meet-semilattice of Morse sets to φ . ByProposition 6.1 this leads to a continuous operation Π r C Att s : Π r Att s ‚ Π r Att s ! Π r Morse s ` p φ, A qp φ, A q ˘ ` φ, C Att p A, A q ˘ . The map Π r C Att s will play a role in setting up the appropriate algebraic construction forsheaf cohomology.A variation on the Conley form is the symmetric Conley form which is defined as follows: p A, A q C ˚ p A, A q : “ C Att p A Y A , A ^ A q “ p A X A q Y p A X A ˚ q . For the symmetric Conley form we use the following notation: p A, A q A ` A . Remark 6.3.
The range of the symmetric Conley form is the same as for the standard Con-ley form. Indeed, if A Ă A then C ˚ p A, A q “ C Att p A, A q . For any pair of attractor A, A absorption implies that C Att p A, A q “ C Att p A, A ^ A q which shows that the Conley formcan always be determinded from nested pairs, in which case the standard and symmetricConley forms coincide. Remark 6.4.
Instead of Morse neighborhoods we can also employ Morse tiles which aredefined via attracting blocks and repelling blocks, i.e. a subset T Ă X given by T “ U X V with U P ABlock p φ q and V P RBlock p φ q is called a Morse tile and the meet-semilattice ofMorse tiles is denoted by
MTile p φ q .6.3. The algebra of attractors.
In this section we take a closer look at the algebraic struc-ture of attractors. Algebraic structures and in particular (abelian) group structures areimportant for the (co)homological theory of sheaves. Our starting point is the lattice ofattractors
Att p φ q of a fixed dynamical system φ , which is a bounded, distributive lattice.Before treating the lattice of attractors we first consider bounded, distributive lattices froma more general point of view.Let p L , ^ , _ , , q be bounded, distributive lattice. Then, by the Priestley representationtheorem, L is isomorphic to the lattice O clp p Σ L q of clopen downsets in the ordered topo-logical space Σ L , the spectrum of L whose points are the prime ideals in L . The latter is azero-dimensional, compact Hausdorff space. The Boolean algebra BL : “ P clp p Σ L q of clopensets in Σ L is called the Booleanization , or minimal Boolean extension of L and j : L ! BL is alattice-embedding given by j p a q “ t I P Σ L | a R I u . This construction is functorial; we havethe Booleanization functor B : BDLat ! Bool . ONTINUATION SHEAVES IN DYNAMICS 23
Boolean algebras can be given the structure of a ring, i.e. given a Boolean algebra p B , ^ , _ , c , , q define a ` b : “ p a ^ b c q _ p b ^ a c q (symmetric difference) and a ¨ b : “ a ^ b. Then, p B , ` , ´ , , q is a commutative, idempotent ring (idempotency with respect to mul-tiplication). One retrieves the Boolean algebra structure via a _ b “ a ` b ` a ¨ b . We canformulate this as a faithful functor I : Bool ! Ring from the category of Boolean algebrasto the category of rings.
BDLat Bool Ring
B I
Define the ring obtained from Booleanization of L as the (Boolean) lattice ring of L : RL : “ p I ˝ B qp L q (11)the composition is also denoted by R : “ I ˝ B . This is the natural way to give an abelianstructure to a bounded distributive lattice L . We note that RL is in general not free asadditive Z -module (vector space), nor as multiplicative monoid. Since L may be regardedas a (commutative) monoid with respect to ^ we can use the monoid ring construction,cf. [23, 2, 38], to define the Z -algebra Z L , which is a free Z -module (vector space). Theelements of Z L is finite formal sums ř i a i , a i P L , with the additional requirement that a “ a ` a “ . Multiplication is given by a ¨ b : “ a ^ b . We refer to Z L as the lattice algbera of L which first introduced in the context of minimal Boolean extension by MacNeille [26].The lattice algebra Z L is clearly a Boolean ring as is the lattice ring. The Z -monoid ringconstruction defines a covariant functor BDLat Ring . Z The analog of the homomorphism j : L ! BL is now given by the ring homomorphism: j : Z L ! RL , j ´ÿ i a i ¯ : “ ÿ i j p a i q “ ÿ i α i . By definition j p a ^ b q “ j p a q X j p b q , which makes j an algebra homomorphism. The imageof the generators of Z L in RL are downsets in ΣL and via the induced _ operation thelattice L can be retrieved. By construction j p a q ` j p b q “ ` j p a q Y j p b q ˘ r ` j p a q X j p b q ˘ “ j p a _ b q r j p a ^ b q“ C σ p a _ b, a ^ b q , When b Ă a , then j p a q ` j p b q “ C σ p a, b q and thus the sums j p a q ` j p b q exhaust the rangeof the Conley form C σ : L ˆ L ! RL . We define the set CL : “ C σ p a, b q | a, b P L ( as the convexity monoid : for σ, σ P CL we have σ ¨ σ “ C σ p a, b q X C σ p a , b q “ C σ p a ^ a , b _ b q P CL and σ ¨ “ C σ p a, b q X C σ p , q “ C σ p a ^ , b _ q “ C σ p a, b q “ σ . Clearly the embedding i : CL ! RL is a monoid homomorphism. Lemma 6.5.
The ring homomorphism j : Z L ! RL is surjective.Proof. Let γ P RL , then by a property of the Priestley topology we can express γ as finiteunion of the form: γ “ Ť i α i r α i , with α i “ j p a i q , α i “ j p a i q and a i , a i P L . The objectiveis to prove that γ is in the range of j . Consider α r β Y γ r δ . We may assume withoutloss of generality that β Ă α and δ Ă γ . Indeed, use α r β “ α r p α X β q . Therefore, α r β Y γ r δ “ p α ` β q Y p γ ` δ q , and p α ` β q Y p γ ` δ q “ α ` β ` γ ` δ ` p α ` β q X p γ ` δ q“ α ` β ` γ ` δ ` p α X γ q ` p α X δ q ` p β X γ q ` p β X δ q , which corresponds to a sum of attractors. We conclude that γ “ Ť i α i r α i “ ř k ˜ α k “ ř k j p ˜ a k q , ˜ a k P L , which proves that γ is in the range of j .We now have the following short exact sequence: j Z L RL , / / / / / / Ă / / / / j / / (12)and since the kernel ker j is an ideal in Z L the first isomorphism theorem for rings yields RL – Z L ker j , where the isomorphism is given ř i a i ` ker j ř i α i . If we regard j as a module (vectorspace) homomorphism then both ker j and Z Att p φ q are free Z -modules. The ideal ker φ can be characterized as follows. Lemma 6.6. ker j is the ideal freely generated by elements of the form a _ b ` a ` b ` a ¨ b .Proof. For an element a _ b ` a ` b ` a ¨ b we have that j p a _ b ` a ` b ` a ¨ b q “ j p a _ b q ` j p a q ` j p b q ` j p a ¨ b q“ j p a q Y j p b q ` j p a q ` j p b q ` j p a q X j p b q“ ` j p a q Y j p b q ˘ “ ∅ , which proves that finite sums of elements of the form a _ b ` a ` b ` a ¨ b are contained in ker j . Let j ´ř i a i ¯ “ ř i α i “ ∅ , then the sum must have an even number of terms. Wecan rearrange the sequence to an filtration α Ă ¨ ¨ ¨ Ă α m such that ř i α i “ ř i α i “ ∅ .Consequently α k ´ ` α k “ ∅ for k “ , ¨ ¨ ¨ , m , i.e. α k ´ “ α k for all k . In order to havedistinct elements mapping to α j ´ “ α j we have j p b k ` c k q “ α k ´ “ α k “ j p b k _ c k ` b k ¨ c k q , which proves that element in ker j is contained in the set of formal sums generated by termsof the form a _ b ` a ` b ` a ¨ b . ONTINUATION SHEAVES IN DYNAMICS 25
Let us return to the lattice of attractors
Att p φ q . Define the attractor ring of a dynamicalsystem φ as RAtt p φ q : “ p I ˝ B q ` Att p φ q ˘ as the Boolean lattice ring of Att p φ q . This is thenatural way to give an abelian structure to the attractors of a dynamical system. Via themonoid ring construction we obtain the algebra Z Att p φ q which is called the free attractorring over Z .7. Continuation sheaves.
From an abstract continuation frame we have shown how tobuild an ´etal´e space which encodes the continuation of the unstable structure of interest.This ´etal´e space Π r G s connects the topology of the base space to the algebraic structure of G . To study this connection, we shift our attention to the sheaves of sections generatedby the ´etal´e spaces of continuation frames. While sheaves and ´etal´e spaces are equivalentfrom a categorical viewpoint, the theory of sheaves contributes a rich algebraic toolkit toour study of continuation. Perhaps most prominent is the idea of sheaf cohomology. Definition 7.1.
Let p G , E , w q be a continuation frame and let p Π r G s , π q be the associated´etal´e space over D . A section in Π r G s over an open set Ω in D is a continuous map σ : Ω ! Π r G s such that π ˝ σ “ id . The set of all sections over Ω is denoted by S G p Ω q .The presheaf S G : O p D q ! Set , Ω S G p Ω q , where O p D q is the posetal category of open sets in D , is in fact a sheaf over D and is calledthe sheaf of sections , cf. [39, Sect. 2.2C]. A stalk of the sheaf S G at φ P D is the object G p φ q .By considering sections in Π r E s we obtain the sheaf of sections S E and stalks in S E aredenoted by E p φ q , cf. [39, Prop. 3.6]. Lemma 7.2.
Let p G , E , w q be a continuation frame and let σ : Ω ! Π r G s be a map with propertythat π ˝ σ “ id on Ω (open). If for every open set Ω Ă Ω and U P E p φ q , φ P Ω such that σ ˇˇ Ω “ Θ r w ; U s ˇˇ Ω , then σ is a section in Π r G s .Proof. This follows immediately from the definition of sheaves.Sections therefore act locally like Θ r w ; U s . Following this intuition, observe that Θ r w ; U s is a section in Π r G s over Φ r E ; U s .The above lemma means we only need to verify that a candidate section locally agreeswith Θ r w ; U s for a particular U P E p φ q for some φ P ob p D q , rather than all such U . By thesame token sections σ : Ω ! Π r E s are given locally by Θ r id ; U s , i.e. σ p φ q “ p φ, U q . From the construction of the sheaves S E and S G we have the following property of thenatural transformation w : S E p Ω q S G p Ω q S E p Ω q S G p Ω q (cid:15) (cid:15) ρ Ω , Ω / / w p Ω q (cid:15) (cid:15) ρ Ω , Ω / / w p Ω q where w p Ω q : S E p Ω q ! S G p Ω q is defined by σ Π r w s ˝ σ , Ω open, and similarly for Ω Ă Ω . The maps ρ Ω , Ω are the restriction maps. The latter defines a morphism of sheaves w : S E ! S G . Since w yields the stalk wise surjections ω φ : E p φ q ։ G p φ q we say that themorphism w : S E ! S G is surjective. Proposition 7.3 (cf. [14]) . The surjectivity of w is equivalent to the following condition: for everyopen set Ω Ă D and for every section σ P S G p Ω q there exists an open covering t Ω i u of Ω andsections σ i P S E p Ω i q , such that σ | Ω i “ w p σ i q , for all i . As a consequence of the above characterization of the surjective morphisms w : S E ։ S G we have that sections are locally given by φ Θ r w ; U sp φ q as indicated in Lemma 7.2.In Section 6.1 we constructed ´etal´e spaces in categories of bounded, distributive lat-tice, semi-lattices, etc. The above construction via sheaves of sections creates sheaveswith values in bounded, distributive lattices, semi-lattices, etc. For the relevant C -structures in this paper we obtain the following sheaves. The C -structure p Att , ANbhd , ω q yields the ´etal´e morphism Π r ω s : Π r ANbhd s ։ Π r Att s and the the BDLat -valued sheaves S ANbhd : O p DS p T , X qq ! BDLat and S Att : O p DS p T , X qq ! BDLat the following funda-mental morphism of sheaves: ω : S ANbhd ։ S Att , that assigns to every section σ : Ω ! Π r ANbhd s the section Π r ω sp σ q : Ω ! Π r Att s . Thefundamental sheaf S Att is called the attractor lattice sheaf over DS p T , X q . Similarly, wehave the sheaves α : S RNbhd ։ S Rep , where S Rep is the dual (isomorphic) repeller lattice sheaf . By the same token duality between
Att and
Rep and
ANbhd and
RNbhd yields the following commutative diagram of sheaves: S ANbhd S RNbhd S Att S Rep (cid:15) (cid:15) (cid:15) (cid:15) ω o o / / c (cid:15) (cid:15) (cid:15) (cid:15) α o o / / ˚ The Conley form on ´etal´e spaces in Section 6.2 gives rise to the
MLat -valued sheaf S Morse : O p DS p T , X qq ! MLat . For the above lattice-valued and meet lattice-valuedsheaves we need to construct a suitable abelian structure in order to define their sheaf
ONTINUATION SHEAVES IN DYNAMICS 27 cohomology. In general, C be a concrete category and let K : C ! Ring a covariant func-tor. Moreover, let S : O p D q ! C be a C -valued sheaf over a topological category D . Forevery open set Ω in D we have the ring K S p Ω q and the ring-valued presheaf K S : O p D q ! Ring , Ω K S p Ω q . Via the sheafification functor : PrSh C p X q ! Sh C p X q , cf. Appendix C, we then obtainthe sheaf K : “ ` K S ˘ : O p D q ! Ring . In Section 6.3 we consider two functors that that take values in the category of rings. On onehand the functor the
Boolean ring functor R “ I ˝ B : BDLat ! Ring and on the other handthe monoid ring functor Z : Monoid ! Ring , where
Monoid is the category of monoids.In the case of the sheaf of attractors we obtain the abelian sheaves: A : “ ` R S Att ˘ : O p DS p T , X qq ! Ring , (13)and Att : “ ` Z S Att ˘ : O p DS p T , X qq ! Ring , (14)which are called the attractor sheaf and the free attractor sheaf over DS p T , X q respectively.The same construction can be carried out for other sheaves mentioned above. The construc-tion via the functor Z : Monoid ! Ring works for all of the above examples since bothbounded, distributive lattices and semilattices compose subcategories of in the category of(commutative) monoids. Of particular interest is the free Morse sheaf
Morse : “ ` Z S Morse ˘ : O p DS p T , X qq ! Ring . The short exact sequence in (12) for
Att yields:
Definition 7.4.
The fundamental short exact sequence of the attractor sheaf is given by: j Att A , / / / / / / Ă / / / / j / / (15)where the stalks A φ “ RAtt p φ q , and Att φ “ Z Att p φ q , are the attractor ring at φ and the free attractor ring over Z at φ respectively.The fundamental exact sequence allows us to relate the sheaves A and Morse . Thegenerators of
Att and the ring structure of A recover the sheaf of attractors S Att . Remark 7.5.
An alternative way to define the sheaves A and Att is a direct defini-tion via ´etal´e spaces. In the case of
Att we define an ´etal´e space Π r Z Att s using thestable C -structure ` Z Att , Z ANbhd , Z p ω q ˘ via the monoid ring functor. The stabil-ity follows from the fact that stability is preserved under free sums. We obtain the´etal´e space π : Π r Z Att s ! DS p T , X q and the associated sheaf of sections S Z Att . It holds that S Z Att – Att . For the Boolean ring functor it is more involved to prove that ` RAtt , RANbhd , R p ω q ˘ is a stable C -structure. By the same token S RAtt – A .8. Parameter spaces and pullbacks.
In this section we discuss continuation frames forparametrized families of dynamical systems and how the associated sheaves can be con-structed.8.1.
Parametrized dynamical systems.
Let Λ be a topological space. We emphasize thatno requirements on the topology are made at this point. Definition 8.1.
Let X be a compact topological space. A parametrized dynamical system over Λ on X is a continuous map φ : T ˆ X ˆ Λ ! X such that φ λ : “ φ p¨ , ¨ , λ q P DS p T , X q forall λ P Λ .The category of dynamical systems DS p T , X q is a function space equipped with thecompact-open topology. For a parametrized dynamical system φ we define the transpose φ ˚˚˚ : Λ ! DS p T , X q by: φ ˚˚˚ p λ q “ φ λ : “ φ p¨ , ¨ , λ q . The transpose φ ˚˚˚ : Λ ! DS p T , X q is a continuous map without additional assumptions onthe topological spaces Λ and X , cf. Appendix B.For the continuation frame p Att , ANbhd , ω q on DS p T , X q a parametrized dynamical sys-tem yields a pullback ´etal´e space on Λ : φ ´ ˚˚˚ Π r Att s : “ p λ, φ, A q P Λ ˆ Π r Att s | φ ˚˚˚ p λ q “ π p φ, A q ( , i.e. the follows diagram commutes φ ´ ˚˚˚ Π r Att s Π r Att s Λ DS p T , X q p λ,φ,A q p φ,A qp λ,φ,A q λ π φ ˚˚˚ where φ ´ ˚˚˚ Π r Att s is the pullback in the category of topological spaces, cf. [4, Sect. I.3]. From[3, Prop. 2.4.9] it follow that φ ´ ˚˚˚ Π r Att s ! Λ is an ´etal´e space over Λ . The binary operationson Π r Att s can be verified to be continuous on the inverse image ´etal´e space. As before weobtain the following BDLat -valued pullback sheaf φ ´ ˚˚˚ S Att : O p Λ q ! BDLat , as the sheaf of section in φ ´ ˚˚˚ Π r Att s . Applying the boolean ring functor R to the sheaf ofsections yields a ring valued sheaf: A φ ˚˚˚ : “ p R φ ´ ˚˚˚ S Att q : O p Λ q ! Ring . ONTINUATION SHEAVES IN DYNAMICS 29
The ringed space p Λ , A φ ˚˚˚ q encodes the continuation data of attractors for the parametrizeddynamical system. Similarly, for the Monoid ring functor Z ¨ we obtain: Att φ ˚˚˚ : “ p Z φ ´ ˚˚˚ S Att q : O p Λ q ! Ring . where the multiplication is inherited from the monoidal structure of Att , cf. Section 6.3.
Remark 8.2.
The sheaf A φ ˚˚˚ can be alternatively defined as follows. The Boolean ringfunctor yields the ´etal´e space Π r RAtt s and the associated sheaf of sections φ ´ ˚˚˚ S RAtt – φ ´ ˚˚˚ A – A φ ˚˚˚ , which defines the sheaf as the pullback sheaf with respect to φ ˚˚˚ .8.2. Conjugate dynamical systems and homeomorphic ´etal´e spaces.
We start off with thebasic notion of conjugacy in dynamical systems.
Definition 8.3.
Let X and Y be compact topological spaces, and let φ ˚˚˚ : Λ ! DS p T , X q and ψ ˚˚˚ : Λ ! DS p T , Y q be parametrized dynamical systems. A conjugacy between φ ˚˚˚ and ψ ˚˚˚ is a continuous map h : Λ ˆ X ! Y such that(i) h λ : “ h p λ, ¨q is a conjugacy in hom p φ λ , ψ λ q for all λ P Λ ;(ii) h λ p X i q “ Y i uniformly for all λ P Λ , where X i and Y i are the connected componentsof X and Y respectively.If a conjugacy h exists, then φ ˚˚˚ and ψ ˚˚˚ are said to be conjugate parametrized dynamical sys-tems . Remark 8.4.
Assumption (ii) is always satisfied pointwise for λ by appropriately indexingthe components of X and Y . The uniformity in the above definition is not guaranteed sinceno restrictions on the topology of Λ are required. For specific topologies on Λ Condition(ii) may be superfluous.
Remark 8.5.
One can also consider quasiconjugacies between parametrized dynamical sys-tems over Λ .Since h λ is a conjugacy we know from Remark 3.5 that the push-forward U λ h λ p U λ q is an attracting neighborhood for ψ λ and simmilarly, the push-forward A λ h λ p A λ q is anattractor for ψ λ . Lemma 8.6.
The following diagram commutes:
ANbhd p φ λ q ANbhd p ψ λ q Att p φ λ q Att p ψ λ q (cid:15) (cid:15) (cid:15) (cid:15) ω φλ (cid:15) (cid:15) (cid:15) (cid:15) ω ψλ / / o o – / / o o – Proof.
Indeed, the maps from above we have U λ h λ p U λ q ω ψ λ ` h λ p U λ q ˘ . From belowyields U λ A λ “ ω φ λ p U λ q h λ p A λ q . Since h is a conjugacy it follows from Remark 3.5that Lemma 3.2 applies to both h λ and p h λ q ´ . This gives: ω ψ λ ` h λ p U λ q ˘ “ ω ψ λ ` h λ ` ω φ λ p U λ q ˘˘ “ ω ψ λ ` h λ p A λ q ˘ “ h λ p A λ q , (16)which proves commutativity.Lemma 8.6 holds for all λ P Λ and which provides stalk wise isomorphisms between theassociated sheaves of attractors. This however does not give isomorphic sheaves necessar-ily! Theorem 8.7 (Conjugacy Invariance Theorem) . Let X , Y be compact metric spaces. Suppose φ ˚˚˚ : Λ ! DS p T , X q and ψ ˚˚˚ : Λ ! DS p T , Y q are conjugate parametrized dynamical systems.Then, the ´etal´e spaces φ ´ ˚˚˚ Π r Att s and ψ ´ ˚˚˚ Π r Att s are homeomorphic.Proof. From Lemma 8.6 we have the following commutative diagram of maps: φ ´ ˚˚˚ Π r Att s ψ ´ ˚˚˚ Π r Att s Λ / / h ˚ $ $ ❏❏❏❏❏❏❏ π z z ttttttt π where h ˚ is defined by p λ, φ λ , A λ q h ˚ p λ, φ λ , A λ q : “ ` λ, ψ λ , h λ p A λ q ˘ . It is sufficient toshow continuity, since if h ˚ is continuous, then h ˚ is a local homeomorphism, in whichcase h ´ ˚ is a also a local homeomorphism ( h ˚ is a bijection), cf. [3, Prop. 2.4.8]. This provesthat φ ´ ˚˚˚ Π r Att s and ψ ´ ˚˚˚ Π r Att s are homeomorphic.In order to prove continuity we argue as follows. Consider the following commutativediagram φ ´ ˚˚˚ Π r Att s φ ´ ˚˚˚ Φ r ANbhd ; U s Λ (cid:15) (cid:15) π : : ttttttttttttt φ ´ ˚˚˚ Θ r ω ; U s / / Ă where φ ´ ˚˚˚ Φ r ANbhd ; U s “ λ | U P ANbhd p φ λ q ( and φ ´ ˚˚˚ Θ r ω ; U sp λ q “ ` λ, φ λ , ω φ λ p U q ˘ . Let D Ă Λ be an open neighborhood of λ P Λ and let ψ ´ ˚˚˚ Θ “ ω ; h λ p U λ q ‰ p D q “ "´ λ, ψ λ , ω ψ λ ` h λ p U λ q ˘¯ | λ P D * be an open neighborhood of h ˚ ` λ , φ λ , A λ ˘ “ ` λ , ψ λ , h λ p A λ q ˘ in ψ ´ ˚˚˚ Π r Att s for somecompact U λ P ANbhd p φ λ q . In order to establish continuity we seek a neighborhood D Ă ONTINUATION SHEAVES IN DYNAMICS 31 D Ă Λ such that h ˚ ´ φ ´ ˚˚˚ Θ r ω ; U λ sp D q ¯ “ "´ λ, ψ λ , h λ ` ω φ λ p U λ q ˘¯ | λ P D * “ "´ λ, ψ λ , ω ψ λ ` h λ p U λ q ˘¯ | λ P D * Ă ψ ´ ˚˚˚ Θ “ ω ; h λ p U λ q ‰ p D q , where the second equality follows from Lemma 8.6, Eqn. (16). This is equivalent to saying ω ψ λ ` h λ p U λ q ˘ “ ω ψ λ ` h λ p U λ q ˘ , @ λ P D . For notational convenience we write U : “ h λ p U λ q P ANbhd p ψ λ q , and A “ h λ p A λ q “ ω ψ λ p U q P Att p ψ λ q . We rephrase the above condition as: ω ψ λ ` h λ p U λ q ˘ “ ω ψ λ p U q , @ λ P D . (17)For U λ “ ∅ , or for U λ “ Ů i X i Ă X , any union of connected components of X , Eqn.(17) is obviously satisfied by the uniform conjugacy condition in Defn. 8.3(ii), cf. Remark8.4. For the remainder of the proof we assume U λ “ ∅ and U λ “ Ů i X i , for all unions ofconnected components of X . Therefore, we may carry out the arguments for the compo-nents U λ i “ U λ X X i “ ∅ , X i .Choose a compact attracting neighborhood U P ANbhd p ψ λ q such that U Ă int U and ω ψ λ p U q “ A . Indeed, since A is an attractor cl U c X A “ ∅ , cf. [18, Lemma 3.23]. Thereforethere exists open sets N, N such that A Ă N , cl U c Ă N and N X N “ ∅ . As a matterof fact cl N X N “ ∅ . Define U “ cl N . By construction A ˚ Ă U c Ă cl U c Ă N andthus U X A ˚ “ ∅ which proves that (i) ω ψ λ p U q “ A , (ii) A Ă N Ă U , (iii) U “ cl N Ă N c Ă ` cl U c ˘ c “ int U , and thus U is an attracting neighborhood satisfying the propertiesstated above, cf. [18, Lemma 3.21]. From the fact that U “ Ů i Y i , a union of components, itfollows that int U Ĺ U . Thus by Property (iii) there exists a δ ą such that B δ p U q Ă U and therefore d H p U, U q ě δ ą , where d H is the Hausdorff metric on the space H p X q ofcompact subsets of X , cf. Appendix B.By the same token we can choose a compact repelling neighborhood V P RNbhd p ψ λ q such that V X U “ ∅ and ω ψ λ p V c q “ A . Indeed, repeat the above arguments starting with U X A ˚ “ ∅ . V is compact, so there exists a δ ą such that d H p U, V q ě δ ą .Since, ψ ´ ˚˚˚ Θ “ ω ; U ‰ , ψ ´ ˚˚˚ Θ “ ω ; U ‰ and ψ ´ ˚˚˚ Θ “ ω ; V c ‰ define local sections in ψ ´ ˚˚˚ Π r Att s over ψ ´ ˚˚˚ Φ r ANbhd ; U s , ψ ´ ˚˚˚ Φ r ANbhd ; U s and ψ ´ ˚˚˚ Φ r ANbhd ; V c s respectively, and since ψ ´ ˚˚˚ Θ “ ω ; U ‰ p λ q “ ψ ´ ˚˚˚ Θ “ ω ; U ‰ p λ q “ ψ ´ ˚˚˚ Θ “ ω ; V c ‰ p λ q there exists an open set E Ă Λ on which three sections coincide, i.e. B λ : “ ω ψ λ p U q “ ω ψ λ p U q “ ω ψ λ p V c q , @ λ P E , and B λ Ă int U , B λ Ă int U and B λ Ă int V c for all λ P E .Let r U be any compact neighborhood such that d H p U, r U q ă δ “ min t δ , δ u{ and let λ P E . Then, B λ Ă U Ă r U , r U X p B λ q ˚ Ă r U X V “ ∅ , which by [18, Lemma 3.21] implies that ω ψ λ p r U q “ B λ for all λ P E .Finally, using the continuity of h λ H in Lemma B.1, choose an open sets D Ă E X D suchthat d H ` h λ p U λ q , U ˘ ă δ for all λ P D . By the previous we choose r U “ h λ p U λ q whichproves that ω ψ λ ` h λ p U λ q ˘ “ B λ “ ω ψ λ p U q , @ λ P D , establishing (17) and thereby the theorem. Remark 8.8.
The condition that the spaces X and Y are compact metric spaces is used atseveral places in the proof and in particular for using the Hausdorff metric. The character-izations of attracting and repelling neighborhoods via attractors and dual repellers at leastworks in compact Hausdorff spaces.Theorem 8.7 can be extended to other structures. Since φ ´ ˚˚˚ Π r Att s is homeomorphic(as a sheaf of sets) to φ ´ ˚˚˚ Π r Rep s , we can get a homeomorphism between Π r Rep φ ˚˚˚ s and Π r Rep ψ ˚˚˚ s . There is the following commutative diagram for Morse sets: φ ´ ˚˚˚ Π r Att s ‚ φ ´ ˚˚˚ Π r Att s ψ ´ ˚˚˚ Π r Att s ‚ ψ ´ ˚˚˚ Π r Att s φ ´ ˚˚˚ Π r Morse s ψ ´ ˚˚˚ Π r Morse s Λ p λ,φ λ ,A q , p λ,φ λ ,A q p λ,ψ λ ,h λ p A qq , p λ,ψ λ ,h λ p A qq Π r C Att s Π r C Att sp λ,φ λ ,M q p λ,ψ λ ,h λ p M qq which, using a similar argument to Propositon 4.9, establishes that the ´etal´e spaces φ ´ ˚˚˚ Π r Morse s and ψ ´ ˚˚˚ Π r Morse s are homeomorphic. Corollary 8.9.
Let X and Y be homeomorphic compact metric spaces and let Att X and Att Y be theattractor functors on DS p T , X q and DS p T , Y q respectively. Then, the ´etal´e spaces Π r Att X s and Π r Att Y s are homeomorphic.Proof. Let h : X ! Y be a homeomorphism and let Λ “ DS p T , X q . Then, φ ˚˚˚ is the identitymap. The map ψ ˚˚˚ : Λ ! DS p T , Y q is defined as follows: Λ Q φ h ˝ φ ˝ h ´ “ ψ . Then, h ` φ t p x q ˘ “ h ´ φ t ` h ´ p y q ˘¯ “ ψ t p y q “ ψ t ` h p x q ˘ , which proves that φ ˚˚˚ and ψ ˚˚˚ are conjugate parametrized dynamical systems. ONTINUATION SHEAVES IN DYNAMICS 33 Bifurcations and sheaf cohomology.
Sheaves attach both local and global data to atopological space. In our setting of continuation, they encode how dynamical structuresvary with parameter values on open sets. Oftentimes, given an open cover of the topo-logical space, one can glue together the local information on each element of the cover toobtain global information. However, sometimes local information fails to extend globally.Sheaf cohomology, which can be viewed as a generalization of singular cohomology, is apowerful tool for studying this. An interpretation for singular cohomology groups is thatthey constitute obstructions to a topological space being contractible. Sheaf cohomologygeneralizes this by representing barriers for local sections to extend to global sections.One can always solve an attractor’s continuation locally using an attracting neighbor-hood. But this problem is sometimes impossible globally . Sheaf cohomology provides aframework for quantifying when and how this occurs. Together with the conjugacy invari-ance theorem, this will build an algebraic invariant for parametrized dynamical systems,which can be used to study bifurcations. A description of how sheaf cohomology is con-structed, and then computed, is in Appendix D.Recall that a parametrized dynamical system on a topological space Λ is a continuousmap φ ˚˚˚ : Λ ! DS p T , X q such that φ ˚˚˚ p λ q : T ˆ X ! X is a dynamical system for all λ P Λ .In principal Λ may be DS p T , X q but in practice simpler topological spaces for Λ are used.In this section, to utilize Theorem 8.7, we assume X is a compact metric space.Definition 9.1. A parametrized dynamical system φ ˚˚˚ : Λ ! DS p T , X q is stable at a point λ P Λ if there exists an open neighborhood Λ Q λ such that φ ˚˚˚ ˇˇ Λ is conjugate to theconstant parametrization θ ˚˚˚ : Λ ! DS p T , X q , given by λ φ ˚˚˚ p λ q for all λ P Λ . If λ isnot stable, it is called a bifurcation point . A parametrized dynamical system φ ˚˚˚ is stable on asubset Λ Ă Λ if it is stable at every point in Λ Ă Λ .If a parametrized dynamical system φ ˚˚˚ : Λ ! DS p T , X q is conjugate to the constantparametrization θ ˚˚˚ : Λ ! DS p T , X q on Λ it is called uniformly stable .In general stability of a parametrized dynamical system does not imply uniform sta-bility. For instance if Λ is not connected then φ ˚˚˚ need not be conjugate to a fixed constantsystem θ ˚˚˚ . This example indicates that stability does not imply uniform stability in generalif Λ is disconnected. See Example 9.5 for an counter example with a connected space Λ .9.1. Locally constant sheaves.
Let φ ˚˚˚ : Λ ! DS p T , X q be a parametrized dynamical sys-tem. From the previous we have the induced attractor sheaf and free attractor sheaf over Λ : A φ ˚˚˚ : O p Λ q ! Ring , Att φ ˚˚˚ : O p Λ q ! Ring . The ringed space p Λ , A φ ˚˚˚ q and p Λ , Att φ ˚˚˚ q encode the continuation data of attractors forthe parametrized dynamical system. At a later stage we also include the attracting neigh-borhood sheaf and free attracting neighborhood sheaf N and ANbhd respectively.
Lemma 9.2.
Let θ ˚˚˚ : Λ ! DS p T , X q be a constant parametrization. Then, the sheaves A θ ˚˚˚ and Att θ ˚˚˚ are constant sheaves.Proof. The pullback ´etal´e space θ ´ ˚˚˚ Π r Att s is given by θ ´ ˚˚˚ Π r Att s “ Λ ˆ A , where A “ Att p φ λ q , for some λ P Λ , is given the discrete topology. Therefore the sheafof sections θ ´ ˚˚˚ S Att is a constant sheaf, cf. C.9. Consequently, A θ ˚˚˚ and Att θ ˚˚˚ are alsoconstant sheaves. Lemma 9.3.
Let φ ˚˚˚ : Λ ! DS p T , X q be stable. Then, the sheaves A φ ˚˚˚ and Att φ ˚˚˚ are locallyconstant sheaves.Proof. Pick a point λ P Λ . Since φ ˚˚˚ is stable there exists a neighborhood Λ Q λ such that φ ˚˚˚ | Λ is conjugate to the constant parametrization. By the Conjugacy Invariance Theoremin 8.7 we have that A φ ˚˚˚ | Λ – A θ ˚˚˚ | Λ as sheaves. The latter is a constant sheaf over Λ andtherefore A φ ˚˚˚ | Λ is a constant sheaf over Λ by definition. We conclude that A φ ˚˚˚ is locallyconstant. The same applies to Att φ ˚˚˚ . Remark 9.4. If φ ˚˚˚ is uniformly stable then φ ˚˚˚ is conjugate to a constant parametrization θ ˚˚˚ on Λ . The associated ´etal´e spaces are homemorphic by Theorem 8.7 and thus the sheaves A φ ˚˚˚ and Att φ ˚˚˚ are constant sheaves is this case. Example 9.5.
Let X be the 2-point compactification of the line and consider the followingfamily of differential equations x “ sin p x ` λ q , x P R , λ P S “ R { π Z . The above system defines a 1-parameter family of flows φ ˚˚˚ : Λ ! DS p R , X q of flows on X over parameter space Λ “ S . Via the conjugacy x x ´ λ we conclude that φ ˚˚˚ is stable andthus the attractor sheaf A φ ˚˚˚ is a locally constant sheaf as indicated by Lemma 9.3. Since ˘8 are not attractors, the only global sections in A φ ˚˚˚ are ∅ and X . The stalks of A φ ˚˚˚ areinfinite complete, atomic Boolean algebras which proves that A φ ˚˚˚ is not a constant sheaf.The above example shows that even if Λ is connected, then a stable system need notbe uniformly stable. Indeed, φ ˚˚˚ in Example 9.5 allows a conjugacy over Λ “ S , thenthe attractor sheaf A φ ˚˚˚ is constant which contradicts above statement that A φ ˚˚˚ is locallyconstant but not constant. A locally constant sheaf is the sheaf of sections of a coveringspace. With additional conditions on Λ such sheaves may be constant sheaves. ONTINUATION SHEAVES IN DYNAMICS 35
Proposition 9.6 (cf. [15], Prop. 4.20 and [40], Prop. 7.5) . Let Λ be a simply connected and locallypath connected topological space, and let F be a locally constant sheaf of rings on Λ . Then, F is aconstant sheaf. The same statement holds for contractible spaces Λ , cf. [21, Exer. II.4]. We can applythe above proposition to the attractor sheaf A φ ˚˚˚ and free attractor sheaf Att φ ˚˚˚ for simpleparametrized systems φ ˚˚˚ . Corollary 9.7.
Let φ ˚˚˚ : Λ ! DS p T , X q be stable and let Λ be a simply connected and locally pathconnected topological space. Then, A φ ˚˚˚ and Att φ ˚˚˚ are constant sheaves. For constant sheaves the sheaf cohomology can be related to singular cohomology whichis a useful tool in our treatment of bifurcations.
Proposition 9.8 (cf. [31], Thm. 9) . Let Λ be a locally contractible topological space and let R bean arbitrary ring. If R denotes the constant sheaf with values in R , then H k p Λ; R q – H k sing p Λ; R q for all k . If we combine Lemma 9.3, Corollary 9.7 and Proposition 9.8 we obtain a results that de-termines the sheaf cohomology of the attractor sheaves for simple parametrized dynamicalsystems.
Corollary 9.9.
Let φ ˚˚˚ : Λ ! DS p T , X q be stable and let Λ be a locally contractible and simplyconnected topological space. Then, H k p Λ; A φ ˚˚˚ q – H k sing p Λ; A φ ˚˚˚ λ q , @ k, where A φ ˚˚˚ λ P Ring is a stalk at any λ P Λ . A similar statement holds for H k p Λ; Att φ ˚˚˚ q .Proof. Lemma 9.3 implies that A φ ˚˚˚ is a constant sheaf. A locally contractible space is lo-cally simply connected and locally path connected, but not necessarily simply connected.In combination with the condition of simple connectedness we can combine Corollary 9.7and Proposition 9.8, which completes the proof.9.2. Sufficient conditions.
Contractible spaces are path connected and simply connectedbut not necessarily locally contractible. Contractible subsets of a manifold (or CW-complex) are locally contractible and simply connected which yields the following corol-lary. We refer such spaces a contractible manifolds (or contractible CW-complexes). These A topological space Λ is locally path connected at λ if for every neighborhood Λ Q λ there exists a pathconnected neighborhood Λ Ă Λ containing λ . A space is locally path connected if it is locally path connected atevery point, cf. [30, Sect. 25] A topological space Λ is locally contractible at λ if for every neighborhood Λ Q λ there exists a neighborhood Λ Ă Λ containing λ which is contractible in Λ . A space is locally contractible if it is locally contractible at everypoint, cf. [35, Exer. Ch. 1]. statements about sheaf cohomology imply the following sufficient condition for bifurca-tions to exist. We formulate the theorems for contractible manifolds but they can also beformulated for contractible CW-complexes. Moreover, the theorems stated for the attractorsheaf A φ ˚˚˚ can also be stated for the free attractor sheaf Att φ ˚˚˚ . Theorem 9.10.
Let Λ be a contractible manifold. Suppose that H k p Λ; A φ ˚˚˚ q ‰ , for some k ą . Then, there exist a bifurcation point in λ P Λ .Proof. Suppose there are no bifurcation points. This implies that φ ˚˚˚ is stable which byCorollary 9.9 implies that H k p Λ; A φ ˚˚˚ q – H k sing p Λ; R q for all k . For a contractible manifold(or contractible CW-complex) Λ we have that H k sing p Λ; R q “ for all k ą . Combiningthese statements give that H k p Λ; A φ ˚˚˚ q – H k sing p Λ; R q “ for all k ą , which contradictsthe above assumptions.As we will see in Section 10 the above criterion does not always detect bifurcations.In order to get a more in depth look into local bifucations we consider its relative sheafcohomology for A φ ˚˚˚ . We use the following lemma about long exact sequences in sheafcohomology. Lemma 9.11.
Let F be a sheaf of rings on Λ and let Λ i ã −! Λ . Assume that the induced homo-morhisms i k ˚ : H k p Λ; F q ! H k p Λ ; F q are isomorphisms for all k ě . Then, H k p Λ , Λ ; F q – , @ k ě . Proof.
For triple p Λ , ∅ q i ã −! p Λ , ∅ q j ã −! p Λ , Λ q we have the long exact sequence, H p Λ , Λ ; F q H p Λ; F q H p Λ ; F q H p Λ , Λ ; F q H p Λ; F q H p Λ ; F q ¨ ¨ ¨ , δ j ˚ i ˚ δ j ˚ i ˚ δ cf. Appendix D. For the exactness of the maps and the isomorphisms i k ˚ we have: ker j ˚ “ im δ “ , which proves that j ˚ is injective. Furthermore, since i ˚ is an isomorphism wehave ker i ˚ “ “ im j ˚ and thus H p Λ , Λ ; F q – . The remaining relative homologygroups are determined as follows: ker δ “ im i ˚ “ H p Λ ; F q – H p Λ; F q . Therefore, ker j ˚ “ im δ “ , which shows that j ˚ is injective. Furthermore, ker i ˚ “ “ im j ˚ ,consequently H p Λ , Λ ; F q – . The same argument can be repeated now for all other k . As an immediate consequence of the long exact sequence we have the following corol-lary if we apply Lemma 9.11 to the attractor sheaf A φ ˚˚˚ . ONTINUATION SHEAVES IN DYNAMICS 37
Corollary 9.12.
Suppose H k p Λ , Λ ; A φ ˚˚˚ q “ for some k . Then, there exist k ě for which theinclusion i does not imply an isomorphism i ˚ k : H k p Λ; A φ ˚˚˚ q ! H k p Λ ; A φ ˚˚˚ q . The relative sheaf cohomology can be used to formulate an analogous criterion as The-orem 9.10.
Theorem 9.13.
Let Λ be a contractible manifold and Λ Ă Λ be a submanifold which is a deforma-tion retract of Λ with φ ˚˚˚ stable on Λ . Suppose that H k p Λ , Λ ; A φ ˚˚˚ q ‰ , for some k ě . Then, there exist a bifurcation point in λ P Λ r Λ .Proof. Suppose there are no bifurcation points in Λ r Λ . This implies that φ ˚˚˚ is stableon Λ . Since Λ is a contractible manifold it is simply connected and locally path con-nected and it follows from Proposition 9.7 that A φ ˚˚˚ is a constant sheaf on Λ . Since Λ is a deformation retract of Λ the same holds for Λ and A φ ˚˚˚ | Λ – A φ ˚˚˚ . This implies that H p Λ; A φ ˚˚˚ q – H p Λ ; A φ ˚˚˚ q . By Corollary 9.9, since H k p Λ; R q – H k p Λ ; R q for all k , wehave that H k p Λ; A φ ˚˚˚ q – H k p Λ ; A φ ˚˚˚ q – for all k ě . Combining these statements gives H k p Λ; A φ ˚˚˚ q – H k p Λ ; A φ ˚˚˚ q for all k . This implies by Lemma 9.11 that H k p Λ , Λ ; A φ ˚˚˚ q – for all k , which contradicts the assumption that H k p Λ , Λ ; A φ ˚˚˚ q “ for some k . Therefore, φ ˚˚˚ is not simple on Λ r Λ and there exists a bifurcation point λ P Λ r Λ .10. Examples of one-parameter bifurcations.
In this section we discuss a number of stan-dard one-parameter bifurcations such as a saddle-node bifurcation and a pitchfork bifur-cation. We will also examine bifurcation at multiple bifurcation points. The objective isto show that sheaf cohomology picks up bifurcarions. At a later stage we will discuss themore practical side of computing sheaf cohomology from limited data.10.1.
One-parameter bifurcations at a single parameter value.
In this subsection we listthree fundamental bifurcations in one-parameter systems. We apply the above results tocompute the sheaf cohomology and to compare the criteria.For example if Λ “ R or Λ “ I , a bounded interval, then the above theorem applies.This is of interest for one-parameter bifurcations. The following lemma addresses the casewhere φ ˚˚˚ has one bifurcation point on R , which will assist in computations. Lemma 10.1.
Let F be a sheaf of rings on Λ “ R , such that F is a constant sheaf on both p´8 , λ q and p λ , for some λ P R . Then, F is acyclic, i.e. H k p Λ , F q “ for all k ě .Proof. Let ǫ ą and let B ǫ denote the interval p λ ´ ǫ, λ ` ǫ q . There is a restriction coho-momorphism r : F F ˇˇ B ǫ . We will show this induces an isomorphism of cohomology: r ˚ : H ˚ p R ; F q ! H ˚ ´ B ǫ ; F ˇˇ B ǫ ¯ . (18) First we address global sections. Because F is constant on p´8 , λ q and p λ , , sectionsin Γ p F ˇˇ B ǫ q extend uniquely to sections in Γ p F q . Thus, r ˚ : Γ p F q ! Γ p F ˇˇ B ǫ q is an iso-morphism. For k ą , H k p R ; F q and H k ´ B ǫ ; F ˇˇ B ǫ ¯ vanish, since intervals have coveringdimension 1, cf. [21, Lemma 2.7.3 and Proposition 3.2.2]. So the maps r ˚ k : H k p R ; F q ! H k ´ B ǫ ; F ˇˇ B ǫ ¯ are trivially isomorphisms. Now we consider k “ . Let R ˚ “ R r t λ u , so that B ǫ and R ˚ form a cover of R . Note that F ˇˇ R ˚ is locally constant, with vanishing higher cohomologygroups. There is a Mayer-Vietoris exact sequence: p F q Γ ´ F ˇˇ B ǫ ¯ ‘ Γ ´ F ˇˇ R ˚ ¯ Γ ´ F ˇˇ B ǫ X R ˚ ¯ H p R ; F q H ` B ǫ ; F ˇˇ B ǫ ˘ ‘ H ` R ˚ ; F ˇˇ R ˚ ˘ H ` B ǫ X R ˚ ; F ˇˇ B ǫ X R ˚ ˘ . α βδ Since H p R ; F ˇˇ R ˚ q and H p B ǫ X R ˚ ; F ˇˇ B ǫ X R ˚ q vanish the sequence simplifies to: p F q Γ ´ F ˇˇ B ǫ ¯ ‘ Γ ´ F ˇˇ R ˚ ¯ Γ ´ F ˇˇ B ǫ X R ˚ ¯ H p R ; F q H p B ǫ ; F ˇˇ B ǫ q . α βδr ˚ The map β is surjective, since the restriction from Γ ´ F ˇˇ R ˚ ¯ to Γ ´ F ˇˇ B ǫ X R ˚ ¯ is surjective.Following the sequence yields Im δ “ ker r ˚ “ . im r ˚ “ ker 0 “ H p B ǫ ; F ˇˇ B ǫ q , so r ˚ isalso surjective. This implies that the restriction cohomomorphism r : F F ˇˇ B ǫ inducesan isomorphism on cohomology and establishes (18). Indeed, for ǫ ă ǫ , the restrictioncohomomorphism from F ˇˇ B ǫ to F ˇˇ B ǫ is an isomorphism, again giving an isomorphism ofcohomology. So, H ˚ ` R ; F ˘ « lim −! ǫ ą H ˚ ` B ǫ ; F ˇˇ B ǫ ˘ . We can compute the limit using [4, Theorem 10.6]: lim −! ǫ ą H ˚ ` B ǫ ; F ˇˇ B ǫ ˘ « H ˚ ` t λ u ; F ˇˇ t λ u ˘ . Since F ˇˇ t λ u is flasque (restriction maps are surjective), it is acyclic, completing the proof.The same results hold for Λ “ I , a bounded, or semi-bounded interval. In the applica-tions below Λ is typically the real line. Lemma 10.2.
Let F be a sheaf of rings on Λ and let Λ i ã −! Λ . Assume that F and F | Λ areacyclic. If ONTINUATION SHEAVES IN DYNAMICS 39 (i) i ˚ : H p Λ; A φ ˚˚˚ q ! H p Λ ; A φ ˚˚˚ q is injective, then im i ˚ – H p Λ; F q and H p Λ , Λ ; F q – H p Λ ; F q im i ˚ , and H k p Λ , Λ ; F q – , for k “ (ii) i ˚ : H p Λ; A φ ˚˚˚ q ! H p Λ ; A φ ˚˚˚ q is surjective, then H p Λ , Λ ; F q – ker i ˚ , and H k p Λ , Λ ; F q – , for k “ . Proof.
As before for triple p Λ , ∅ q i ã −! p Λ , ∅ q j ã −! p Λ , Λ q we have the long exact sequence, H p Λ , Λ ; F q H p Λ; F q H p Λ ; F q H p Λ , Λ ; F q H p Λ; F q H p Λ ; F q ¨ ¨ ¨ . δ j ˚ i ˚ δ j ˚ i ˚ δ Since, by Lemma 10.1, F is acyclic we obtain the truncated sequence H p Λ , Λ ; F q H p Λ; F q H p Λ ; F q H p Λ , Λ ; F q . δ j ˚ i ˚ δ j ˚ (19)Since i ˚ is injective and thus ker i ˚ “ “ im j ˚ . Moreover, ker j ˚ “ im δ “ , whichimplies that H p Λ , Λ ; F q – . Consequently, we have the short exact sequence H p Λ; F q H p Λ ; F q H p Λ , Λ ; F q , j ˚ i ˚ δ j ˚ from which the result for H p Λ , Λ F q follows. The cohomology H k p Λ , Λ F q – , for k ě follows from Lemma 9.11, which completes the proof of (i).As for (ii) we have the truncated exact sequence in (19). Now i ˚ is surjective whichimplies that ker δ “ im i ˚ “ H p Λ ; F q . Therefore, ker j ˚ “ im δ “ and thus j ˚ isinjective. Consequently, H p Λ , Λ F q – . We now have the short exact sequence H p Λ , Λ ; F q H p Λ; F q H p Λ ; F q , δ j ˚ i ˚ δ which implies that H p Λ , Λ ; F q – ker i ˚ . The relative homology for k ě follows fromLemma 9.11.10.1.1. The pitchfork bifurcation.
Consider a parametrized dynamical system on X “ R Yt´8 , , the 2-point compactification of R , experiencing a pitchfork bifurcation, cf. Figure10.1.1. The parametrized flow is defined via the differential equation x “ λx ´ x , x P R , λ P R . Before the bifurcation point at λ “ there are two repelling fixed points at `8 and ´8 and a single attracting fixed point at x “ . After λ “ , there are two additional at-tracting fixed points x “ ˘ x λ and x “ has changed to a repelling fixed point. We fix aparametrization: ζ ˚˚˚ : Λ ! DS p T , X q , Λ0 X NW F IGURE
2. In the pitchfork bifurcation, the section on W Ă Λ defined by σ p λ q “ ` λ, φ λ , ω φ λ p N q ˘ fails to extend globally.where Λ “ R is parameter space, T “ R ` is the time space and X is the 2-point compacti-fication of R . Proposition 10.3.
Let Λ : “ r a, . If a ą , then H k p Λ , Λ ; A ζ ˚˚˚ q – Z for k “ , and H k p Λ , Λ ; A ζ ˚˚˚ q “ otherwise. When a ď , all relative cohomology groups vanish.Proof. The global sections are H p Λ; A ζ ˚˚˚ q – Γ ` A ζ ˚˚˚ ˘ – Z . For a ą , we have H p Λ ; A ζ ˚˚˚ q – Γ ´ A ζ ˚˚˚ ˇˇ Λ ¯ – Z , and injectivity of i ˚ , which by Lemma 10.2(i) yields H p Λ , Λ ; A ζ ˚˚˚ q – Z . For a ď , we have H p Λ ; A ζ ˚˚˚ q – Γ ´ A ζ ˚˚˚ ˇˇ Λ ¯ – Z , which implies H p Λ , Λ ; A ζ ˚˚˚ q “ . Since by Lemma 10.1 both A ζ ˚˚˚ and A ζ ˚˚˚ | Λ are acyclic, the higherorder relative cohomology groups vanish by Lemma 10.2(i). Proposition 10.4.
Let Λ : “ p´8 , a s . Then, H k p Λ , Λ ; A ζ ˚˚˚ q – for all k and for all a P R .Proof. Note that Γ ` A ζ ˚˚˚ ˘ – Γ ´ A ζ ˚˚˚ ˇˇ Λ ¯ – Z for all a P R . Therefore, H p Λ; A ζ ˚˚˚ q – H p Λ ; A ζ ˚˚˚ q for all a P R and thus by Lemma 10.2(i) H k p Λ , Λ ; A ζ ˚˚˚ q – for all k . Theorem 10.5.
Let φ ˚˚˚ be a parametrized dynamical system over Λ conjugate to the above canonicalparametrization ζ ˚˚˚ for the pitchfork bifurcation. Then, A φ ˚˚˚ is acyclic and H p Λ; A φ ˚˚˚ q – Z . Moreover, there exists a value λ P R such that H k ` Λ , Λ ; A φ ˚˚˚ ˘ – $&% Z if k “ and a ą λ ;0 if k “ or a ď λ , If Λ does not contain a bifurcation point then acyclicity follows from the fact that A ζ ˚˚˚ | Λ is a constant sheafon a contractible manifold. ONTINUATION SHEAVES IN DYNAMICS 41 where Λ “ r a, , Furthermore, for Λ : “ p´8 , a s , then H k ` Λ , Λ ; A φ ˚˚˚ ˘ – for all k and for all a P R .Proof. This follows immediately from Theorem 8.7, Lemma 10.1, and Propositions 10.3 and10.4.This theorem can be applied locally in parameter space. If φ ˚˚˚ : R ! DS p R ; I q is someparametrized dynamical system such that φ ˚˚˚ experiences a pitchfork bifurcation on anopen set U , then A φ ˚˚˚ ˇˇ U has the above cohomology groups. Another important observa-tion is that the relative cohomology in the example below is the same for a local pitchforkbifuraction. Example 10.6.
Let φ ˚˚˚ be a parametrized flow over Λ “ R on the interval X “ r´ , s with a single attracting fixed point at x “ for λ ď . This system is a semi-flow with T “ R ` . For λ ě the system undergoes a pitchfork bifurcation with two branches ˘ x λ of attracting fixed points converging to ˘ respectively as λ ! `8 , cf. Figure 10.1.1. If werepeat the analysis in Propositions 10.3 and 10.4 the sheaf cohomology over Λ is different: A φ ˚˚˚ is acyclic and H p Λ; A φ ˚˚˚ q – Z . On the other hand the relative sheaf cohomologies H k ` R , r a, ; A φ ˚˚˚ ˘ and H k ` R , p´8 , a s ; A φ ˚˚˚ ˘ are the same.10.1.2. The saddle-node bifurcation.
Consider a parametrized dynamical system on X “ R Yt´8 , , the 2-point compactification of R , experiencing a saddle-node bifurcation. Theparametrized flow is defined via the differential equation x “ λ ´ x , x P R , λ P R , and `8 and ´8 are a repelling and attracting fixed point respectively. Before the bifur-cation point at λ “ , the entire interval flows from `8 to ´8 . After λ “ , there is anadditional attracting and repelling fixed point. We fix a parametrization: ψ ˚˚˚ : Λ ! DS p T , X q , where Λ “ R is parameter space, T “ R is the time space and X is the 2-point compacti-fication of R . Lemma 10.1 again shows that the attractor sheaf A ψ ˚˚˚ has vanishing higherorder cohomology, but relative cohomology recognizes the bifurcations. Proposition 10.7.
Let Λ “ r a, . If a ą , then H k p Λ , Λ ; A ψ ˚˚˚ q – Z for k “ , and vanishesotherwise. When a ď , then H k p Λ , Λ ; A ψ ˚˚˚ q “ for all k .Proof. The global sections are H p Λ; A ψ ˚˚˚ q – Γ ` A ψ ˚˚˚ ˘ – Z . For a ą , we have H p Λ ; A ψ ˚˚˚ q – Γ ´ A ψ ˚˚˚ ˇˇ Λ ¯ – Z . The injectivity of i ˚ and Lemma 10.2(i) yields H p Λ , Λ ; A ψ ˚˚˚ q – Z . For a ď , we have H p Λ ; A ψ ˚˚˚ q – Γ ´ A ψ ˚˚˚ ˇˇ Λ ¯ – Z , which im-plies H p Λ , Λ ; A ψ ˚˚˚ q “ . As before the higher order relative cohomology groups vanishby Lemma 10.2(i). F IGURE
3. A saddle-node bifurcation.
Proposition 10.8.
Let Λ “ p´8 , a s . If a ą , then H k p Λ , Λ ; A ψ ˚˚˚ q – for all k . When a ă ,then H p Λ , Λ ; A ψ ˚˚˚ q – Z and vanishes otherwise.Proof. As before the global sections are H p Λ; A ψ ˚˚˚ q – Γ ` A ψ ˚˚˚ ˘ – Z . For a ą , wehave H p Λ ; A ψ ˚˚˚ q – Γ ´ A ψ ˚˚˚ ˇˇ Λ ¯ – Z . The injectivity of i ˚ and Lemma 10.2(i) yields H p Λ , Λ ; A ψ ˚˚˚ q – . For a ă , we have H p Λ ; A ψ ˚˚˚ q – Γ ´ A ψ ˚˚˚ ˇˇ Λ ¯ – Z . The surjectivityof i ˚ and Lemma 10.2(ii) then implies that H p Λ , Λ ; A ψ ˚˚˚ q – Z . The higher order relativecohomology groups vanish by Lemma 10.2(i) and (ii). Theorem 10.9.
Let φ ˚˚˚ be a parametrized dynamical system over Λ conjugate to the above canonicalparametrization ψ ˚˚˚ for the saddle-node bifurcation. Then, A φ ˚˚˚ is acyclic and H p Λ; A φ ˚˚˚ q – Z . Moreover, there exists a value λ P R such that H k p Λ , Λ ; A φ ˚˚˚ q – $&% Z if k “ and a ą λ k “ , or a ď λ , with Λ “ r a, ,H k p Λ , Λ ; A φ ˚˚˚ q – $&% Z if k “ and a ă λ k “ , or a ě λ , with Λ “ p´8 , a s . Proof.
Apply Theorem 8.7, Lemma 10.1 and Propositions 10.7 and 10.8.
Remark 10.10.
The generator of H k ` R , p´8 , a s ; A ψ ˚˚˚ ˘ when a ă λ is the sum of twoglobal sections of attractors: the bottom fixed point and the maximal attractor. The twocoincide before the bifurcation point, which leaves their sum zero. Afterwards, however,this corresponds to the Morse set between the top two fixed points. Continuation of thisMorse set to the empty set via a global section yields nontrivial relative cohomology. ONTINUATION SHEAVES IN DYNAMICS 43
Example 10.11.
Consider a saddle-node bifurcation in the system described in Figure10.1.2. We impose an attracting fixed point at the bottom of Figure 10.1.2, such that wemay restrict phase space to a forward-invariant compact interval X “ r , s . Call thisparametrized dynamical system φ ˚˚˚ : R ! DS p R ` , X q . Lemma 10.1 again shows that A φ ˚˚˚ has vanishing higher cohomology. However, H p Λ; A φ ˚˚˚ q – Z which differs from theabove example. The relative cohomology groups are the same as in the above example asis the case for the pitchfork bifurcation.10.1.3. The transcritical bifurcation.
Consider a parametrized dynamical system on X “ R Yt´8 , , the 2-point compactification of R , experiencing a transcritical bifurcation. Theparametrized flow is defined via the differential equation x “ λx ´ x , x P R , λ P R , and `8 and ´8 are a repelling and attracting fixed points respectively. As before we fix aparametrization: η ˚˚˚ : Λ ! DS p T , X q , where Λ “ R is parameter space, T “ R is the time space and X is the 2-point compacti-fication of R . Lemma 10.1 again shows that the attractor sheaf A η ˚˚˚ has vanishing higherorder cohomology. Proposition 10.12.
Let Λ “ r a, . If a ą , then H k p Λ , Λ ; A η ˚˚˚ q – Z for k “ , and vanishesotherwise. When a ď , then H k p Λ , Λ ; A η ˚˚˚ q “ for all k .Proof. The global sections are H p Λ; A η ˚˚˚ q – Γ ` A η ˚˚˚ ˘ – Z . For a ą , we have H p Λ ; A η ˚˚˚ q – Γ ´ A η ˚˚˚ ˇˇ Λ ¯ – Z . The injectivity of i ˚ and Lemma 10.2(i) yields H p Λ , Λ ; A η ˚˚˚ q – Z . For a ď , we have H p Λ ; A η ˚˚˚ q – Γ ´ A η ˚˚˚ ˇˇ Λ ¯ – Z , which im-plies H p Λ , Λ ; A η ˚˚˚ q “ . As before the higher order relative cohomology groups vanishby Lemma 10.2(i). Proposition 10.13.
Let Λ “ p´8 , a s . If a ě , then H k p Λ , Λ ; A η ˚˚˚ q – for all k . When a ă ,then H p Λ , Λ ; A η ˚˚˚ q – Z and vanishes otherwise.Proof. As before the global sections are H p Λ; A η ˚˚˚ q – Γ ` A η ˚˚˚ ˘ – Z . For a ě , wehave H p Λ ; A η ˚˚˚ q – Γ ´ A η ˚˚˚ ˇˇ Λ ¯ – Z . The injectivity of i ˚ and Lemma 10.2(i) yields H p Λ , Λ ; A η ˚˚˚ q – . For a ă , we have H p Λ ; A η ˚˚˚ q – Γ ´ A η ˚˚˚ ˇˇ Λ ¯ – Z . The injectivityof i ˚ and Lemma 10.2(i) then implies that H p Λ , Λ ; A η ˚˚˚ q – Z . The higher order relativecohomology groups vanish by Lemma 10.2(i). Theorem 10.14.
Let φ ˚˚˚ be a parametrized dynamical system over Λ conjugate to the above canon-ical parametrization for the transcritical bifurcation. Then, A φ ˚˚˚ is acyclic and H p Λ; A φ ˚˚˚ q – Z . F IGURE
4. An S-shaped bifurcation.
Moreover, there exists a value λ P R such that H k p Λ , Λ ; A φ ˚˚˚ q – $&% Z if k “ and a ą λ k “ , or a ď λ , with Λ “ r a, ,H k p Λ , Λ ; A φ ˚˚˚ q – $&% Z if k “ and a ă λ k “ , or a ě λ , with Λ “ p´8 , a s . Proof.
Apply Theorem 8.7, Lemma 10.1 and Propositions 10.7 and 10.8.
Remark 10.15.
Note the subtle difference in the relative sheaf cohomology for the saddle-node and transcritical bifurcations. For the latter we only find relative cohomology at k “ for different choices of Λ , as for the saddle-node we have cohomology at k “ and k “ for various choices of Λ .10.2. One-parameter bifurcations at multiple bifurcation points.
In this subsection weconsider a bifurcation that occur at multiple points.10.2.1.
The S-shaped bifurcation.
Now we study the S-shaped bifurcation, as in Figure 10.2.1.Consider a parametrized dynamical system on X “ R Y t´8 , , the 2-point compacti-fication of R , experiencing an S-shaped bifurcation. The parametrized flow is defined viathe differential equation x “ λ ` x ´ x , x P R , λ P R , and `8 and ´8 are a repelling fixed points. As before we fix a parametrization: ξ ˚˚˚ : Λ ! DS p T , X q , where Λ “ R is parameter space, T “ R is the time space and X is the 2-point compactifi-cation of R . Here, there are two bifurcation points at λ “ ´ and λ “ ` . Proposition 10.16. A ξ ˚˚˚ is acyclic. ONTINUATION SHEAVES IN DYNAMICS 45
Proof.
Pick λ ă a ă b ă λ , such that Λ : “ p´8 , b s and Λ : “ r a, cover R . Considerthe Mayer-Vietoris exact sequence: p A ξ ˚˚˚ q Γ ´ A ξ ˚˚˚ ˇˇ Λ ¯ ‘ Γ ´ A ξ ˚˚˚ ˇˇ Λ ¯ Γ ´ A ξ ˚˚˚ ˇˇ r a,b s ¯ H p Λ; A ξ ˚˚˚ q H ´ Λ ; A ξ ˚˚˚ ˇˇ Λ ¯ ‘ H ´ Λ ; A ξ ˚˚˚ ˇˇ Λ ¯ H ´ r a, b s ; A ξ ˚˚˚ ˇˇ r a,b s ¯ , δ α ˚ β ˚ δ α ˚ β ˚ since H p R ; A ξ ˚˚˚ q – , which uses the fact that intervals have covering dimension 1, cf. [21,Lemma 2.7.3 and Proposition 3.2.2]. We can compute the global sections: Γ ` A ξ ˚˚˚ ˘ – Z , Γ ´ A ξ ˚˚˚ ˇˇ Λ ¯ – Γ ´ A ξ ˚˚˚ ˇˇ Λ ¯ – Z , Γ ´ A ξ ˚˚˚ ˇˇ r a,b s ¯ – Z . Since im δ – and ker δ – we have ker α ˚ “ im δ – . Consequently, im α ˚ – Z .Similarly, ker β ˚ “ im α ˚ – Z which implies that im β ˚ – Z . Furthermore, ker δ “ im β ˚ – Z and thus im δ – . Since Λ , Λ both contain only one bifurcation point, wecan apply Lemma 10.1 to conclude that H ´ Λ ; A ξ ˚˚˚ ˇˇ Λ ¯ and H ´ Λ ; A ξ ˚˚˚ ˇˇ Λ ¯ vanish forall k ě . Hence, im δ “ ker α ˚ “ H p Λ; A ξ ˚˚˚ q , which proves that H p R , A ξ ˚˚˚ q is zero. Theremaining sheaf cohomology vanishes due to the dimension restriction on Λ .The S-shaped bifurcation is an example where A ξ ˚˚˚ and Att ξ ˚˚˚ , the attractor sheaf andfree attractor sheaf respectively, have differing cohomologies. Proposition 10.17.
Let
Att ξ ˚˚˚ be the free attractor sheaf associated to ξ ˚˚˚ . H k p Λ; Att ξ ˚˚˚ q – $’’’&’’’% Z if k “ Z if k “ if k ě Proof.
Let Λ “ p´8 , λ q and Λ “ p λ , be an open covering for Λ “ R . We build the ordered ˇCech complex from this cover: C pt Λ , Λ u ; Att ξ ˚˚˚ q C pt Λ , Λ u ; Att ξ ˚˚˚ q C pt Λ , Λ u ; Att ξ ˚˚˚ q . . . δ δ δ , which in our case is: Att ξ ˚˚˚ p Λ q ‘ Att ξ ˚˚˚ p Λ q Att ξ ˚˚˚ p Λ X Λ q , ρ , ´ ρ , where ρ , denotes the restriction map from Att ξ ˚˚˚ p Λ q to Att ξ ˚˚˚ p Λ X Λ q , and ρ , from Att ξ ˚˚˚ p Λ q . We get cohomology groups from the above chain complex: ˇ H pt Λ , Λ u ; Att ξ ˚˚˚ q “ ker δ – Z , ˇ H pt Λ , Λ u ; Att ξ ˚˚˚ q “ ker δ { Im δ – Z , ˇ H k pt Λ , Λ u ; Att ξ ˚˚˚ q “ for k ą . Since Λ X Λ contains no bifurcation points, Att ξ ˚˚˚ is locally constant on Λ X Λ and there-fore acyclic on Λ X Λ . We now use Leray’s Theorem to determine the sheaf cohomologyof Att ξ ˚˚˚ , cf. Appendix D, which yields the desired result.The result of Proposition 10.17 is an example where Theorem 9.10 applies. The shaefcohomology og Att ξ ˚˚˚ picks up bifurcations. For the sheaf cohomology of A ξ ˚˚˚ Theorem9.10 does not apply.
Proposition 10.18.
Let Λ “ r a, . If a P p λ , λ s , then H k p Λ , Λ ; A ξ ˚˚˚ q – Z for k “ , andvanishes otherwise. When a R p λ , λ s , then H k p Λ , Λ ; A ξ ˚˚˚ q “ for all k .Proof. We achieve a truncated long exact sequence from Proposition 10.16 and Proposition9.3: H p Λ , Λ ; A ξ ˚˚˚ q H p Λ; A ξ ˚˚˚ q H p Λ ; A ξ ˚˚˚ q H p Λ , Λ ; A ξ ˚˚˚ q . j ˚ i ˚ δ The map i ˚ is injective and thus H p Λ , Λ ; A ξ ˚˚˚ q – . Lemma 10.2 then yields: H p Λ , Λ ; A ξ ˚˚˚ q – H p Λ ; A ξ ˚˚˚ q im i ˚ Note that im i ˚ – H p Λ; A ξ ˚˚˚ q “ Γ ` A ξ ˚˚˚ ˘ – Z . For b P p λ , λ s , we have H p Λ ; A ξ ˚˚˚ q “ Γ ´ A ξ ˚˚˚ ˇˇ Λ ¯ – Z , which implies H p Λ , Λ ; A ξ ˚˚˚ q – Z . Otherwise, H p Λ , Λ ; A ξ ˚˚˚ q “ . Proposition 10.19.
Let Λ “ p´8 , a s . If a P r λ , λ q , then H k p Λ , Λ , A ξ ˚˚˚ q – Z for k “ , andis zero otherwise. When a R r λ , λ q , then H k p Λ , Λ ; A ξ ˚˚˚ q “ for all k .Proof. An identical argument as in the proof of Proposition 10.18.
Theorem 10.20.
Let φ ˚˚˚ be a parametrized dynamical system conjugate to the above parametriza-tion ξ ˚˚˚ of the S-shaped bifurcation. Then, A φ ˚˚˚ is acyclic and H p Λ; A φ ˚˚˚ q – Z . Moreover, there exists λ , λ P R such that H k p Λ , Λ ; A φ ˚˚˚ q – $&% Z if k “ and a P p λ , λ s otherwise. with Λ “ r a, ,H k p Λ , Λ ; A φ ˚˚˚ q – $&% Z if k “ and a P r λ , λ q otherwise. with Λ “ p´8 , a s . Proof.
Apply Theorem 8.7, Proposition 10.16, and Propositions 10.18 and 10.19.
Remark 10.21.
Note that the relative cohomologies H k p Λ , Λ ; A φ ˚˚˚ q are the same for thetrans-critical and S-shaped bifurcation. ONTINUATION SHEAVES IN DYNAMICS 47
Remark 10.22.
If we consider the S-shaped bifurcation on an interval X “ I “ r´ c, c s , c " , with time space T “ R ` and parameter space Λ “ r´ λ , λ s , with λ “ ´ c ` c ´ ǫ , ă ǫ ! we obtain the following sheaf cohomology: A φ ˚˚˚ is acyclic and H p Λ; A φ ˚˚˚ q – Z . Moreover, there exists a value λ P R such that H k p Λ , Λ ; A φ ˚˚˚ q – $&% Z if k “ and a P p λ , λ s otherwise. with Λ “ r a,
8q X Λ ,H k p Λ , Λ ; A φ ˚˚˚ q – $&% Z if k “ and a P r λ , λ q otherwise. with Λ “ p´8 , a s X Λ . For free attractor sheaf we have: H k p Λ; Att ξ ˚˚˚ q – $’’’&’’’% Z if k “ Z if k “ if k ě , which is clearly not acyclic.10.3. Comparing the attractor and free attractor sheaves.
In the above treatment of thepitchfork, the saddle-node and transcritical bifurcations we have only used the attractorsheaf. If we consider the same examples using the free attractor sheaf the cohomologygroups will be different only for the cohomology groups that are non-zero. We can illus-trate this best by looking at the saddle-node bifurcation as given in Propositions 10.7 and10.8. Let ψ ˚˚˚ : Λ ! DS p T , X q , be the parametrized system for the saddle-node bifurcation as given in 10.1.2, where Λ “ R is parameter space, T “ R is the time space and X is the 2-point compactification of R .For the free attractor sheaf we have: Att ψ ˚˚˚ is acyclic and H p Λ; Att ψ ˚˚˚ q – Z . Moreover, H k p Λ , Λ ; Att ψ ˚˚˚ q – $&% Z if k “ and a ą k “ , or a ď , with Λ “ r a, ,H k p Λ , Λ ; Att ψ ˚˚˚ q – $&% Z if k “ and a ă k “ , or a ě , with Λ “ p´8 , a s . The cohomologies are different but H k p Λ , Λ ; Att ψ ˚˚˚ q is trivial if and only if H k p Λ , Λ ; A ψ ˚˚˚ q is trivial. REFERENCES [1] Z. Arai, W. Kalies, H. Kokubu, K. Mischaikow, H. Oka, and P. Pilarczyk. A database schema for the analysisof global dynamics of multiparameter systems.
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Appendix A. Functorial properties of attractors.
Proof of Lemma 3.2.
For t ě , we have φ t p f ´ p U qq Ă p f ´ ˝ f ˝ φ t qp f ´ p U qq . Since f is aquasiconjugacy, we have p f ´ ˝ f ˝ φ t qp f ´ p U qq “ f ´ p ψ : t pp f ˝ f ´ qp U qqq Ă f ´ p ψ : t p U qq and thus φ t p f ´ p U qq Ă f ´ p ψ : t p U qq , @ t ě . The inequality for ω now follows from elementary properties of inverse images and clo-sures: ω φ p f ´ p U qq “ č t ě cl ď s ě t φ s ` f ´ p U q ˘ Ă č t ě cl ď s ě t f ´ ` ψ : s p U q ˘ “ č t ě cl f ´ ´ ď s ě t ψ : s p U q ¯ Ă č t ě f ´ ´ cl ď s ě t ψ : s p U q ¯ “ f ´ ˆ č t ě cl ď s ě t ψ : s p U q ˙ “ f ´ ˆ č t ě cl ď x P U ď σ ě τ p t,x q ψ σ p x q ˙ “ f ´ ˆ č τ ě cl ď σ ě τ ψ σ p U q ˙ “ f ´ p ω ψ p U qq , which uses the invertibility of the parametrization function τ . Finally applying ω φ weobtain ω φ p f ´ p U qq “ ω φ p ω φ p f ´ p U qqq Ă ω φ p f ´ p ω ψ p U qqq Ă ω φ p f ´ p U qq so that ω φ p f ´ p U qq “ ω φ p f ´ p ω ψ p U qqq , (20)which completes the proof. Proof of Remark 3.3.
To deal with negative times we define τ p´ t, x q : “ τ p t, x q in which case ψ :´ t “ ψ ` τ p´ t, ¨q , ¨ ˘ “ ψ ` ´ τ p t, ¨q , ¨ ˘ “ ` ψ : t ˘ ´ . Let x P φ ´ t p f ´ p U qq so that φ t p x q P f ´ p U q . Then, by the quasiconjugacy condition f p φ t p x qq “ ψ : t p f p x qq P U , and therefore f p x q P ψ :´ t p U q . This yields x P f ´ p ψ :´ t p U qq .Summarizing we have φ ´ t p f ´ p U qq Ă f ´ p ψ :´ t p U qq , @ t ě . The remainder of the proof is similar to the proof of Lemma 3.2.
Proof of Proposition 3.4.
Since A is an attractor for ψ , there exists an attracting neighborhood U such that ω ψ p U q “ A . By Eqn. (20) we have ω φ ` f ´ p U q ˘ “ ω φ ` f ´ p ω ψ p U qq ˘ “ ω φ ` f ´ p A q ˘ , which proves that ω φ ` f ´ p A q ˘ is an attractor for φ , since we already know f ´ p U q is anattracting neighborhood for φ .Therefore, for a quasiconjugacy τ ˆ f P hom p φ, ψ q , the map ω φ ˝ f ´ : Att p ψ q ! Att p φ q iswell defined. It remains to show that the latter is a lattice homomorphism. Preservation ofjoins is clear, cf. Property (v) for omega limit sets. Let A, A P Att p ψ q , then ω φ p f ´ p A ^ A qqq “ ω φ p f ´ p ω ψ p A X A qqq Ă ω φ p f ´ p A X A qq “ ω φ ` f ´ p A q X f ´ p A q ˘ “ ω φ ` ω φ ` f ´ p A q X f ´ p A q ˘˘ Ă ω φ ` ω φ p f ´ p A qq X ω φ p f ´ p A qq ˘ “ ω φ p f ´ p A qq ^ ω φ p f ´ p A qq Idempotency of ω φ and Equation (1) imply ω φ p f ´ p A qq ^ ω φ p f ´ p A qq “ ω φ ` ω φ p f ´ p A qq X ω φ p f ´ p A qq ˘ Ă ω φ ` f ´ p ω ψ p A qq X f ´ p ω ψ p A qq ˘ “ ω φ ` f ´ p ω ψ p A q X ω ψ p A qq ˘ “ ω φ p f ´ p A q X f ´ p A qq “ ω φ p f ´ p A X A qq“ ω φ ` ω φ p f ´ p A X A qq ˘ Ă ω φ p f ´ p ω ψ p A X A qqq“ ω φ p f ´ p A ^ A qqq , which proves that ω φ ` f ´ p A ^ A qq ˘ “ ω φ p f ´ p A qq ^ ω φ p f ´ p A qq , and thus ω φ ˝ f ´ : Att p ψ q ! Att p φ q is a lattice homomorphism. Proof of Remark 3.5. If τ ˆ f P hom p φ, ψ q is a conjugacy, then f ` φ t p x q ˘ “ ψ : t ` f p x q ˘ . (21)Define y “ f p x q and s “ τ ` t, f ´ p y q ˘ . Since f is a homeomorphism, we obtain τ ´ p s, y q ,and therefore φ : s ` f ´ p y q ˘ “ f ´ ` ψ s p y q ˘ , (22)where φ : s “ ψ ` τ ´ p s, ¨q , ¨ ˘ . This proves that τ ´ ˆ f ´ P hom p ψ, φ q is a conjugacy. ONTINUATION SHEAVES IN DYNAMICS 51
Let A P Att p ψ q , then by Proposition 3.4, we have ω φ ` f ´ p A q ˘ P Att p φ q . By Equation (22)we have φ : s ` f ´ p A q ˘ “ f ´ ` ψ s p A q ˘ “ f ´ p A q for all s ě , which proves invariance of f ´ p A q . Furthermore, since f is a homeomorphism, it follows that f ´ p A q is closed, andthus ω φ ` f ´ p A q ˘ “ f ´ p A q , which proves that f ´ p A q P Att p φ q . Similarly, f p A q P Att p ψ q for all A P Att p φ q . Proof of Remark 4.2.
Given a morphism h : φ ! φ , by the functoriality of the diagrams in(3), we have the commutative diagram E p φ q E p φ q G p φ q G p φ q (cid:15) (cid:15) w φ (cid:15) (cid:15) w φ o o E p h q o o G p h q which implies: G p h q ` w φ p U q ˘ “ w φ ` E p h qp U q ˘ . By the condition E p h qp U q “ U we obtain G p h q ` w φ p U q ˘ “ w φ ` U ˘ . A morphism h : φ ! φ in Φ r E ; U s satisfies E p h qp U q “ U , andtherefore G p h q ` w φ p U q ˘ “ w φ ` U ˘ , which proves that Θ r w ; U sp h q defined by ` φ, w φ p U q ˘ ! ` φ , w φ p U q ˘ is indeed a morphismin Π r G s . Appendix B. Function spaces and the compact-open topology.
We recall some basic factsabout topologies on function spaces of continuous functions. Let X and Y be arbitrarytopological spaces and let C p X, Y q the denote the set of all continuous maps f : X ! Y . Atopology on C p X, Y q which is of particular importance is the compact-open topology whichis defined a subbasis of sets of the form O p K, U q : “ f | f p K q Ă U for K compact in X and U open in Y ( , where K ranges over all compact subsets in X and U ranges over all open subsets in Y ,cf. [11]. If X is a locally compact, Hausdorff space then the compact-open topology is theweakest topology such that the map p f, x q f p x q , f P C p X, Y q , is continuous, cf. [27,Cor. 1.2.4]. If X is compact and Y is a metric space with metric d , then the compact-opentopology corresponds with the metric topology on C p X, Y q given by the metric: d p f, g q “ sup x P X d p f p x q , g p x qq , f, g P C p X, Y q , cf. [29, 34].Let Λ be an arbitrary topological space. For a continuous map h : Λ ˆ X ! Y we definethe transpose of h by: h ˚˚˚ : Λ ! C p X, Y q , λ h ˚˚˚ p λ q “ h λ : “ h p λ, ¨q . Following the terminology in [10] we say that a topology on C p X, Y q is weak if continu-ity of h implies continuity of the transpose h ˚˚˚ , and a topology is strong if continuity ofthe transpose h ˚˚˚ implies continuity of h . For arbitrary topological spaces X, Y and Λ thecompact-open topology is a weak topology on C p X, Y q , i.e. h continuous implies that h ˚˚˚ is continuous, cf. [11, Lemma 1],[29]. If X is regular and locally compact (in particularfor locally compact, Hausdorff spaces), then the compact-open topology is is both weakand strong, i.e. h is continuous if and only if h ˚˚˚ is continuous, cf. [11, Theorem 1], [29].This implies that for regular and locally compact spaces X the compact-open topology on C p X, Y q is both weak and strong, which is also referred to as an exponential topology , cf.[10]. The latter is unique. Finally, the map h h ˚˚˚ is an embedding when both Λ and X are Hausdorff spaces. The map is a homeomorphism when Λ is Hausdorff and X is locallycompact, Hausdorff, cf. [29].For a compact metric space p X, d q define ` H p X q , d H ˘ to be the metric space of compactsubsets of X equipped with the Hausdorff metric d H . Every continuous function f : X ! Y induces a continuous function f H : H p X q ! H p Y q , which sends compact subsets to theirimage under the function f . Recall the Hausdorff metric: d H p K, K q : “ max " sup x P K inf x P K d p x, x q , sup x P K inf x P K d p x, x q * , K, K P H p X q . Lemma B.1.
Let X , Y be compact metric spaces, Λ a topological space, and h : Λ ˆ X ! Y continuous map. Then, the function h H : Λ ˆ H p X q ! H p Y q p λ, K q h H ` t λ u ˆ K ˘ is continuous.Proof. We will first prove the assignment D : C p X, Y q ! C p H p X q , H p Y qq f D p f q : “ f H , is continuous. Let f, g P C p X, Y q . Then, d C p f, g q “ sup x P X d p f p x q , g p x qq and d C H p f H , g H q “ sup K P H p X q d H ` f p K q , g p K q ˘ . Since d p y, y q ď sup x P X d p f p x q , g p x qq “ d p f, g q for any choice of y P f p K q , y P g p K q it fol-lows that d C H p f H , g H q ď d C p f, g q . Moreover, since points are compact subsets, the reversedinequality holds as well: d C p f, g q “ sup x P X d H ` f pt x uq , g pt x uq ˘ ď d C H p f H , g H q , which proves that D is an isometry implying its continuity. The metric topology on C p H p X q , H p Y qq coincides with the compact-open topology and therefore h H is continuous ONTINUATION SHEAVES IN DYNAMICS 53 if and only if its transpose h ˚˚˚ H , given by h ˚˚˚ H : Λ ! C ` H p X q , H p Y q ˘ , λ h ˚˚˚ H p λ q “ h λ H : “ h H ` t λ u , ¨ ˘ , is continuous, [11, 10]. Note that h ˚˚˚ H “ D ˝ h ˚˚˚ which proves that h ˚ H is continuous, whichcompletes the proof. Remark B.2.
In this paper we abuse notation by writing h p K q , K P H p X q denoting h H p K q in accordance with the analogous notation for h p U q “ t y “ h p x q | x P U u , U Ă X . Appendix C. Basic sheaf theory.
In this section we outline the basic definitions ofpresheaves and sheaves as well as the standard sheafification procedure. Consider a topo-logical space X and O p X q be the poset of open subsets treated as small category where themorphisms are inclusions of sets: posetal category. Definition C.1. A presheaf on a topological space X with values in a category C is a con-travariant functor F : O p X q ! C . To be more explcit F is characterized by the followingingredients:(i) for every open subset U Ă X there is an object F p U q in C — sections —;(ii) for every pair of open sets V Ď U in X , there are restriction morphisms ρ V,U : F p U q ! F p V q , which satisfy(a) for all open sets U in X , it holds that ρ U,U “ id F p U q ;(b) for all triples of open set W Ď V Ď U in X it holds that ρ W,U “ ρ W,V ˝ ρ V,U . Remark C.2.
Besides the notation F p U q to indicate sections, there is an alternative notationthat is used frequently: Γ p F , U q . The elements in F p U q are sections of F over U and aredenoted by σ P F p U q . In this notation the restriction ρ V,U σ is also denoted by σ | V .Another notion that is important in the theory of presheaves and sheaves is that of stalks.Let x P X and consider all open set U in X such that x P U . In F there exists an object F p U q for each such U and morphisms ρ V,U : F p V q ! F p U q for any open subset V Ď U .Therefore, the sets t U | x P U Ă X u , with the sections t F p U qu , form a directed system. Definition C.3.
For any x P X the stalk of F at x is defined as F x : “ lim ÝÝÝ ! x P U F p U q . The elements in F x are called germs of sections in F p U q , i.e. F p U q ! F x .Between presheaves we can also consider natural maps, or morphisms. Let F and F be presheaves over X (with values in C ), then a morphism f : F ! F is given by maps f p U q : F p U q ! F p U q , for any open set U in X such that for any open set V Ď U (in X ) it holds that ρ V,U ˝ f p U q “ f p V q ˝ ρ V,U , where ρ V,U are the restriction morphisms in F . Compositions of presheaf morphisms andidentity are defined in the obvious way which makes the presheaves over X with valuesin C into a category. This category is denoted by PrSh C p X q .For different topological spaces X and Y we can define maps between the categories PrSh C p X q and PrSh C p Y q . Let f : X ! Y be a continuous mapping. Then define thefunctor f ˚ : PrSh C p X q ! PrSh C p Y q via the relation p f ˚ F qp V q : “ F ` f ´ p V q ˘ , for any open set V in Y . The functor f ˚ is called the direct image functor . One can alsodefine the notion of pull back or inverse image. This construction is a bit more involved.Obviously, for any open set U in X , the image f p U q is not necessarily open in Y . We definethe functor f ˚ : PrSh C p Y q ! PrSh C p X q via approximation. Let G over Y and U an openset in X , then U p f ˚ G qp U q : “ lim V ÝÝ ! Ě f p U q G p U q , V open in Y, define a presheaf over X . The functor f ˚ is called the inverse image functor . In the nextsection we also discuss these notions for sheaves. The construction of inverse image doesnot give a sheaf in general. In that case an additional construction is needed.Sheaves can be defined via presheaves by adding additional axioms on the restrictionmorphisms with respect to coverings of X . Definition C.4. A sheaf F over X with values in a category C is a presheaf which satisfiesthe following hypotheses:(s1) (Normalization) F p ∅ q is the terminal object in C ;(s2) (Mono-presheaf) Let t U i u i P I be an open covering of an open set U in X , and let σ, σ P F p U q be sections such that σ | U i “ σ | U i , @ i P I, then σ “ σ ;(s3) (Gluing) Let t U i u i P I be an open covering of an open set U in X , and let t σ i u i P I , σ i P F p U i q , be a family of sections such that σ i | U i X U j “ σ j | U i X U j , @ i, j P I, then there exists a section σ P F p U q with the property that σ i “ σ | U i .The section σ given by (s3) is called a gluing of sections σ i , consistent with the overlaps. ONTINUATION SHEAVES IN DYNAMICS 55
As before we can talk about stalks F x in F and germs over sections. We regard sheafmorphisms f : F ! F as presheaf morphisms. With the notion of sheaf morphisms weobtain a category of sheaves over X with values in C . This category is denoted by Sh C p X q .There are two important functors between PrSh C p X q and Sh C p X q which reveal impor-tant constructions to turn presheaves into sheaves and vice versa. The construction consistsof a number of steps. The first step concerns the definition of sheaf spaces. Definition C.5. A ´etal´e space over a topological space X with values in C is a pair p Π , π q ,where Π is a topological space and π : Π ! X a continuous map such that(i) π is a local homeomorphism;(ii) for each x P X the fiber π ´ p x q is an object in C ;(iii) for each open set U in X the set of sections Γ p U, Π q is an object in C .The set of sections Γ p U, Π q consists of continuous maps σ : U ! Π , which satisfy π ˝ σ “ id U .The morphisms of ´etal´e spaces are maps f : p Π , π q ! p Π , π q such that π “ π ˝ f . Asfor presheaves and sheaves, the ´etal´e spaces over X form a category, denoted by Et C p X q .´Etal´e spaces give rise to sheaves in a natural way. Let p S , π q be a ´etal´e space, then thepresheaf ΓΠ is defined as follows. Let U be an open set in X , then O p X q ! C , U p ΓΠ qp U q : “ Γ p U, Π q , (23)gives a presheaf over X with values in C .Morphisms of ´etal´e spaces p Π , π q yield morphisms of presheaves ΓΠ . Let f : p Π , π q ! p Π , π q be a ´etal´e morphism, then Γ f : ΓΠ ! ΓΠ , defines a presheaf morphism via the relation σ f ˝ σ . Proposition C.6.
Let p Π , π q be a sheaf space over X with values in C . Then ΓΠ is a sheaf over X with values in C and is a called the sheaf of sections of p Π , π q . This construction defines the functor
Γ : Et C p X q ! Sh C p X q . The functor Γ also mapsto presheaves. The construction of considering sheaves of sections can be extended topresheaves, which is the second step in establishing the above correspondence.Let F : O p X q ! C be a presheaf over X with values in C and let F x be the stalk over apoint x P X . Define the space L F : “ ğ x P X F x , and π : L F ! X the projection such that π ´ p x q “ F x . We now put a topology on L F such that p L F , π q is an ´etal´e space. Let U be an open set in X and let σ P F p U q be a section. Then, define the map p σ : U ! L F via x σ x P F x . Declare p σ p U q “ t σ x | x P U u as opensets and define p σ p U q | σ P F p U q ( as a basis for the topology on L F . With the above defined topology, that p L F , π q is a´etal´e space in the sense of Definition C.5. Presheaf morphisms f : F ! F yield mor-phisms L f : L F ! L F between ´etal´e spaces via f x : F x ! F x . Remark C.7.
If we restrict to sheaves and ´etal´e spaces then we have the functors
Γ : Et C p X q ! Sh C p X q , L : Sh C p X q ! Et C p X q . It is straightforward to prove that if p Π , π q is a ´etal´e space, then LΓ S and S are canonicallyisomorphic as ´etal´e spaces. Similarly, if F is a sheaf, then F and ΓL F are isomorphic. Thisshows that ´etal´e spaces and sheaves are essentially the same. Some texts don’t distinguishand refer to ´etal´e spaces also as sheaf. This characterization will be useful at times.Starting from a presheaf F the consecutive application of Γ and L provides a sheaf F : “ LΓ F . This is called the sheafification of F which defines a covariant functor : PrSh C p X q ! Sh C p X q . From the previous remark we have that if F is a sheaf it iscanonically isomorphic to to its sheafification. For the application of sheaf theory in thistext the category C that is involved often are small categories of algebraic structures suchas semi-lattices, lattices, abelian group, rings, etc. The following proposition gives an im-portant criterion for checking the conditions of being a sheaf with values in a category. Letus consider case where C is category of meet semi-lattices MLat . For an ´etal´e space define Π ˝ Π : “ p σ, σ q P Π ˆ Π | π p σ q “ π p σ q ( . Proposition C.8.
Let p Π , π q be a ´etal´e space over X with values in the category MLat of meetsemi-lattices. Then Definition C.5(iii) is equivalent to ‚ the mapping Π ˝ Π ! Π given by p σ, σ q σ ^ σ is continuous; Another way to word this proposition is to say that the composition law in
MLat iscontinuous in Π . Example C.9.
Let E P Ab be a abelian group. The presheaf E : O p X q ! Ab defined by E p U q : “ σ : U ! E constant ( , U Ă X open, is called the constant presheaf over X withvalues in E . The sheafification E : “ E is called the constant sheaf over X with values in E . The constant sheaf can be characterized as the sheaf of locally constant functions withvalues in E , i.e. E p U q “ σ : U ! E locally constant ( , U Ă X, open . ONTINUATION SHEAVES IN DYNAMICS 57
If we equip E with the discrete topology then such functions a continuous function σ : U ! E . This corresponds to the sheaf of section of the ´etal´e space X ˆ E , with E equippedwith the discrete topology, cf. [39, Sect. 2.4], [40, Ex. 3.31 and 3.40]. If U Ă X is open andconnected then E p U q – E . For an open set whose connected components are open the E p U q is isomorphic to a direct product of copies of E , one for each connected component, cf. [14,Ex. 1.0.3].A abelian sheaf F is called locally constant is there exists an open covering U “ t U i u of X such that F | U i is a constant sheaf for all i . This is equivalent to saying that everypoint allows a neighborhood U Ă X such that F | U is constant, cf. [2, Defn. I.1.9]. Locallyconstant sheaves are sheaves of sections of covering spaces, [40, Ex. 3.41]. Appendix D. Sheaf cohomology.
Let us recall the most important principles of sheaf co-homology. Let F : O p X q ! Ab be a sheaf of abelian groups over a topological space X ,where O p X q is the posetal category of open sets in X . Here we review the construction ofthe Godement resolution for computing sheaf cohomology. A more detailed descriptioncan be found in [4]. Consider the following presheaf on X : C p X ; F q : O p X q ! Ab , U ź x P U F x , where F x is the stalk of F at x with the restriction maps being the canonical projections.It holds that C p X ; F q is again a sheaf. There is a corresponding injection of sheaves ǫ : F ! C p X ; F q , which sends sections on an open set U Ă X to the product of theirgerms in ś x P U F x . We let K p X ; F q be the cokernel sheaf of this mapping, accompaniedby the quotient map B : C p X ; F q ! K p X ; F q . This gives us an exact sequence F C p X ; F q K p X ; F q . ǫ B Define inductively C n p X ; F q “ C p X ; K n p X ; F qq , K n ` p X ; F q “ K p X ; K n p X ; F qq . Then, by letting d n “ ǫ n ` ˝ B n , we obtain a long exact sequence: F C p X ; F q C p X ; F q C p X ; F q ... ǫ d d d which is called the Godement resolution for F . The functor Γ : Sh Ab p X q ! Ab whichassigns to F the group of global sections F p X q on X is left-exact and yields the cochaincomplex ` F ˘ Γ ` C p X ; F q ˘ Γ ` C p X ; F q ˘ Γ ` C p X ; F q ˘ ... ǫ d d d which we denote by ´ C k p X ; F q , d k ¯ , where C k p X ; F q : “ Γ ` C k p X ; F q ˘ . The cohomologygroups of the above cochain complex comprise the sheaf cohomology H k p X ; F q : “ H k ´ C ˚ p X ; F q , d ˚ ¯ . For any subset U Ă X we define H k p U ; F q : “ H k ` U ; F | U ˘ . In [4, Sect. I.6] ‘family ofsupports’ are used. Here we choose the family of supports to be all closed subsets of X .The support of a global section σ P F p X q is defined as | σ | “ x P X | σ p x q “ ( which is aclosed subset and therefore an element of the family of supports.Let N Ă X be a closed subset. Following [14, Exer. III.2.3] we define the functor Γ N : Sh Ab p X q ! Ab which assigns to F the group of sections σ with support in N . Us-ing the above resolution for F we obtain the sheaf cohomology with support in E which isdenoted by H kN p X ; F q : “ H k ´ Γ N ` C ˚ p X ; F q ˘ , d ˚ N ¯ . The latter is also referred to as local sheaf cohomology . For local sheaf cohomology we havethe following long exact sequence −! H N p X ; F q −! H p X ; F q −! H p X r N ; F q −! H N p X ; F q −! ¨ ¨ ¨ In [4, Sect. II.12] yet another variation on sheaf cohomology is defined by considering rela-tive sections in a sheaf. For a subset U Ă X the embedding i : U ã ! X implies the homo-morphisms of sheaves i k : C k p X ; F q ! i C k p U ; F | U q . Define C k p X, U ; F q : “ ker i k whichimplies the following resolution F C p X, U ; F q C p X, U ; F q C p X, U ; F q ... ǫ d d d The latter induces relative sheaf cohomology H k p X, U ; F q : “ H k ´ Γ ` C ˚ p X, U ; F q ˘ , d ˚ X,U ¯ . As before we have the following long exact sequence −! H p X, U ; F q −! H p X ; F q −! H p U ; F q −! H p X, U ; F q −! ¨ ¨ ¨ cf. [4, Sect. II.12]. There is a relation between local and relative sheaf cohomology: For aclosed subset U Ă X we have H kU p X ; F q – H k p X, X r U ; F q . A number of properties of relative sheaf cohomology can be summarized as follows:(i) For a triple V Ă U Ă X we have the long exact sequence −! H p X, U ; F q −! H p X, V ; F q −! H p U, V ; F q −! H p X, U ; F q −! ¨ ¨ ¨ (ii) For any cl V Ă int U we have H k p X, U ; F q – H k p X r V, U r V ; F q ; ONTINUATION SHEAVES IN DYNAMICS 59 (iii) For an exact sequence of sheaves ! F ! F ! F ! we have the long exactsequence −! H p X, U ; F q −! H p X, U ; F q −! H p X, U ; F q −! H p X, U ; F q −! ¨ ¨ ¨ Sheaf cohomology may be hard to compute in concrete situations. The ˇCech cohomol-ogy construction provides a good approximation of sheaf cohomology and is isomorphicto sheaf cohomology when X is a paracompact, Hausdorff topological space. For the ˇCechconstruction we use coverings of X .Let W “ t W i u i P I be a open covering for X and denote intersections of elements in W by W i ...i n “ Ş nk “ W i k . The notation W i ... x i m ...i n : “ W i ...i m ´ i m ` ...i n omits W i m from theintersection. Note that W i ...i n Ă W i ... x i m ...i n and ρ mi ...i n : “ F p ι q : F p W i ... x i m ...i n q ! F p W i ...i n q is the associated restriction map. Define the ˇCech cochain groups C k p W ; F q : “ ź p i ...i k qP I k ` F p W i ...i k q , and associated ˇCech coboundary operators δ k F : C k p W ; F q ! C k ` p W ; F q , given by δ k F p σ q i ...i k ` “ k ` ÿ j “ p´ q j ρ ji ...i k p σ i ,... p i j ...i k ` q . This defines the ˇCech cohomology of W is defined as H k p W ; F q : “ H k ´ C ˚ p W ; F q , δ ˚ F ¯ . Ifone has a total ordering on I , one can define the ordered ˇCech complex C k p W ; F q : “ ź i ă ... ă i k F p W i ...i k q , with the same coboundary operator. The cohomology of this complex is isomorphic to thatof the standard ˇCech cohomology. The usual construction of ˇCech cohomology then yields ˇCech sheaf cohomology of X as ˇ H k p X ; F q : “ lim −! W H k p W ; F q . There exists a natural homomorphism H k p X ; F q ! ˇ H k p X ; F q , which is an isomorphismfor k ď and for all k is X is a paracompact, Hausdorff topological space. For sufficientlynice covers of X , Leray’s Theorem yields immediate convergence of the limit. Theorem D.1 (cf. [4], Thm. 4.13) . Let F be a sheaf on a paracompact space X and W an opencovering of X such that F ˇˇ W sn is acyclic for all s n P N p W q , the nerve of W . Then there is acanonical isomorphism H ˚ p X ; F q – ˇ H ˚ p W ; F q . The ˇCech construction can also be used to define relative ˇCech sheaf cohomology, cf.[36, 37]. For an open subset U Ă X . A covering W for X induces a covering W “ t W i u i P I for U from sets W i X U “ ∅ , i P I . The pair p W , W q is a covering of p X, U q . The realtivechain groups are defined as C k p W , W ; F q : “ ! σ P C k p W ; F q | σ i ¨¨¨ i k “ if i , ¨ ¨ ¨ , i k P I ) . Via restriction we obtain the coboundary operator δ k F : C k p W , W ; F q ! C k ` p W , W , F q and the associated relative ˇCech sheaf cohomologies H k p W , W ; F q and ˇ H k p X, U ; F q . E-mail address : [email protected] E-mail address : [email protected] E-mail address ::