Reversing orientation homeomorphisms of surfaces
RReversing orientation homeomorphismsof surfaces
Iryna Kuznietsova, Sergiy Maksymenko
Abstract.
Let M be a connected compact orientable surface, f : M → R be a Morse function, and h : M → M be a diffeomorphism which preserves f in the sense that f ◦ h = f . We will show that if h leaves invariant eachregular component of each level set of f and reverses its orientation, then h is isotopic to the identity map of M via f -preserving isotopy.This statement can be regarded as a foliated and a homotopy analogueof a well known observation that every reversing orientation orthogonalisomorphism of a plane has order 2, i.e. is a mirror symmetry with respectto some line. The obtained results hold in fact for a larger class of mapswith isolated singularities from connected compact orientable surfaces tothe real line and the circle. Introduction
The present paper describes several foliated and homotopy variants ofa “rigidity” property for reversing orientation linear motions of the planeclaiming that every such motion has order
2. Though it is motivated bystudy of deformations of smooth functions on surfaces (and we prove thecorresponding statements), the obtained results seem to have an indepen-dent interest.Let D n = { r, s | r n = s = 1 , rs = sr − } be the dihedral group, i.e. groupof symmetries of a right n -polygon. Then each “reversing orientation”element is written as r k s for some k and has order 2:( r k s ) = r k sr k s = r k r − k ss = 1 . More generally, let SO − (2) = (cid:8)(cid:0) sin t cos t cos t − sin t (cid:1) | t ∈ [0; 2 π ) (cid:9) be the set ofreversing orientation orthogonal maps of R , i.e. the adjacent class of O (2) /SO (2) distinct from SO (2). Then again one easily checks that eachelement of SO − (2) has order 2.Another counterpart of this effect is that every motion of R which re-verses orientation is given by the formula: f ( x ) = a − x for some a ∈ R ,and therefore it has order 2, i.e. f ( f ( x )) = x .Notice that such a property does not hold for “non-rigid motions”, likearbitrary homeomorphisms or diffeomorphisms of R or S . For example, Keywords : Diffeomorphism, Morse function, dihedral group a r X i v : . [ m a t h . G T ] F e b I. Kuznietsova, S. Maksymenko let h : R → R be given by h ( x ) = − x . It reverses orientation, but h ( h ( x )) = − ( − x ) = x , and therefore it is not the identity.Nevertheless, counterparts of the above rigidity effects for homeomor-phisms can still be obtained on a “homotopy” level.For instance, let H ( S ) be the group of all homeomorphisms of the circle S and H + ( S ), (resp. H − ( S )), be the subgroup (resp. subset) consistingof homeomorphisms preserving (resp. reversing) orientation. Endow thesespaces with compact open topologies. Notice that we have a natural in-clusion O (2) ⊂ H ( S ), which consists of two inclusions SO − (2) ⊂ H − ( S )and SO (2) ⊂ H + ( S ) between the corresponding path components. Itis well known and is easy to see SO − (2) (resp. SO (2)) is a strong de-formation retract of H − ( S ) (resp. H + ( S )). This implies that the map sq : H − ( S ) → H + ( S ) defined by sq ( h ) = h is null homotopic.The aim of the present paper is to prove a parametric variant of theabove “rigidity” statements for self-homeomorphisms of open subsets oftopological products X × S preserving first coordinate (Theorem 6.2).That result will be applied to diffeomorphisms preserving a Morse functionon an orientable surface and reversing certain regular components of someof its level-sets (Theorems 3.3, 3.5, and 3.6).2. Preliminaries
Action of the groups of diffeomorphisms.
Let M be a compactsurface and P be either the real line R or the circle S and D ( M ) be thegroup of C ∞ diffeomorphisms of M . Then there is a natural right actionof D ( M ) on the space of smooth maps C ∞ ( M, P ) defined by the followingrule: ( h, f ) (cid:55)→ f ◦ h , where h ∈ D ( M ), f ∈ C ∞ ( M, P ). Let S ( f ) = { h ∈ D ( M ) | f ◦ h = f } , O ( f ) = { f ◦ h | h ∈ D ( M ) } , be the stabilizer and the orbit of f ∈ C ∞ ( M, P ) with respect to the aboveaction. It will be convenient to say that elements of S ( f ) preserve f .Endow the spaces D ( M ), C ∞ ( M, P ) with Whitney C ∞ -topologies, and theirsubspaces S ( f ), O ( f ) with induced ones. Denote by S id ( f ) the identitypath component of S ( f ) consisting of all h ∈ S ( f ) isotopic to id M by some f -preserving isotopy.For orientable M we denote by D + ( M ) the group of its orientation pre-serving diffeomorphisms. Also for f ∈ C ∞ ( M, P ) we put S + ( f ) = D + ( M ) ∩ S ( f ) , S − ( f ) = S ( f ) \ S + ( f ) , so S − ( f ) is another adjacent class of S ( f ) by S + ( f ) distinct from S + ( f ). eversing orientation homeomorphisms of surfaces 3 Homogeneous polynomials on R without multiple factors. Itis well known and is easy to prove that every homogeneous polynomial f : R → R is a product of finitely many linear and irreducible over R quadratic factors: f = l (cid:81) i =1 L i · q (cid:81) j =1 Q j , where L i ( x, y ) = a i x + b i y and Q j ( x, y ) = c i x + d i xy + e i y .Suppose deg f ≥
2. Then the origin 0 is a unique critical point of f ifand only if f has no multiple (non-proportional each other) linear factors.Structure of level sets of f near 0 is shown in Figure 2.1. a) local extreme b) quasi-saddle c) non-degenerate d) saddlesaddle f = L · · · L l Q · · · Q q f = Q · · · Q q f = L Q · · · Q q f = L L l ≥ l = 0, q ≥ l = 1, q ≥ l = 2, q = 0 Figure 2.1.
Topological structure of level sets of a homo-geneous polynomial f : R → R without multiple factorsWe will restrict ourselves with the following space of smooth maps. Definition 2.2.1.
Let F ( M, P ) be the set of C ∞ maps f : M → P satis-fying the following conditions:(A1) The map f takes constant value at each connected component of ∂M and has no critical points in ∂M .(A2) For every critical point z of f there is a local presentation f z : R → R of f near z such that f z is a homogeneous polynomial R → R withoutmultiple factors.A map f ∈ C ∞ ( M, P ) will be called
Morse if f satisfies (A1) and all itscritical points are non-degenerate. Notice that due to Morse lemma eachMorse map f satisfies the condition (A2) with homogeneous polynomials f z = ± x ± y for each critical point z . Denote by Morse( M, P ) the set ofall Morse maps M → P . Then we have the following inclusion:Morse( M, P ) ⊂ F ( M, P ) ⊂ C ∞ ( M, P ) . Then Morse(
M, P ) is open and everywhere dense in the subset of smoothfunctions from C ∞ ( M, P ) satisfying the condition (A2). Hence F ( M, P )consists of even more that typical maps M → P . I. Kuznietsova, S. Maksymenko
Several constructions associated with f ∈ F ( M, P ) . In whatfollows we will assume that f ∈ F ( M, P ). Let Σ f be the set of criticalpoints of f . Then condition (A2) implies that each z ∈ Σ f is isolated. Inthe case when P = S one can also say about local extremes of f , and evenabout local minimums or maximums if we fix an orientation of S .A connected component K of a level-set f − ( c ), c ∈ P , will be called a leaf (of f ). We will call K regular if it contains no critical points. Other-wise, it will be called critical .For ε > N ε be the connected component of f − [ c − ε, c + ε ] containing K . Then N ε will be called an f -regular neighborhood of K if ε is so smallthat N ε \ K contains no critical points of f and no boundary componentsof ∂M .A submanifold U ⊂ M will be called f -adapted if U = a ∪ i =1 A i , whereeach A i is either a regular leaf of f or an f -regular neighborhood of some(regular or critical) leaf of f .2.3.1. Graph of f . Let Γ f be the partition of M into leaves of f , and p : M → Γ f be the natural map associating to each x ∈ M the corre-sponding leaf of f containing x . Endow Γ f with the quotient topology,so a subset (a collection of leaves) U ⊂ Γ f is open iff p − ( U ) (i.e. theirunion) is open in M . It follows from axioms (A1) and (A2) that Γ f has anatural structure of 1-dimensional CW-complex, whose 0-cells correspondto boundary components of M and critical leaves of f . It is known as Reebor Kronrod-Reeb or Lyapunov graph of f , see [1, 9, 3, 2]. We will call Γ f simply the graph of f .The following statement is evident: Lemma 2.3.2.
For f ∈ F ( M, P ) the following conditions are equivalent: (a) every regular leaf of f in Int M separates M ; (b) the graph Γ f of f is a tree.For instance, those conditions hold if M is either of the surfaces: -disk,cylinder, -sphere, M¨obius band, projective plane. (cid:3) Singular foliation of f . Consider finer partition Ξ f of M whose ele-ments are of the following three types:(i) regular leaves of f ;(ii) critical points of f ;(iii) connected components of the sets K \ Σ f , where K runs over allcritical leaves of f (evidently, each such component is an open arc).In other words, each critilal leaf of f is additionally partitioned by criticalpoints. We will call Ξ f the singluar foliation of f . eversing orientation homeomorphisms of surfaces 5 Hamiltonian like flows of f ∈ F ( M, P ) . Suppose M is orientable.A smooth vector field F on M will be called Hamiltonian-like for f if thefollowing conditions hold true.(a) F ( z ) = 0 if and only if z is a critical point of f . (b) df ( F ) ≡ everywhere on M , i.e. f is constant along orbits of F . (c) Let z be a critical point of f . Then there exists a local representationof f at z in the form of a homogeneous polynomial g : R → R withoutmultiple factors as in Axiom (A2) , such that in the same coordinates ( x, y ) near the origin in R we have that F = − g (cid:48) y ∂∂x + g (cid:48) x ∂∂y . It follows from (a) and Axiom (A1) that the orbits of F precisely elementsof the singular foliation Ξ f .By [6, Lemma 5.1] or [7, Lemma 16] every f ∈ F ( M, P ) admits aHamiltonian-like vector field.The folowing statement is a principal technical result established in aseries of papers by the second author, see [8, Lemma 6.1(iv)] for details:
Lemma 2.3.5.
Let F : M × R → M be a Hamiltonian like flow for f and h ∈ S ( f ) . Then h ∈ S id ( f ) if and only if there exists a C ∞ function α : M → R such that h ( x ) = F ( x, α ( x )) for all x ∈ M . Moreover, in thiscase the homotopy H : M × [0; 1] → M given by H ( x, t ) = F ( x, tα ( x )) isan isotopy between H = id M and H = h in S ( f ) . Main results
Let M be a compact surface, h : M → M a homeomorphism, and γ ⊂ M a submanifold of M . Then γ is h -invariant , whenever h ( γ ) = γ .Moreover, suppose γ is connected and orientable h -invariant submanifoldwith dim γ ≥
1. Then γ is h + -invariant (resp. h − -invariant ) whenever therestriction h | γ : γ → γ preserves (resp. reverses) orientation of γ . If γ is afixed point of h , then we assume that γ is mutually h + - and h − -invariant.We will be interesting in the structure of diffeomorphisms preserving f ∈ F ( M, P ) and reversing orientations of some regular leaves of f . Thefollowing easy lemma can be proved similarly to [6, Lemma 3.5]. Lemma 3.1.
Let M be a connected orientable surface and h ∈ S ( f ) besuch that every regular leaf of f is h -invariant. Then every critical leaf of f in also h -invariant and the following conditions are equivalent: (1) some regular leaf of f is h + -invariant; (2) all regular leaves of f are h + -invariant; (3) h preserves orientation of M . I. Kuznietsova, S. Maksymenko
Counterexample 3.2.
Lemma 3.1 fails for non-orientable surfaces. Let M be a M¨obius band, and f : M → R be a Morse function having twocritical points: one local extreme z and one saddle y . Let K be the criticalleaf of f containing y , and D and E be the connected component of M \ K containing z and ∂M respectively. Then it is easy to construct a diffeomor-phism h : M → M (called slice along central circle of M¨obius band ) whichis fixed on ∂M , and for which D is h − -invariant. Then regular leaves of f in D are h − -invariant, while regular leaves of f in E are h + -invariant.Denote by ∆ − ( f ) the subset of S ( f ) consisting of diffeomorphisms h suchthat every regular leaf of f is h − -invariant .Let h ∈ ∆ − ( f ). Then by Lemma 3.1 every critical leaf K of f is h − -invariant, however, h may interchange critical points of f in K and theleaves of Ξ f contained in K (i.e. connected components of K \ Σ f ). Noticealso that in general ∆ − ( f ) can be empty.The following theorem can be regarded as a homotopical and foliated variant of the above rigidity property for diffeomorphisms preserving f . Theorem 3.3.
Let M be a connected compact orientable surface, f ∈F ( M, P ) , and h ∈ ∆ − ( f ) . Then h ∈ S id ( f ) . In particular, for every h ∈ ∆ − ( f ) each leaf of Ξ f is ( h , +) -invariant. Corollary 3.4.
Let or : π S ( f ) → Z = { , } be the orientation ho-momorphism defined by or ( S + ( f )) = 0 and or ( S − ( f )) = 1 . Then for each h ∈ ∆ − ( f ) the map s : Z → π S ( f ) , defined by s (0) = [id M ] and s (1) = [ h ] is a homomorphism satisfying or ◦ s = id Z . In other words, s is a sectionof or , whence π S ( f ) is a certain semidirect product of π S + ( f ) and Z . Proof.
One should just mention that Theorem 3.3 implies that [ h ] = [id M ]in π S ( f ). (cid:3) Now consider the situation, when not all regular leaves are h − -invariant.To clarify the situation we first formulate a particular case of the generalstatement for maps on 2-disk and cylinder. Theorem 3.5.
Let M be a connected compact orientable surface, f ∈F ( M, P ) , V be a regular leaf of f , and h ∈ S ( f ) . Suppose every regularleaf of f in Int M separates M (this holds e.g. when M is a -disk, cylinderor a -sphere, see Lemma 2.3.2), and that V is h − -invariant. Then thereexists g ∈ S ( f ) which coincide with h on some neighborhood of V and suchthat g ∈ S id ( f ) . Emphasize that Theorem 3.5 does not claim that g ∈ ∆ − ( f ), though g reverses orientation of V and its square belong to S id ( f ). eversing orientation homeomorphisms of surfaces 7 General result.
Theorems 3.3 and 3.5 are consequences of the follow-ing Theorem 3.6 below. Let M be a connected compact (not necessarilyorientable) surface, f ∈ F ( M, P ), and h ∈ S ( f ). Let also • A be the union of all h − -invariant regular leaves of f , • K , . . . , K k be all the critical leaves of f such that A ∩ K i (cid:54) = ∅ ; • for i = 1 , . . . , k , let R K i be an f -regular neighborhood of K i chosenso that R K i ∩ R K j = ∅ for i (cid:54) = j and Z := A (cid:91) (cid:16) k ∪ i =1 R K i (cid:17) . Evidently, Z is an f -adapted subsurface of M and each of its con-nected components intersects A . Lemma 3.5.1.
Suppose Z is orientable. Let also γ be a boundary compo-nent of Z . Consider the following conditions: (1) γ ⊂ Int M ; (2) γ = ∂U ∩ ∂Z for some connected component U of M \ Z ; (3) h ( γ ) ∩ γ = ∅ ; (4) h ( γ ) (cid:54) = γ .Then (1) ⇔ (2) ⇒ (3) ⇔ (4) . Proof. (1) ⇔ (2) is evident, and (3) ⇔ (4) follows from the observation that γ is a regular leaf of h and h permutes leaves of f .(2) ⇒ (4) Suppose h ( γ ) = γ . Let Z (cid:48) be a connected component of Z containing γ . Then by the construction Z (cid:48) must intersect A , whence h changes orientation of some regular leaves in Z (cid:48) . Since Z (cid:48) is also orientable,we get from Lemma 3.1 that h also changes orientation of γ . This meansthat γ ⊂ A , and therefore there exists an open neighborhood W ⊂ A of γ consisting of regular leaves of f . In particular, the regular leaves in W ∩ Int U must be contained in A which contradicts to the assumption thatInt U ⊂ M \ Z ⊂ M \ A . (cid:3) Thus h interchanges boundary components of Z belonging to the interiorof M . We will introduce the following property on Z :(B) every connected component of ∂Z ∩ Int M separates M . This condition means that there is a bijection between boundary com-ponents of ∂Z ∩ Int M and connected components of M \ Z , whence byLemma 3.5.1 there will be no h -invariant connected components of M \ Z . Theorem 3.6. If Z is non-empty, orientable and has property (B) , thenthere exists g ∈ S ( f ) such that g = h on Z and g ∈ S id ( f ) . I. Kuznietsova, S. Maksymenko
Proof of Theorem 3.3.
Suppose M is orientable and h ∈ ∆ − ( f ). Thenin the notation of Theorem 3.6, Z = M , and by that theorem there exists g ∈ S ( f ) such that g = h on Z and g ∈ S id ( f ). This means that h = g and h ∈ S id ( f ). (cid:3) Proof of Theorem 3.5.
The assumption that every regular leaf of f inInt M separates M implies condition (B). (cid:3) Structure of the paper.
In Section 4 we discuss a notion of a shiftmap along orbits of a flow which was studied in a series of papers by thesecond author, and extend several results to continuous flows. Section 5devoted to reversing orientation families of homeomorphisms of the circle.In Section 6 we study flows without fixed point, and in Section 7 recallseveral results about passing from a flow on the plane to the flow writtenin polar coordinates. Section 8 introduces a certain subsurfaces of a surface M associated with a map f ∈ F ( M, P ) and called chipped cylinders . Weprove Theorem 8.3 describing behaviour of diffeomorphisms reversing reg-ular leaves of f contained in those chipped cylinders. In Section 9 we proveLemma 9.1 allowing to change f -preserving diffeomorphisms so that its fi-nite power will be isotopic to the identity by f -preserving isotopy. Finally,in sections 9 and 10 we prove Theorem 3.6.4. Shifts along orbits of flows
In this section we extend several results obtained in [4, 8] for smoothflows to a continuous situation. Let X be a topological space. Definition 4.1.
A continuous map F : X × R → X is a (global) flow on X , if F = id X and F s ◦ F t = F s + t for all s, t ∈ R , where F s : X → X isgiven by F s ( x ) = F ( x, s ). For x ∈ X the subset F ( x × R ) ⊂ X is called the orbit of x .It is well known that a C r , 1 ≤ r ≤ ∞ , vector field F of a smooth compactmanifold X tangent to ∂X always generates a flow F .Assume further that F is a flow on a topological space X . Let also V ⊂ X be a subset. Say that a continuous map h : V → X preserves orbitsof F on V if h ( γ ∩ V ) ⊂ γ for every orbit γ of F . The latter means thatfor each x ∈ U there exists a number α x ∈ R such that h ( x ) = F ( x, α x ).Notice that in general α x is not unique and does not continuously dependon x .Conversely, let α : V → R be a continuous function such that its graphΓ α = { ( x, α ( x ) | x ∈ V } is contained in W . For a global flow any continuous α : V → R satisfies that condition. Then one can define the following map eversing orientation homeomorphisms of surfaces 9 F α : V → X by F α ( z ) := F ( z, α ( z )) = F α ( z ) ( z ) . We will call F α a shift along orbits of F by the function α , while α will becalled a shift function for F α .Notice that F α preserves orbits of F on V , and in general is not a home-omorphism. Definition 4.2.
Let x be a non-fixed point of F , and Y be a topologicalspace. Let also U be an open neighborhood of x , and φ = ( ζ, ρ ) : U → Y × R be an open embedding. Then the pair ( φ, U ) will be called a flow-box chart at x , if there exist an ε > V of x in X suchthat φ ◦ F ( z, t ) = (cid:0) ζ ( z ) , ρ ( z ) + t (cid:1) (4.1)for all ( z, t ) ∈ V × ( − ε ; ε ).In other words, F is locally conjugated to the flow G : ( Y × R ) × R → Y × R defined by G ( y, s, t ) = ( y, s + t ), since the identity (4.1) can be written as φ ◦ F ( z, t ) = (cid:0) ζ ( z ) , ρ ( z ) + t (cid:1) = G (cid:0) ζ ( z ) , ρ ( z ) , t (cid:1) = G ( φ ( z ) , t ) , i.e. φ ◦ F t ( z ) = G t ◦ φ ( z ) . (4.2)It is well known that each C flow (generated by some C vector field) ona smooth manifold admits flow-box charts at each non-fixed point. Lemma 4.3. If ( φ, U ) is a flow box chart at a non-fixed point x ∈ X of F , then for any τ ∈ R the pair ( φ ◦ F − τ , F τ ( U )) is a flow box chart at thepoint y = F ( x, τ ) . Proof.
Denote U (cid:48) = F τ ( U ) and φ (cid:48) = φ ◦ F − τ . Let also ε and V be asin Definition 4.1, and V (cid:48) = F τ ( V ) be a neighborhood of y . Then for any( z, t ) ∈ V (cid:48) × ( − ε ; ε ) we have that φ (cid:48) ◦ F t ( z ) = ( φ ◦ F − τ ) ◦ F t ( z ) = ( φ ◦ F t − τ )( z ) == ( φ ◦ F t (cid:0) F − τ ( z ) (cid:1) (4.1) == G t ◦ φ ( F − τ ( z )) = G t ◦ φ (cid:48) ( z ) . (cid:3) The following lemma shows that for flows admitting flow box charts (e.g.for smooth flows) every orbit preserving map admits a shift function neareach non-fixed point. Moreover, such a function is locally determined byits value at that point.
Lemma 4.4. (cf. [8, Lemma 6.1(i)]).
Suppose a flow F : X × R → X has a flow-box ( φ, U ) at some non-fixed point x . Let also h : U → X bea continuous map preserving orbits of F and such that h ( x ) = F ( x, τ ) for some τ ∈ R . Then there exists an open neighborhood V ⊂ U of x and aunique continuous function α : V → R such that (1) α ( x ) = τ ; (2) h ( z ) = F ( z, α ( z )) for all z ∈ V .If in addition X is a manifold of class C r , (1 ≤ r ≤ ∞ ) , F and h are C r ,and φ is a C r embedding, then α is C r as well. Proof.
The proof almost literally repeats the arguments of [8, Lemma 6.2]proved for smooth flows and based on existence of flow box charts. Forcompleteness we present a short proof for continuous situation.1) First suppose τ = 0, so h ( x ) = x . Let V and ε be the same as inDefinition 4.2. Decreasing V one can also assume that V ⊂ U ∩ h − ( U ),so in particular h ( V ) ⊂ U . Denote ˆ V := φ ( V ) and ˆ U := φ ( U ). Then thesesets are open, and we have a well-defined map ˆ h = φ ◦ h ◦ φ − : ˆ V → ˆ U whichpreserves orbits of G due to (4.2). This means that ˆ h ( y, t ) = (cid:0) y, η ( y, t ) (cid:1) for some continuous function Y × R ⊃ ˆ V η −−→ R . Define another continuousfunction α (cid:48) : ˆ V → R by α (cid:48) ( y, t ) = η ( y, t ) − t . Thenˆ h ( y, t ) = (cid:0) y, t + α (cid:48) ( y, t ) (cid:1) = G (cid:0) y, t, α (cid:48) ( y, t ) (cid:1) = G α (cid:48) ( y, t ) , that is φ ◦ h ◦ φ − = ˆ h = G α (cid:48) , whence for each z ∈ V we have that h ( z ) = φ − ◦ G α (cid:48) ◦ φ ( z ) = φ − ◦ G α (cid:48) ◦ φ ( z ) ◦ φ ( z ) = F α (cid:48) ◦ φ ( z ) ( z ) = F α (cid:48) ◦ φ ( z ) . Thus one can put α = α (cid:48) ◦ φ : V → R . It follows that α is continuous.Moreover, let ˆ y, ˆ t ) = φ ( x ). Since h ( x ) = x , we get that ˜ h (ˆ y, ˆ t ) = ˆ y, ˆ t ),whence η (ˆ y, ˆ t ) = ˆ t , whence α (cid:48) (ˆ y, ˆ t ) = 0, and thus α ( x ) = α (cid:48) ◦ φ ( x ) = α (cid:48) (ˆ y, ˆ t ) = 0 .
2) If τ (cid:54) = 0, then consider the map ˜ h : U → X given by ˜ h = F − τ ◦ h .Then ˜ h ( x ) = F − τ ◦ h ( x ) = F ( F ( x, τ ) , − τ ) = x , whence by 1) ˜ h = F ˆ α for aunique continuous function ˆ α : V → R such that ˆ α ( x ) = 0. Put α = ˆ α + τ .Then α ( x ) = τ and h ( z ) = F τ ◦ ˜ h ( z ) = F ( F ( z, ˆ α ( z )) , τ ) = F ( z, ˆ α ( z ) + τ ) = F ( z, α ( z )) . (cid:3) The formulas for α imply that if X , F , φ , and h are C r , (1 ≤ r ≤ ∞ ),then α is C r as well. Corollary 4.5.
Suppose U ⊂ X is an open connected subset such that every x ∈ U is non-fixed and admits a flow-box. Let also α, α (cid:48) : U → R be twocontinuous functions such that F α = F α (cid:48) on U . If α ( x ) = α (cid:48) ( x ) at some x ∈ U (this holds e.g. if F has at least one non-periodic point in U ), then α = α (cid:48) on U . eversing orientation homeomorphisms of surfaces 11 Proof.
The set A = { α ( y ) = α (cid:48) ( y ) | y ∈ U } is closed and by assumptionnon-empty (contains x ). Moreover, by Lemma 4.4 this set is open, whence A = U . (cid:3) Corollary 4.6.
Let F : X × R → X be a continuous flow, U ⊂ X be anopen subset such that every point x ∈ U is non-fixed and non-periodic andadmits a flow box chart, and h : U → X be an orbit preserving map. Thenthere exists a unique continuous function α : U → R such that h ( x ) = F ( x, α ( x )) for all x ∈ U .If in addition X is a manifold of class C r , (0 ≤ r ≤ ∞ ) , F is C r andadmits C r flow box charts, and h is C r , then α is C r as well. Proof.
Since every point x ∈ U is non-fixed and non-periodic, there exists aunique number α ( x ) such that h ( x ) = F ( x, α ( x )). Moreover, since F admitsflow box chart at x , it follows from Lemma 4.4 that the correspondence x (cid:55)→ α ( x ) is a continuous function α : U → R , which is also C r under thecorresponding smoothness assumptions on X , F , φ , and h . (cid:3) The following statement extends some results established in [8] for smoothflows to continuous flows having flow box charts at each non-fixed point onarbitrary topological spaces.
Lemma 4.7. (c.f. [8, Lemmas 6.1-6.3])
Let F : X × R → X be a flow, p : ˜ X → X a covering map, and ξ : ˜ X → ˜ X a covering transformation, i.e.a homeomorphism such that p ◦ ξ = p . Then the following statements hold. (1) F lifts to a unique flow ˜ F : ˜ X × R → ˜ X such that p ◦ ˜ F t = F t ◦ p forall t ∈ R . (2) For each continuous function α : X → R the map ˜ F α ◦ p : ˜ X → R isa lifting of F α , that is (3) ˜ F commutes with ξ in the sense that ˜ F t ◦ ξ = ξ ◦ ˜ F t for all t ∈ R .More generally, for any function β : ˜ X → R we have ˜ F β ◦ ξ = ξ ◦ ˜ F β ◦ ξ . (4) For a continuous function β : ˜ X → R and a point z ∈ ˜ X considerthe following (“global” and “point”) conditions: (g1) β = β ◦ ξ ; (g2) ˜ F β = ˜ F β ◦ ξ ; (g3) ˜ F β ◦ ξ = ξ ◦ ˜ F β ; (p1) β ( z ) = β ◦ ξ ( z ) ; (p2) ˜ F β ( z ) = ˜ F β ◦ ξ ( z ) ; (p3) ˜ F β ◦ ξ ( z ) = ξ ◦ ˜ F β ( z ) . Then we have the following diagram of implications: (g1) (cid:43) (cid:51) (cid:11) (cid:19) (g2) (cid:107) (cid:115) (2) (cid:43) (cid:51) (cid:11) (cid:19) (g3) (cid:11) (cid:19) (p1) (cid:43) (cid:51) (p2) (cid:107) (cid:115) (2) (cid:43) (cid:51) (p3) (4.3)(i) If z is non-fixed and non-periodic point for ˜ F , then (p2) ⇒ (p1) . (ii) If ˜ X is path connected and ˜ F β is a lifting of some continuous map h : X → X , that is h ◦ p = p ◦ ˜ F β , then (p3) ⇒ (g3) , whence allconditions in the right square of (4.3) are equivalent. (iii) If ˜ X is path connected, and ˜ F has no fixed points and admits flow boxcharts at each point of ˜ X , then (g2)&(p1) ⇒ (g1) . Proof. (1) Consider the following homotopy (with “open ends”) G = F ◦ ( p × id R ) : ˜ X × R → X, G ( z, t ) = F ( p ( z ) , t ) , and let ˜ F : ˜ X × → ˜ X be given by ˜ F ( z,
0) = z . Then ˜ F is a lifting of G | ˜ X × , that is p ◦ ˜ F ( z,
0) = p ( z ) = F ( p ( z ) ,
0) = G ( z, F extendsto a unique lifting ˜ F : ˜ X × R → ˜ X of G . One easily checks that this liftingis a flow on ˜ X .(2) Let z ∈ ˜ X and t = α ◦ p ( z ). Then p ◦ ˜ F α ◦ p ( z ) ( z ) = p ◦ ˜ F t ( z ) (1) = F t ◦ p ( z ) = F α ◦ p ( z ) ◦ p ( z ) = F α ◦ p ( z ) . (3) Notice that the map ˜ F (cid:48) : ˜ X × R → ˜ X defined by ˜ F (cid:48) t = ξ ◦ ˜ F t ◦ ξ − isalso a flow on ˜ X . Moreover, p ◦ ˜ F (cid:48) t = p ◦ ξ ◦ ˜ F (cid:48) t ◦ ξ − = p ◦ ˜ F t ◦ ξ − = F t ◦ p ◦ ξ − = F t ◦ p. Thus ˜ F (cid:48) and ˜ F are two liftings of F which coincide at t = 0, and therefore˜ F (cid:48) = ˜ F by uniqueness of liftings. Therefore for any continuous function˜ α : ˜ X → R and z ∈ ˜ X we have, (cf. [8, Eq. (6.8)])˜ F ˜ α ◦ ξ ( z ) = ˜ F (cid:0) ξ ( z ) , ˜ α ( ξ ( z )) (cid:1) = ˜ F ˜ α ( ξ ( z )) ◦ ξ ( z )= ξ ◦ ˜ F ˜ α ◦ ξ ( z ) ( z ) = ξ ◦ ˜ F ˜ α ◦ ξ ( z ) . (4) The implication in Diagram (4.3) are trivial. Assume that ˜ X (andtherefore X ) are path connected.(i) If z is non-fixed and non-periodic, then ˜ F ( z, a ) = ˜ F ( z, b ) implies that a = b for any a, b . In particular, this hold for a = β ( z ) and b = β ◦ ξ ( z ).(ii) Suppose that ˜ F β is a lifting of some continuous map h : X → X .Then p ◦ ˜ F β ◦ ξ = h ◦ p ◦ ξ = h ◦ p and p ◦ ξ ◦ ˜ F β = p ◦ ˜ F β = h ◦ p , i.e. both˜ F β ◦ ξ and ξ ◦ ˜ F β are liftings of h . eversing orientation homeomorphisms of surfaces 13 Now if (p3) holds, i.e. ˜ F β ◦ ξ ( z ) = ξ ◦ ˜ F β ( z ) at some point z ∈ ˜ X , thenthese liftings must coincide on all of ˜ X , which means condition (g3).(iii) Suppose ˜ F has no fixed points and conditions (g2) and (p1) hold,that is ˜ F ( y, β ( y )) = ˜ F ( y, β ◦ ξ ( y )) for all y ∈ ˜ X and β ( z ) = β ◦ ξ ( z ) forsome z ∈ ˜ X . Notice that the set A = { y ∈ ˜ X | β ( y ) = β ◦ ξ ( y ) } is close.Moreover, by the local uniqueness of shift-functions (Corollary 4.5) A isalso open. Since ˜ X is connected, A is either ∅ or ˜ X . But z ∈ A , whence A = ˜ X , i.e. condition (g1) holds. (cid:3) Let p : ˜ X → X be a regular covering map with path connected ˜ X and X , G be the group of covering transformation, F : X × R → X be a flowon X , and ˜ F : ˜ X × R → ˜ X be its lifting as in Lemma 4.7(1). Corollary 4.8. (c.f. [8, Lemma 6.3])
Suppose all orbits of ˜ F are non-fixedand non-closed and ˜ F has flow box charts at all points of ˜ X . Let also h : X → X be a continuous map admitting a lifting ˜ h : ˜ X → ˜ X , i.e. p ◦ ˜ h = h ◦ p , such that ˜ h leaves invariant each orbit γ of ˜ F and commuteswith each ξ ∈ G . Then there exist a unique continuous function α : X → R such that h = F α .Again if ˜ X , X , p , h , F and its flow box charts are C r , (1 ≤ r ≤ ∞ ) , then α is also C r . Proof.
Due to assumptions on ˜ F we get from Corollary 4.6 that thereexists a unique continuous function β : ˜ X → R such that ˜ h = ˜ F β . Let ξ ∈ G . Since ˜ h commutes with ξ , i.e. condition (g3) of Lemma 4.7 holds,we obtain the following implications:(g3) (cid:107) (cid:115) (cid:43) (cid:51) (g2) (cid:43) (cid:51) (p2) (i) (cid:43) (cid:51) (p1) (iii) (cid:43) (cid:51) (g1) . meaning that β ◦ ξ = β . Thus β is invariant with respect to all ξ ∈ G , andtherefore it induces a unique function α : X → R such that β = α ◦ p . Since p is a local homeomorphism, it follows that α is continuous. Moreover, byLemma 4.7(2), ˜ h = ˜ F β = ˜ F α ◦ p is a lifting of F α . But ˜ h is also a lifting of h , whence h = F α .Statements about smoothness of α follows from the corresponding smooth-ness parts of used lemmas. We leave the details for the reader. (cid:3) Self maps of the circle
Let S = { z ∈ C | | z | = 1 } be the unit circle in the complex plane, p : R → S be the universal covering map defined by p ( s ) = e πis and ξ ( s ) = s + 1 be a diffeomorphism of R generating the group of coveringslices Z . Denote by C k ( S , S ), k ∈ Z , the set of all continuous maps h : S → S of degree k , i.e. maps homotopic to the map z (cid:55)→ z k . Then { C k ( S , S ) } k ∈ Z is a collection of all path components of C ( S , S ) with respect to thecompact open topology.For a map h : X → X it will be convenient to denote the composition h ◦ · · · ◦ h (cid:124) (cid:123)(cid:122) (cid:125) n by h n for n ∈ N . A point x ∈ X is fixed for h , whenever h ( x ) = x . Lemma 5.1.
Let h : S → S be a continuous map and ˜ h : R → R be anylifting of h with respect to p , i.e. a continuous map making commutativethe following diagram: R p (cid:15) (cid:15) ˜ h (cid:47) (cid:47) R p (cid:15) (cid:15) S h (cid:47) (cid:47) S that is p ◦ ˜ h = h ◦ p , or e πi ˜ h ( s ) = h ( e πis ) for s ∈ R . For a ∈ Z definethe map ˜ h a : R → R by ˜ h a = ξ a ◦ ˜ h , that is ˜ h a ( s ) = ˜ h ( s ) + a . Then thefollowing conditions are equivalent: (a) h ∈ C k ( S , S ) ; (b) ˜ h ◦ ξ = ξ k ◦ ˜ h = ˜ h k , that is ˜ h ( s + 1) = ˜ h ( s ) + k for all t ∈ R .Moreover, let k be the degree of h . Then (i) h has at least | k − | fixed points; (ii) { ˜ h ak } a ∈ Z is the collection of all possible liftings of h ; (iii) ˜ h a ◦ ˜ h b = ξ a + kb ◦ h for any a, b ∈ Z ; (iv) if k = − , then ˜ h a = ˜ h for all a ∈ Z , in other words, for any lifting ˜ h a of h its square ˜ h a does not depend on a . Moreover, if A is the set of fixedpoints of h , then p − ( A ) is the set of fixed points of ˜ h . Proof.
All statements are easy. Statement (i) is a consequence of interme-diate value theorem, and (ii) follows from (b).(iii) ˜ h a ◦ ˜ h b = ξ a ◦ ˜ h ◦ ξ b ◦ ˜ h = ξ a ◦ ξ kb ◦ ˜ h = ξ a + kb ◦ ˜ h .(iv) If k = − h a ◦ ˜ h a = ξ a − a ◦ ˜ h = ˜ h .Hence one can put g := ˜ h = ˜ h a and this map does not depend on a ∈ Z .Let also ˜ A be the set of fixed points of g . We have to show that ˜ A = p − ( A ).Let s ∈ ˜ A , and z = p ( s ). Then s = g ( s ) = ˜ h a ( s ) implies that z = p ( s ) = p ◦ ˜ h a ( s ) = h ◦ p ( s ) = h ( z ) , so z ∈ A , that is p ( ˜ A ) ⊂ A and thus ˜ A ⊂ p − ( A ). eversing orientation homeomorphisms of surfaces 15 Conversely, let z ∈ A and s ∈ R be such that z = p ( s ). Then there existsa unique lifting ˜ h a of h such that ˜ h a ( s ) = s . But then g ( s ) = ˜ h a ( s ) = s ,whence s ∈ ˜ A . (cid:3) The following example shows that the effect described in the state-ment (iv) of Lemma 5.1 includes the rigidity property of reflections of thecircle mentioned in the introduction.
Example 5.2.
Let h ( z ) = ze − πφ e πφ = ¯ ze φ be a reflection of the complexplane with respect to the line passing though the origin and constitutingan angle φ with the positive direction of x -axis. Then h is an involutionpreserving the unit circle and the restriction h | S : S → S is a map ofdegree − SO − (2). Moreover, each its lifting h a : R → R isgiven by ˜ h a ( s ) = a + φ − s . But then ˜ h a = id R and does not depend on a .Moreover, the set of fixed points of ˜ h a is R which coincides with p − ( S ),where S is the set of fixed points of h = id S .Another interpretation of the above results can be given in terms of shiftfunctions . Corollary 5.3.
Let F : S × R → S be a flow on the circle S having nofixed points, so S is a unique periodic orbit of F of some period θ . (1) Let h : S → S be a continuous map. Then h ∈ C ( S , S ) if andonly if there exists a continuous function α : S → R such that h = F α .Such a function is not unique and is determined up to a constant summand nθ for n ∈ Z . If F and h are C r , (0 ≤ r ≤ ∞ ) , then so is α . (2) For every h ∈ C − ( S , S ) there exists a unique continuous function α : S → R such that (a) h ( z ) = F ( z, α ( z )) for all z ∈ S ; (b) α ( z ) = 0 for some z ∈ S iff h ( z ) = z , (due to Lemma 5.1 (i) thereexists at least two such points); (c) if F and h are C r , (0 ≤ r ≤ ∞ ) , then so is α . (3) Let α k : S → R , k = 0 , , be two continuous functions, and for each t ∈ [0; 1] let α t : S → R and h t : S → S be defined by α t = (1 − t ) α + tα , h t ( z ) = ze πiα k ( z ) . If h and h are homeomorphisms (diffeomorphsms of class C r , ≤ r ≤ ∞ ),then so is h t for each t ∈ [0; 1] . Proof.
Let ˜ F : R × R → R be a unique lifting of F with respect to thecovering p . Then R is a unique non-periodic orbit of ˜ F . (1) If α : S → R is any continuous function, then the map F α ishomotopic to the identity by the homotopy { F tα } t ∈ [0;1] , whence F α hasdegree 1 (the same as id S ).Conversely, let h ∈ C ( S , S ) and let ˜ h be any lifting of h . Then byCorollary 4.6 there exists a unique continuous function β : R → R such that˜ h = ˜ F β . Moreover, since h is a map of degree 1, we get from Lemma 5.1(b),that ˜ h commutes with ξ , i.e. condition (g3) of Lemma 4.7 holds. Since˜ h = F β is a lifting of h , Lemma 4.7(ii) implies that condition (g1) ofLemma 4.7 also holds, i.e. β ◦ ξ = β . Hence β induces a unique function α : S → R such that β = α ◦ p and ˜ h = ˜ F α ◦ p . Therefore By Lemma 4.7(2),˜ h is a lifting of F α . But it is also a lifting of h , whence h = F α .(2) Since h ∈ C − ( S , S ), it follows that h ∈ C ( S , S ), whence (a)and (c) directly follow from (1).(b) Denote by A the set of fixed points of h . We should prove that A = α − (0). Evidently, if α ( y ) = 0, then h ( y ) = F ( y, α ( y )) = h ( y ) = F ( y,
0) = y. Conversely, let y ∈ A , so h ( y ) = y , and let x ∈ R be any point with p ( x ) = y . Then there exists a unique lifting ˜ h of h with ˜ h ( x ) = x and byLemma 5.1(iv), p − ( A ) is the set of fixed points of ˜ h . Moreover, since ˜ h has degree 1, we get from (1) that ˜ h = ˜ F α ◦ p . Since x is a fixed point of ˜ h as well, we must have that 0 = α ◦ p ( x ) = α ( y ).(3) Consider two flows of R and S respectively:˜ F : R × R → R , ˜ F ( x, t ) = x + t, F : S × R → S , F ( z, t ) = ze πit . Evidently, ˜ F is a lifting of F , and h t = F α t . Then by Lemma 4.7(2) themap ˜ h t := ˜ F α t ◦ p : R → R , ˜ h t ( x ) = x + (1 − t ) α ( x ) + tα ( x )is a lifting of h t . Evidently, ˜ h t = (1 − t )˜ h + t ˜ h . Since h and h arehomeomorphisms (diffeomorphisms of class C r ), it follows that so are ˜ h and˜ h are homeomorphisms. Therefore so are their convex linear combination ˜ h t and the induced map h t . (cid:3) Flows without fixed points
Let X be a Hausdorff topological space and F : Y × R → Y a continuousflow on an open subset Y ⊂ X × S satisfying the following conditions:(Φ1) the orbits of F are exactly the connected components of the intersec-tions ( x × S ) ∩ Y for all x ∈ X . eversing orientation homeomorphisms of surfaces 17 (Φ2) F admits flow box charts at each point ( y, s ) ∈ Y ; (Φ3) the set B = { x ∈ X | x × S ⊂ Y } is dense in X ; (Φ4) for each x ∈ X , the intersection ( x × S ) ∩ Y has only finitely manyconnected components (being by (Φ1) orbits of F ). Thus every orbit of F is either x × S or some arc in x × S , and, inparticular, F has no fixed points. Also condition (Φ3) implies that the setof periodic orbits of F is dense in X × S . Then the complement to B : A := X \ B = { x ∈ X | x × S (cid:54)⊂ Y } consists of x ∈ X for which x × S contains a non-closed orbit of F .It will also be convenient to use the following notations for x ∈ X : L x := ( x × S ) ∩ Y, (cid:101) L x := p − ( L x ) = ( x × R ) ∩ (cid:101) Y .
Example 6.1.
Let X be a smooth manifold, and G be a vector field on X × S defined by G ( x, s ) = ∂∂s , so its orbits are the circles x × S . Let Y ⊂ X × S . Then it is well known that there exists a non-negative C ∞ function α : X × S → [0; + ∞ ) such that Y = ( X × S ) \ α − (0). Defineanother vector field F on X × S by F = αG . Let also F : ( X × S ) × R → X × S be the flow on X × S generated by F . Then Y is the set of non-fixedpoints of F and the induced flow on Y satisfies conditions (Φ1) and (Φ2).Another two conditions (Φ3) and (Φ4) depend only on a choice of Y . Forinstance they will hold if the complement ( X × S ) \ Y is finite. Theorem 6.2.
Suppose F satisfies (Φ1) - (Φ4) . Let also h : Y → Y be ahomeomorphism having the following properties. (1) h ( L x ) ⊂ x × S for each x ∈ X , so h preserves the first coordinate, andin particular, leaves invariant each periodic orbit of F , thought it mayinterchange non-periodic orbits contained in x × S . (2) For each x ∈ B the restriction h : x × S → x × S has degree − as aself-map of a circle.Then there exists a unique continuous function α : Y → R such that (a) h ( x, s ) = F ( x, s, α ( x, s )) for all ( x, s ) ∈ Y , in particular h pre-serves every orbit of F ; (b) α ( x, s ) = 0 iff h ( x, s ) = ( x, s ) ; (c) if in addition X is a manifold of class C r , (0 ≤ r ≤ ∞ ) , F is C r and admits C r flow box charts, and h is C r , then α is C r as well. Proof.
Let p : X × R → X × S be the infinite cyclic covering of X × S defined by p ( x, s ) = ( x, e πis ) and ξ ( x, s ) = ( x, s + 1) be a diffeomorphism of X × R generating the group Z of covering slices. In particular, p ◦ ξ = p .Denote (cid:101) Y = p − ( Y ) Then the restriction p : (cid:101) Y → Y is a covering of Y .Let ˜ F : (cid:101) Y × R → (cid:101) Y be the lifting of the flow F , so p ◦ ˜ F ( y, t ) = F ( p ( y ) , t ) , ( y, t ) ∈ (cid:101) Y × R . Then the orbits of ˜ F are exactly the connected components of (cid:101) L x for all x ∈ X . In particular, all orbits of ˜ F are non-closed. Evidently, for x ∈ X the following conditions are equivalent: • x × R is an orbit of ˜ F ; • x × R ⊂ (cid:101) Y ; • x × S is an orbit of F ; • x × S ⊂ Y , i.e. x ∈ B. Lemma 6.2.1.
Let ˜ h : (cid:101) Y → (cid:101) Y be any lifting of h , that is p ◦ ˜ h = h ◦ p .Then the following conditions hold. (i) ˜ h ( (cid:101) L x ) ⊂ (cid:101) L x for all x ∈ X ; (ii) if ˜ h is another lifting of h , then ˜ h = ˜ h ; (iii) for each x ∈ X the restriction ˜ h : (cid:101) L x → (cid:101) L x is strictly decreasing inthe sense that if h ( x, s ) = ( x, t ) for some s, t ∈ R , then s > t ; (iv) for each x ∈ X the restriction ˜ h leaves invariant each orbit of ˜ F . Proof. (i) Notice that p ◦ ˜ h ( (cid:101) L x ) = h ◦ p ( (cid:101) L x ) = h ( L x ) ⊂ L x , whence˜ h ( (cid:101) L x ) ⊂ p − (cid:0) h ( L x ) (cid:1) ⊂ p − ( L x ) = (cid:101) L x . (ii) Let x ∈ B , so x × S ⊂ Y is an orbit of F . Then the restriction p | x × R : x × R → x × S is a universal covering map, h | x × S : x × S → x × S is a map of degree −
1, and ˜ h | x × R , ˜ h | x × R : x × R → x × R are two liftingof h | x × S . Hence by Lemma 5.1(iv) ˜ h | x × R = ˜ h | x × R . Thus ˜ h = ˜ h on B × R . But by property (Φ3), B is dense in X , whence B × R is dense in X × R . Since X is Hausdorff, it follows that ˜ h = ˜ h on all of X × R .(iii) If x ∈ B , then h | x × S : x × S → x × S is a map of degree − h | x × R : x × R → x × R reverses orientation of x × R , i.e. is strictlydecreasing.Suppose x ∈ X \ B , i.e. x × R (cid:54)⊂ (cid:101) Y . We need to show that ˜ h | (cid:101) L x : (cid:101) L x → (cid:101) L x is strictly decreasing, that is if ( x, s ) , ( x, s ) ∈ (cid:101) L x are two distinct pointswith s < s , ( x, t ) = ˜ h ( x, s ), and ( x, t ) = ˜ h ( x, s ), then t > t .Since ˜ h is a homeomorphism, it follows that t (cid:54) = t . Suppose that t < t . Then there exist a >
0, and two open neighborhoods V ⊂ U of x in X such that ˜ h (cid:0) V × s i (cid:1) ⊂ U × ( t i − a ; t i + a ) , i = 0 , , (6.1) eversing orientation homeomorphisms of surfaces 19 t + a < t − a. (6.2)Due to property (Φ3), the set B is dense in X , so there exists a point y ∈ B ∩ V (cid:54) = ∅ .Then on the one hand y × R ⊂ (cid:101) Y is an orbit of ˜ F , and ˜ h : y × R → y × R reverses orientation, so if ( y, t (cid:48) i ) = ˜ h ( y, s i ), i = 0 ,
1, then t (cid:48) > t (cid:48) .On the other hand, due to (6.1) t (cid:48) i ∈ ( t i − a ; t i + a ), whence by (6.2): t (cid:48) < t + a < t − a < t (cid:48) which gives a contradition. Hence t > t .(iv) If x ∈ B , then (cid:101) L x = x × R is an orbit of ˜ F and by (i) it is invariantwith respect to ˜ h . Hence it is also invariant with respect to ˜ h .Suppose x ∈ X \ B . Then by property (Φ4), if L x = ( x × S ) ∩ Y consistsof finitely many connected components I , . . . , I n − for some n enumeratedin the cyclical order along the circle x × S . This implies that (cid:101) L x is adisjoint union of countably many open intervals I i , ( i ∈ Z ), being orbits of F which can be enumerated so that ξ ( (cid:101) I k ) = (cid:101) I k + n and p ( (cid:101) I k ) = I k mod n .Since ˜ h is a strictly decreasing homeomorphism of (cid:101) L x , it follows thatthere exists a ∈ Z such that ˜ h ( (cid:101) I k ) = (cid:101) I a − k . Hence˜ h ( (cid:101) I k ) = ˜ h ( (cid:101) I a − k ) = (cid:101) I a − ( a − k ) = (cid:101) I k . (cid:3) Now we can deduce our Theorem from Lemma 6.2.1.(a) By Corollary 4.8 applied to h there exists a unique continuous func-tion α : Y → R such that˜ h ( x, s ) = ˜ F ( x, s, α ◦ p ( x, s )) , h ( x, s ) = F ( x, s, α ( x, s )) . (b) Let ( x, s ) ∈ X × S . If x ∈ B , then x × S ⊂ Y is an orbit of F andby Corollary 5.3(b) α ( x, s ) = 0 iff h ( x, s ) = ( x, s ).Suppose x ∈ X \ B . Then ( x, s ) is a non-periodic point of F , whence h ( x, s ) = F ( x, s, α ( x, s )) = ( x, s ) is possible if and only if α ( x, s ) = 0.(c) Smoothness properties of α follows similarly to the statement (c) ofCorollary 4.8. (cid:3) Polar coordinates
Let H = R × [0; + ∞ ), Int ( H ) = R × (0; + ∞ ), and p : H → C ≡ R , p ( ρ, φ ) = ρe πiφ = ( ρ cos φ, ρ sin φ ) , be the infinite branched coveging map defining polar coordinates . Lemma 7.1. ([8, Lemma 11.1])
Let U ⊂ C be an open neighborhood of ,and h : U → C a C r , ≤ r ≤ ∞ , smooth embedding with h (0) = 0 . Thenthere exists a C r − embedding ˜ h : p − ( U ) → H such that p ◦ ˜ h = h ◦ p . Suppose, in addition, that the Jacobi matrix J ( h, of h at is orthog-onal. Then there exists a ∈ R such that for each s ∈ R ˜ h (0 , s ) = (cid:40) (0 , a + s ) if J ( h, ∈ SO(2) , (0 , a − s ) if J ( h, ∈ SO − (2) . Hence in the second case (when h reverses orientation) ˜ h (0 , s ) = (0 , s ) ,that is ˜ h is always fixed on × R . Let f : R → R be a homogeneous polynomial without multiple factorsand having degree k ≥ F = − ∂f∂y ∂∂x + ∂f∂x ∂∂y be the Hamiltonial vectorfield of f . Since the restriction p : Int ( H ) → R \ F induces a vector field ˜ F on Int ( H ). One can even obtainprecise formulas for ˜ F (see [5, § F ( r, φ ) = (cid:0) ∂∂ρ , ∂∂φ (cid:1) (cid:18) cos φ sin φ − ρ sin φ ρ cos φ (cid:19) (cid:18) − f (cid:48) y ( z ) f (cid:48) x ( z ) (cid:19) , where z = p ( ρ, φ ) = ρe πi . The latter formula can be reduced to a moresimplified form. Define the following function q : C \ → C by q ( z ) = − f (cid:48) y ( z ) + if x ( z ) z = ( − f (cid:48) y ( z ) + if x ( z ))¯ z | z | . Then ˜ F ( r, φ ) = Re( q ( z )) ∂∂ρ + Im( q ( z )) ρ ∂∂φ . Since f is a homogeneous of degree k ≥
2, it follows that ˜ F ( r, φ ) smoothlyextends to H .Let F and ˜ F be the local flows generated by F and ˜ F respectively. Then F t ◦ p ( ρ, φ ) = p ◦ ˜ F t ( ρ, φ ) whenever all parts of that identity are defined. Example 7.2.
1) Suppose 0 is a non-degenerate local extreme of f . Thenone can assume that f ( x, y ) = ( x + y ). In this case F ( x, y ) = − y ∂∂x + x ∂∂y , F ( z, t ) = ze πit , ˜ F ( ρ, φ ) = ∂∂φ , ˜ F ( ρ, φ, t ) = ( ρ, φ + t ) .
2) Let 0 be a non-degenerate saddle, so one can assume that f ( x, y ) = xy .Then F ( x, y ) = − x ∂∂x + y ∂∂y , F ( x, y, t ) = (cid:0) xe − t , ye t (cid:1) , ˜ F ( ρ, φ ) = ρ cos 2 φ ∂∂ρ + sin 2 φ ∂∂φ . Notice that writing down precise formulas for ˜ F is a rather complicatedtask. eversing orientation homeomorphisms of surfaces 21
3) If 0 is a degenerate critical point of f , so deg f ≥
3, then the situationis more complicated. Notice that in this case ˜ F is zero on ∂ H = R × F is fixed on that line. Again the formulas for F and ˜ F are highlycomplicated. Lemma 7.3. (see [5, Theorem 1.6], [8, Proof of Theorem 5.6])
Let U ⊂ R be an open neighborhood of the origin ∈ R , h : U → R an embeddingwhich preserves orbits of F , and α : U \ → R be a C ∞ function such that h ( x ) = F ( x, α ( x )) for all x ∈ U \ . Let also ˜ h : p − ( U ) → H be any liftingof h as in Lemma 7.1. Then α can be defined at x so that it becomes C ∞ in U in the following cases: (a) x is a non-degenerate local extreme of f or a (possibly degenerate)saddle point; (b) x is a degenerate local extreme of f and ˜ h is fixed on R × . (c) x is a degenerate local extreme of f and there exists an open neighbor-hood V ⊂ U and another embedding q : U → R such that q ( V ) ⊂ U , q preserves orbits of F and reverses their orientation, and h = q . Proof.
Cases (a) and (b) are proved in [5, Theorem 1.6], see also proof ofTheorem 5.6 in [8].(c) ⇒ (b) Let (cid:101) q : p − ( U ) → H be any lifting of q as in Lemma 7.1. Then (cid:101) q is a lifting of q = h . Moreover, since (cid:101) q reverse orientation of orbits,we get from Lemma 7.1 that (cid:101) q is fixed on R ×
0. Hence condition (b)holds. (cid:3) Chipped cylinders of a map f ∈ F ( M, P )Let f ∈ F ( M, P ). In what follows we will use the following notations.(i) K , . . . , K k denote all the critical leaves of f , and K = k ∪ i =1 K i . (ii) Let R K i , i = 1 , . . . , k , be an f regular neighborhood of K i choosenso that R K i ∩ R K j = ∅ for i (cid:54) = j .(iii) Let also L , . . . , L l be all the connected components of M \ K ;(iv) For each i = 1 , . . . , l let N i = L i \ Σ f . Then there exist a finite subset Q i ⊂ {− , } × S , and an immersion φ i : (cid:0) [ − , × S (cid:1) \ Q i → N i and a C ∞ embedding η : [0 , → P such that the following diagram is commutative: (cid:0) [0; 1] × S (cid:1) \ Q i φ (cid:47) (cid:47) p (cid:15) (cid:15) N if (cid:15) (cid:15) [0; 1] η (cid:47) (cid:47) P where p is the projection to the first coordinate. Notice that φ can be non-injective only at points of {− , } × S and this can happens only when P = S , see Example 8.1 and Figure 8.2d) below.We will call N i a chipped cylinder of f , see Figure 8.1. Figure 8.1.
It will also be convenient to denote N − i = φ i (cid:0) [ −
1; 0] × S (cid:1) \ Q i (cid:1) , N + i = φ i (cid:0) [0; 1] × S (cid:1) \ Q i (cid:1) , Int N i = φ i (cid:0) ( −
1; 1) × S (cid:1) . We will call N − i and N + i chipped half-cylinders of N i and f , and Int N i the interior of N i .(v) Let also Z i = N i (cid:91) (cid:16) ∪ N i ∩ K j (cid:54) = ∅ R K j (cid:17) be the union of the chipped cylinder N i with f -regular neighborhoods ofcritical leaves of f which intersect the closure N i . We will call Z i an f -regular neighborhood of N i . Example 8.1. a) Let f : [0 , × S → P be a map of class F ([0 , × S , P )having no critical points, see Figure 8.2a). Then it has a unique chippedcylinder N = [0 , × S , which coincides with its f -regular neighborhoodand K = ∅ .b) Let f : D → P be a map of class F ( D , P ) having only one criticalpoint z , see Figure 8.2b). Then K = { z } , f has a unique chipped cylinder N = D \ { z } , and its f -regular neighborhood is all D . eversing orientation homeomorphisms of surfaces 23 c) Let f : S → P be a map of class F ( S , P ) having only two criticalpoints z and z being therefore extremes of f , see Figure 8.2c). Then K = { z , z } , f has a unique chipped cylinder N = S \ { z , z } and its f -regular neighborhood is all S .d) Let M be either a 2-torus or Klein bottle with a hole and f : M → S be a map of class F ( M, S ) schematically shown in Figure 8.2d). It has onlyone critical point z and that point is a saddle, a unique critical leaf K = K ,and two chipped cylinders N and N . It follows from the Figure 8.2d)that N intersects only one K from “both sides”, in the sense that bothintersections N − ∩ K and N +1 ∩ K are non-empty.e) Let f : [0 , × S → R be a Morse function having one minimum z andone saddle point y as in Figure 8.3. Then f has two critical leaves: the point z and a critical leaf K containing y , and three chipped cylinders N , N , N . Let R K be an f -regular neighborhood of K . Then the corresponding f -regular neighborhoods of chipped cylinders are the following ones: Z = N ∪ { z } ∪ R K , Z = N ∪ R K , Z = N ∪ R K . a) b) c) d) Figure 8.2.Figure 8.3.
The following lemma describes simple properties of chipped cylinders.The proof is left for the reader.
Lemma 8.2.
Let N be a chipped cylinder of f , N − and N + be its chippedhalf-cylinders, Z be f -regular neighborhood of N , and p : M → Γ f be theprojection onto the graph of f . Then the following statements hold. (1) N − and N + are orientable manifolds. Moreover, if P = R , then N is orientable as well. However, if P = S , then it is possible to con-struct an example of f having non-orientable chipped cylinder, see Ex-ample 8.1d). (2) Each of the closures N − and N + intersects at most one critical leavesof f , and those intersections consist of open arcs being leaves of thesingular foliation Ξ f . (3) If N ∩ K i = ∅ but N ∩ K i (cid:54) = ∅ , then K i is a critical point of f beinga local extreme. (4) If M is connected and N does not contain any saddle point of f , then M = N = Z and f is one of the maps from a ) -c ) of Example 8.1. (5) N (cid:48) ∩ N (cid:48)(cid:48) ⊂ K for any two distinct chipped cylinders N (cid:48) and N (cid:48)(cid:48) of f . (6) Every regular leaf of f is contained in some chipped cylinder of f . The following theorem is the principal technical tool. Let M be a (possi-bly non-orientable) compact surface, f ∈ F ( M, P ), N be a chipped cylinderof f , and Z = N (cid:83)(cid:16) ∪ N ∩ K j (cid:54) = ∅ R K j (cid:17) be its f -regular neighborhood of N . Theorem 8.3. Let V ⊂ N be a regular leaf of f , and h ∈ S ( f | Z ) adiffeomorphism of Z such that h ( V ) = V and h reverses orientation of V .Then every leaf of the singular foliation Ξ f in Z is ( h , +) -invariant, i.e. h fixes all critical points of f in Z and preserves all other leaves of Ξ f in Z with their orientations. Suppose in addition that Z is orientable and let F be any Hamiltonianlike flow of f | Z on Z . Then there exists a unique C ∞ function α : Z → R such that h = F α on Z and α = 0 at each fixed point of h in Int N andfor each local extreme z of f in Z . Proof.
1) Let us mention that since h reverses orientation of V , it reversesorientations of all regular leaves in N . Therefore those leaves are ( h , +)-invariant, and we should prove the same for all other leaves of Ξ f in Z .First we introduce the following notation. If K ∩ N − (cid:54) = ∅ , then let K − be a unique critical leaf of f intersecting N − , and let R K − be its f -regularneighborhood. Otherwise, when K ∩ N − = ∅ , put K − = R K − = ∅ , Definefurther K + and R K + in a similar way with respect to N + . Then Z = R K − ∪ N − ∪ N + ∪ R K + . As those four sets are invariant with respect to h , it suffices to prove that h preserves leaves of Ξ f with their orientation for each of those sets.1a) Let us show that h preserves all leaves of Ξ f in N − . Since N − is an orientable manifold, one can construct a Hamiltonian likeflow of f on N − . Evidently, F satisfies conditions (Φ1)-(Φ4). Moreover,since h reverses orientation of all periodic orbits of F in N + , we get fromTheorem 6.2 that there exists a unique C ∞ function α : N → R such that eversing orientation homeomorphisms of surfaces 25 h | N − = F α and α vanishes at fixed points of h on regular leaves of f in N − . In particular, each non-periodic orbit of F in N + is ( h , +)-invariantas well.1b) Now let us prove that each leaf of Ξ f in K − is also ( h , +)-invariant.This will imply ( h , +)-invariantness of all leaves of Ξ f in R K − (see theproof of the implication (iii) ⇒ (i) in [8, Lemma 7.4]).If K − = ∅ there is nothing to prove.If K − ∩ N − = ∅ , then by Lemma 8.2(3) K − is a local extreme of f .Hence K is an element of Ξ f and its is evidently invariant with respect h ,and therefore with respect to h .Thus assume that K − ∩ N − (cid:54) = ∅ . Then this intersection contains anon-periodic orbit γ of F . Since γ is ( h , +)-invariant, it follows from [6,Claim 7.1.1] or [8, Lemma 7.4], that all elements of the foliation Ξ f arealso ( h , +) -invariant .Let us recall a simple proof of that fact. Indeed, let v be a vertex of γ being therefore a critical point of f . Then h ( v ) = v , whence h preservesthe set of all edges incident to v . Moreover, as h preserves orientationat v , it must also preserve cyclic order of edges incoming to v . But since γ (being one of those edges) is ( h , +)-invariant, it follows that so are allother edges incident to v . Applying the same arguments to those edges andso on, we will see that h preserves all edges of K with their orientation.The proofs for N + and R K + are similar.2) Assume now that Z is an orientable surface and let F be any Hamil-tonian like flow of f . We know from 1) that h preserves all orbits of F with their orientations.2a) We claim that there exists a unique C ∞ function α : N → R suchthat h | N = F α and α vanishes at fixed points of h on periodic orbits. If both K − and K + are non-empty and distinct, then the restriction of F to N satisfies conditions (Φ1)-(Φ4). Moreover, as h reverses orientationof all periodic orbits of F , the statement follows from Theorem 6.2 as in1a).However, if K − = K + , as in Example 8.1d), the situation is slightlymore complicated: N might be not of the form ([ − , × S ) \ Q for somefinite set Q ⊂ {− , } × S and Theorem 6.2 is not directly applicable.Nevertheless, one can apply that theorem to each of the sets N − , N + , andInt N and construct three functions α − : N − → R , α − : N + → R , α : Int N → R satisfying h | N − = F α − , h | N + = F α + , h | Int N = F α , and vanishing atfixed points of h on periodic orbits. From uniqueness of such functions,we get that α − = α on N − ∩ Int N and α + = α on N + ∩ Int N . A possible problem is that N intersects K − from “both sides”, andtherefore a priori α + and α − can differ on N − ∩ N + ∩ K − . However, N − ∩ N + ∩ K − consists of non-periodic orbits of F , and therefore foreachsuch orbit γ the identity h | γ = F α − | γ = F α + | γ implies that α − = α + on γ . Thus α − = α + on N − ∩ N + ∩ K − , and therefore those functions definea well defined C ∞ function α : N → R satisfying h | Int N = F α and α vanishes at fixed points of h on periodic orbits.2b) It remains to show that α extends to a shift function for h on R K − ∪ R K + and thus on all of Z . It suffices to prove that for R K − .If K − = ∅ , then R K − = ∅ and there is nothing to prove.If K − is a local extreme of f , then by Lemma 7.3 (cases (a) and (c) fornon-degenerate and degenerate critical point) α can be defined at K − sothat it becomes C ∞ .In all other cases K − contains a non-periodic orbit of F . Then by theimplication (ii) ⇒ (iv) of [8, Lemma 7.4], α extends to a C ∞ shift functionfor h on R K − . It remains to prove the following lemma: Lemma 8.3.1. α ( z ) = 0 for every local extreme of f in Z . Proof.
Indeed, it is evident, that arbitrary small neighborhood of z con-tains a periodic orbit γ of F . Since h reverses orientations of γ , we havefrom Lemma 5.1(i) that h always has at least one fixed point x ∈ γ (in factit has even two such points). Hence by Corollary 5.3(b), α ( x ) = 0. Thenby continuity of α we should have that α ( z ) = 0 as well. (cid:3) Theorem 8.3 is completed. (cid:3) Creating almost periodic diffeomorphisms
Let M be a compact orientable surface, f ∈ F ( M, P ), Z be an f -adaptedsubsurface, h ∈ S ( f ) be such that h ( Z ) = Z , and m ≥
2. If h m | Z is isotopicto the identity of Z by f -preserving isotopy, then the following Lemma 9.1gives conditions when one can change h on M \ Z so that its m -power willbe f -preserving isotopic to the identity on all of M . The proof follows theline of [8, Lemma 13.1(3)] in which M is a 2-disk or a cylinder. Lemma 9.1. (cf. [8, Lemma 13.1(3)])
Let M be a connected compact ori-entable surface, f ∈ F ( M, P ) , Z be an f -adapted subsurface, and h ∈ S ( f ) be such that h ( Z ) = Z . Suppose that the following conditions hold. (1) Each component of Z contains at least one saddle point of f . eversing orientation homeomorphisms of surfaces 27 (2) h m is isotopic in S ( f ) to a diffeomorphism τ fixed on some neighborhoodof Z ( by [8, Lemma 7.1] this condition holds if there exists a C ∞ function α : Z → R such that h m | Z = F α ) . (3) There exists m ≥ and a ≥ such that connected components of M \ Z can be enumerated as follows: Y , Y , · · · Y ,m − Y , Y , · · · Y ,m − · · · · · · · · · · · · Y a, Y a, · · · Y a,m − (9.1) so that h ( Y i,j ) = Y i,j +1 mod m for all i, j , that is h cyclically shiftscolumns in (9.1) .Then there exists g ∈ S ( f ) such that g = h on Z and g m ∈ S id ( f ) , thatis g m = F β for some C ∞ function β : M → R . Proof.
Let Y j = a (cid:83) i =1 Y i,j , j = 0 , . . . , m −
1, be the union of componentsfrom the same column of (9.1). Then h ( Y j ) = Y j +1 mod m . Notice thatcondition (3) implies that h m ( Y i,j ) = Y i,j for all i, j .We will show that the the desired diffeomorphism g ∈ S ( f ) can be definedby the formula: g ( x ) = (cid:40) h ( x ) , x ∈ Z ∪ Y ∪ · · · ∪ Y m − ,τ − ◦ h ( x ) , x ∈ Y m − . Indeed, by definition g = h on Z . Moreover, as τ is fixed on some neigh-borhood of Z , it also fixed near Z ∩ Y m − . Therefore g = h = τ − ◦ h near Z ∩ Y m − , and so g is a well defined C ∞ map. It remains to prove that g m = F β ∈ S id ( f ) for some C ∞ function β : M → R .Let F be a Hamiltonian flow for f . Since τ and h m are isotopic in S ( f ),it follows that τ − ◦ h m ∈ S id ( f ). Hence by Lemma 2.3.5, τ − ◦ h m = F α for some C ∞ function α : M → R .Since S id ( f ) is a normal subgroup of S ( f ), it follows that h j ◦ ( τ − ◦ h m ) ◦ h − j = h j ◦ τ − ◦ h m − j ∈ S id ( f ) , j = 0 , . . . , m − , as well. Therefore, again by Lemma 2.3.5, h j ◦ τ − ◦ h m − j = F α j for some C ∞ function α j : M → R .As τ is fixed on some neighborhood of Z , it follows that for each j F α j = h j ◦ τ − ◦ h m − j = τ − ◦ h m = F α on Z. Then the assumption (1) that every connected component Z (cid:48) of Z containsa saddle point, implies that F has a non-closed orbit γ in Z (cid:48) . Therefore α = α j on γ . Since Z (cid:48) is connected, it follows from local uniqueness of shift functions for τ − ◦ h m | Z (see Corollary 4.5) that α = α j near Z (cid:48) forall j = 0 , . . . , m −
1. Hence α = α j near all of Z for all j = 0 , . . . , m − C ∞ function β : M → R given by: β ( x ) = (cid:40) α ( x ) , x ∈ Z,α j ( x ) , x ∈ Y j , j = 0 , . . . , m − . We claim that g m = F β .a) Indeed, if x ∈ Z , then g m ( x ) = h m ( x ) = F α ( x ) = F β ( x ).b) Also notice that g ( Y i,j ) = Y i,j +1 mod m and g ( Y j ) = Y j +1 mod m . Then g m | Y j is the following composition of maps: Y j h −−→ Y j +1 h −−→ · · · h −−→ Y m − τ − ◦ h −−−−−→ Y h −−→ · · · h −−→ Y j . which thus coincides with h j ◦ τ − ◦ h m − j = F α j = F β . (cid:3) Proof of Theorem 3.6
Let M be a connected compact surface, f ∈ F ( M, P ), h ∈ S ( f ), A bethe union of all regular leaves of f being h − -invariant and K , . . . , K k beall the critical leaves of f such that A ∩ K i (cid:54) = ∅ . For i = 1 , . . . , k , let R K i be an f -regular neighborhood of K i chosen so that R K i ∩ R K j = ∅ for i (cid:54) = j and Z := A (cid:91)(cid:16) k ∪ i =1 R K i (cid:17) . Assume that Z is non-empty, orientable and every connected component γ of ∂Z ∩ Int M separates M . We have to prove that there exists g ∈ S ( f )which coincide with h on Z and such that g ∈ S id ( f ). Lemma 10.1.
There exists a unique C ∞ function α : Z → R such that h | Z = F α and α = 0 at each fixed point of h on h − -invariant regularleaves of f . Proof.
Let V be a regular leaf V of f , and N a chipped cylinder of f suchthat V ⊂ Int N . If V is h − -invariant, then so is every other regular leaf V (cid:48) ⊂ Int N . This implies that Z is a union of f -regular neighborhoods Z , . . . , Z l of some chipped cylinders N , . . . , N l of f .By Theorem 8.3, for each i = 1 , . . . , l there exists a unique C ∞ function α i : Z i → R such that h | Z i = F α i and α = 0 at each fixed point of h oneach on h − -invariant regular leaf of f in Z i .Notice that if Z i ∩ Z j (cid:54) = ∅ , then every connected component W of thatintersection always contains a non-periodic orbit γ of F . Therefore byuniqueness of shift functions (Corollary 4.5) we obtain that α i = α j on W . eversing orientation homeomorphisms of surfaces 29 Hence the functions { α i } i =1 ,...,l agree on the corresponding intersections,and therefore they define a unique C ∞ function α : Z → R such that h | Z = F α . Then α = 0 at each on h − -invariant regular leaf of f in Z i . (cid:3) If M = Z , then theorem is proved. Thus suppose that M (cid:54) = Z . Lemma 10.2.
The number of connected components M \ Z is even , andthey can be enumerated by pairs of numbers: Y , Y , . . . Y a, Y , Y , . . . Y a, (10.1) for some a > so that h exchanges the rows in (10.1) , that is h ( Y i, ) = Y i, and h ( Y i, ) = Y i, for each i . Proof.
Let Y , . . . , Y q be all the connected components of M \ Z . Denote γ i := Y i ∩ Z . Then by condition (B), γ i is a unique common boundarycomponent of Y i and Z .Since h ( Z ) = Z , it follows that h induces a permutation of connectedcomponents of { Y i } i =1 ,...,q . Moreover, by Lemma 3.5.1 h ( γ i ) ∩ γ i = ∅ ,whence h ( Y i ) ∩ Y i = ∅ as well. On the other hand, h = F α , whence h ( γ i ) = F α ( γ i ) = γ i , and therefore h ( Y i ) = Y i .Thus { Y i } i =1 ,...,q splits into pairs which are exchanged by h . (cid:3) Now it is enough to apply Lemma 9.1 with m = 2. Then there exists g such that g = h on Z and g ∈ S id ( f ). Notice that each component of Z contains at least one saddle point of f , otherwise by Lemma 8.2 (4) we havethat M = Z . So the first condition (1) of Lemma 9.1 holds. The secondcondition (2) follows from Lemma 10.1 and the third condition (3) followsfrom Lemma 10.2. Theorem 3.6 is completed. (cid:3) References [1] G. M. Adelson-Welsky, A. S. Kronrode. Sur les lignes de niveau des fonctions continuesposs´edant des d´eriv´ees partielles.
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Iryna Kuznietsova, Sergiy Maksymenko
Topology Laboratory, Department of Algebra and Topology, Institute ofMathematics of NAS of Ukraine, Tereshchenkivska str. 3, Kyiv, 01601, Ukraine