aa r X i v : . [ m a t h . D S ] D ec IMAGE OF A SHIFT MAP ALONGTHE ORBITS OF A FLOW
SERGIY MAKSYMENKO
Abstract.
Let ( F t ) be a smooth flow on a smooth manifold M and h : M → M be a smooth orbit preserving map. The followingproblem is studied: suppose that for every z ∈ M there exists agerm α z of a smooth function at z such that h ( x ) = F α ( x ) ( x ) near z ;can the germs ( α z ) z ∈ M be glued together to give a smooth functionon all of M ? This question is closely related to reparametrizationsof flows. We describe a large class of flows ( F t ) for which the aboveproblem can be resolved, and show that they have the followingproperty: any smooth flow ( G t ) whose orbits coincides with theones of ( F t ) is obtained from ( F t ) by smooth reparametrization oftime. Introduction
Let M be a smooth ( C ∞ ), connected, m -dimensional manifold possi-bly non-compact and with or without boundary, F be a smooth vectorfield on M tangent to ∂M and generating a flow F : M × R → M , andΣ be the set of singular points of F . For x ∈ F we will denote by o x the orbit of x and if x is periodic, then Per( x ) is the period of x .Let D ( M ) be the group of all C ∞ diffeomorphisms of M and D ( F )be the subgroup of D ( M ) consisting of all diffeomorphisms of F , thatis diffeomorphisms h : M → M such that h ( o ) = o for every orbit of F . A natural problem, which usually appears in studying functionalspaces, is to find general formulas for elements of D ( F ), or parametrize D ( F ) with elements of a certain space which seems to be simpler , e.g.with functions. In general this question is very difficult. However, ifwe confine ourselves with the path component of D ( F ) with respect tosome natural topology, then in many cases the mentioned problem canbe satisfactory resolved in terms of the flow of F .The aim of the present paper is to give sufficient conditions when the“ local parametrizations of elements of D ( F ) via functions can be gluedto a global one ”, (Theorem 5.25). We also present a class of vectorfields satisfying those conditions (Theorem 8.41). Date : 5/11/2009.1991
Mathematics Subject Classification.
Key words and phrases. orbit preserving diffeomorphism, shift map.
We will now explain the meaning of the previous paragraph. Firstof all it is more convenient to extend D ( F ) and work with the sub-semigroup E ( F ) of C ∞ ( M, M ) consisting of maps h : M → M suchthat(1) h ( o ) ⊂ o for every orbit o of F ;(2) h is a local diffeomorphism at every singular point z ∈ Σ.We will call E ( F ) the semigroup of endomorphisms of F . Evidently, D ( F ) = E ( F ) ∩ D ( M ).For 0 ≤ k ≤ ∞ denote by E id ( F ) k (resp. D id ( F ) k ) the path com-ponent of the identity map id M in E ( F ) (resp. D ( F )) in the weaktopology W k , see § E id ( F ) k consists of all maps h ∈ E ( F ) which are homotopic to id M in E ( F ) via a homotopy which induces a homotopy on the level of k -jets . In particular, E id ( F ) con-sists of all h ∈ E ( F ) homotopic to id M in E ( F ). Moreover, E id ( F ) k for k ≥ contains (but does not coincide with) the space of all h ∈ E ( F )homotopic to id M in E ( F ) via some C k homotopy. Similar descriptionshold for D id ( F ) k .Define also the following map ϕ : C ∞ ( M, R ) → C ∞ ( M, M ) by ϕ ( α )( x ) = F ( x, α ( x )) , α ∈ C ∞ ( M, R ) , which will be called ϕ the shift map along the orbits of F , see [22] .The map h = ϕ ( α ) : M → M will be called the shift via α , which inturn will be a shift function for h . Denote the image of ϕ in C ∞ ( M, M )by Sh ( F ). Lemma 1.1. [22, Lm. 20 & Cor. 21]
Let α ∈ C ∞ ( M, R ) and z ∈ M .Then ϕ ( α ) is a local diffeomorphism at z iff and only if F ( α )( z ) = − ,where F ( α ) is the Lie derivative of α along F . In particular, this holdsfor each z ∈ Σ , as F ( α )( z ) = 0 = − . Finally, consider the following subset of C ∞ ( M, R ):Γ + := { α ∈ C ∞ ( M, R ) : F ( α ) > − } . Lemma 1.2. [26, Lm. 3.6]
The following inclusions hold true (1.1) Sh ( F ) ⊂ E id ( F ) ∞ ⊂ · · · ⊂ E id ( F ) ⊂ E id ( F ) ,ϕ V (Γ + V ) ⊂ D id ( F ) ∞ ⊂ · · · ⊂ D id ( F ) ⊂ D id ( F ) . If D id ( F ) k ⊂ Sh ( F ) for some k = 0 , . . . , ∞ , e.g. if Sh ( F ) = E id ( F ) k ,then ϕ (Γ + ) = D id ( F ) k . The identity Sh ( F ) = E id ( F ) k means that for each h ∈ E ( F ) beinghomotopic to id M via a homotopy in E ( F ) inducing a homotopy of k -jets there exists a smooth function α h : M → R such that(1.2) h ( x ) = F ( x, α h ( x )) , I must warn the reader that my paper [22] contains some misprints and gaps.However, they are not “dangerous” for the present paper, see Remark 1.4 below.
MAGE OF A SHIFT MAP ALONG THE ORBITS OF A FLOW 3 for all x ∈ M . Then the representation (1.2) can be regarded as generalformulas for elements of E id ( F ) r mentioned above.In general, every h ∈ E id ( F ) has a C ∞ shift function λ defined atleast on M \ Σ. This λ can be uniquely reconstructed from a particularhomotopy between h and id M in E ( F ), but it may depend on such ahomotopy and does not always extend to a smooth function on all of M , see [22, Th. 25] and Theorem 4.16. In fact the following statementholds true: if Σ = ∅ , then Sh ( F ) = E id ( F ) and the map ϕ : C ∞ ( M, R ) −→ Sh ( F ) = E id ( F ) satisfies a covering path axiom.Suppose Σ = ∅ . It is shown in [22] (see Corollary 8.38) that if F islinearizable (or more generally a “regular extension” of a linear vectorfield) at each of its singular points, then Sh ( F ) = E id ( F ) . Moreover,in [23, 27] there were given examples of vector fields F on R for which Sh ( F ) = E id ( F ) and that for some of them E id ( F ) = E id ( F ) , seeLemma 9.48. The results of the present paper arise from analysis ofthose examples.In this paper we consider the problem of gluing local shift functionsto a global one. More precisely, let h ∈ E ( F ) and suppose that for each y ∈ Σ there exists a neighbourhood V y and a C ∞ function α y : V y → R such that (1.2) holds for all x ∈ V y . Does there exist α ∈ C ∞ ( M, R ) such that (1.2) holds on all of M , that is whether h ∈ Sh ( F )?Due to Lemma 1.2 it is necessary that h ∈ E id ( F ) k for some k ≥ k ≥ F hold true, then h ∈ Sh ( F ). Theorems 5.24 is based on results of M. Newman, A. Dress,D. Hoffman, and L. N. Mann [6] about lower bounds of diameters of Z p -actions on topological manifolds.In § Remark 1.3.
The idea of substituting a function into a flow-mapinstead of time is not new, see e.g. [16, 3, 45, 20, 39, 18, 19], wherereparametrizations of measure preserving flows and mixing propertiesof such flows are studied. In particular, in those papers the values ofsuch functions and orbit preserving maps on sets of measure zero wereignored, so they allowed to be even discontinuous.There are also analogues of this approach for discrete dynamicalsystems on Cantor set, e.g. [10, 11, 1].In the present paper we consider smooth orbit preserving maps andrequire smoothness of their shift functions. This turned out to be usefulfor the study of homotopical properties of groups of orbit preservingdiffeomorphisms for vector fields, and stabilizers and orbits of certainclasses of smooth functions with respect to actions of diffeomorphism
SERGIY MAKSYMENKO groups, see [22, 24, 25, 28]. The results of this paper will be used toextend [24].
Remark 1.4.
As noted in the introduction, the paper [22] containssome misprints and gaps. However they do not impact on the resultsof the present paper.1) The proof of [22, Pr. 10], used for [22, Th. 12] (see Theorem 3.8), isincorrect: it was wrongly claimed that the map Ψ( x, ∗ ) : I → GL n ( R ) ,associating to each t the Jacobi matrix of the flow map Φ t at x , is ahomomorphism . Nevertheless the statement of [22, Pr. 10] is true, andreparations and extensions of that proposition are given in [29]. Inparticular, Theorem 3.8 is valid.2) [22, Defn. 24] should contain an assumption that f t is a localdiffeomorphism at every z ∈ Σ ∩ V (i.e. belongs to E ( F , V )). Thisassumption was explicitly used in [22, Th. 27 & Lm. 31] but it wasnot verified in [22, Lm. 29] so the proof of that lemma requires simpleadditional arguments. We provide them in 1) of Lemma 7.31.3) [22, Eq. (10)] is misprinted and must be read as follows: α ( x ) = p ◦ f ( x ) − p ◦ Φ( x, a ) + a. See Eq. (4.9) for another variant of this formula.4) Assumptions of [22, Defn. 15] are not enough for the proof of [22,Th. 17]. In fact in [22, Defn. 15] it should be additionally required (inthe notations of that definition) that ϕ V ( M ) is open in ϕ V ( C ∞ ( V, R ))with respect to some topology W r . Such an assumption was explicitlyused in [22, Th. 17]. This implies useless of [22, Lm. 28] being the part(B) of [22, Th. 27] concerning (S)-points and verifying [22, Defn. 15]for linear vector fields, and also incorrectness of the part of [22, Th. 1]claiming that the shift map ϕ is either a homeomorphism or a coveringmap.5) The “division” lemma [22, Lm. 32] also is not true. Let F beeither the field R or C , V be an open neighbourhood of the origin in F ,and Z : C ∞ ( V, F ) → C ∞ ( V, F ) be the multiplication by z map, that is Z ( α )( z ) = zα ( z ) for α ∈ C ∞ ( V, F ). Evidently, Z is an injective map. Itwas wrongly claimed in [22, Lm. 32] that the inverse of Z is continuousbetween topologies W r for all r . In fact, it follows from the Hadamardlemma, that in the case F = R the map Z − is continuous from W r +1 topology to W r for all r ≥
0, while in the case F = C I can only provethat Z − is continuous only between W ∞ topologies. This implies thatthe estimations of the continuity of the local inverses of shift maps areincorrect, see paragraph after [22, Eq. (26)].Incorrect statements mentioned in 4) and 5) are not used in thepresent paper. Their reparation is given in [28].All other results in [22] are correct. In particular, besides Theo-rem 12 and Eq. (10) we will use only the following “safe” statementsfrom [22]: [22, Lm. 5 & 7] (see Lemma 3.7), [22, Lm. 20 & Cor. 21] MAGE OF A SHIFT MAP ALONG THE ORBITS OF A FLOW 5 (see Lemma 1.1), [22, Th. 25] (it will be extended in Theorem 4.16),[22, Lm. 29] (it will also be extended in Theorem 9.47 of the presentpaper), and [22, Eqs. (23)-(26)], see §§ Structure of the paper.
In next section we recall the definitionof Whitney topologies and introduce certain types of deformations.Then in § V ⊂ M and recall their properties established in [22, 29]. In § § ϕ = E id ( F ) k for some k ≥ § the identity Sh ( F ) = E id ( F ) k is preserved ifwe properly decrease our manifold ”, see Lemma 6.29. This statementwill not be used but it illustrates local nature of assumptions of Theo-rem 5.25.Further in § § Sh ( F ) = E id ( F ) k forsome k ≥
0. In particular, we recall certain “extension” properties( E ) d , ( d ≥ F introduced in [22]. Property( E ) guarantees Sh ( F ) = E id ( F ) . On the other hand we also giveexamples of vector fields for which Sh ( F ) = E id ( F ) = E id ( F ) . At theend of § Sh ( F ) = E id ( F ) .Further in § ESD ( l ; d, k ) whichguarantee Sh ( F ) = E id ( F ) k and prove Theorem 8.41.2. Preliminaries
Real Jordan normal form of a matrix.
For k ∈ N let E k bethe unit ( k × k )-matrix, A be a square ( k × k )-matrix, and a, b ∈ R .Define the following matrices: J p ( A ) = (cid:18) A ··· E k A ··· ··· ··· ··· ··· ··· E k A (cid:19) , R ( a, b ) = (cid:0) a b − b a. (cid:1) , J p ( a ± ib ) = J p ( R ( a, b )) . For square matrices
A, B it is also convenient to put A ⊕ B = ( A B ).Now let C be an ( n × n ) matrix over R . Then by the real variantof Jordan’s normal form theorem, e.g. [38], C is similar to a matrix ofthe following form: s ⊕ σ =1 J q σ ( a σ ± ib σ )) ⊕ r ⊕ τ =1 J p τ ( λ τ ) , where a σ ± ib σ ∈ C and λ τ ∈ R are all the eigen values of C . SERGIY MAKSYMENKO
Deformations.
Let T be a topological space and V, S, U be smoothmanifolds. We introduce here special types of deformations which willbe used throughout the paper.Recall that for every k = 0 , . . . , ∞ the space C k ( V, U ) can be endowedwith the so-called weak topology which we will denote by W k , see [14]for details. A topology W coincides with the compact open one, [21],while topologies W k , ( k ≥ J k ( V, U ) be the manifold of k -jets of C k maps h : V → U .Associating to every such h its k -jet prolongation j k h : V → J k ( V, U ),we obtain a canonical embedding C k ( V, U ) ⊂ C k ( V, J r ( V, U )). Endow C k ( V, J k ( V, U )) with the topology W . Then the induced topology on C k ( V, U ) is called W k . Finally, the topology W ∞ is generated by all W k for k < ∞ .For a subset H ⊂ C k ( V, U ) denote by C k ( S, H ) the space of all C k maps Ω : V × S → U such that Ω σ = Ω( · , σ ) : V → U belongs to H foreach σ ∈ S . Thus C k ( S, H ) ⊂ C k ( V × S, U ). Definition 2.5.
Let
H ⊂ C ∞ ( V, U ) be a subset. A continuous map (2.3) Ω : V × S × T → U will be called an ( S ; T, k ) -deformation in H if (1) for every ( σ, τ ) ∈ S × T the map Ω ( σ,τ ) = Ω( · , σ, τ ) : V → U belongs to H ; (2) for every τ ∈ T the map Ω τ = Ω( · , · , τ ) : V × S → U is C ∞ andthe induced mapping ( V × S ) × T → J k ( V × S, U ) , ( x, σ, τ ) j k Ω τ ( x, σ ) is continuous. Roughly speaking S is the space of C ∞ -parameters and T is thespace of “almost” C k -parameters of the deformation Ω. We will usuallydenote the “space of parameters” S × T by P .Evidently, Ω can be regarded as the following map ω : T → C ∞ ( S, H ) ⊂ C ∞ ( V × S, U ) , ω ( τ )( σ )( x ) = Ω( x, σ, τ ) . Let S = ∗ be a point, so Ω : V × T → U and ω : T → H . Supposealso that T is locally compact. Then it is well known, e.g. [21], thatcontinuity of Ω is equivalent to continuity of ω into the topology W of H . Therefore it follows from the above description of W k that a( ∗ ; T, k )-deformation is the same that a continuous map T → H intothe topology W k .A ( ∗ ; T, k )-deformation will also be called a (
T, k ) -deformation . More-over, if T = I , then an ( I, k )-deformation Ω : V × I → U will be calleda k -homotopy . It can be thought as a continuous path I → H into thetopology W k . In particular, a 0-homotopy is a usual homotopy. MAGE OF A SHIFT MAP ALONG THE ORBITS OF A FLOW 7 If T = ∗ is a point, then Ω : V × S → U is a C ∞ map which, bydefinition, belongs to C ∞ ( S, H ). In this case k does not matter and wewill call Ω an ( S, C ∞ ) -deformation .The following lemma is left for the reader. Lemma 2.6.
Let Ω be an ( S ′ ; T ′ , k ) -deformation given by (2.3) . Letalso S ′ and U ′ be smooth manifolds, T ′ be another topological space, µ : S ′ → S and u : U → U ′ be smooth maps, and ρ : T ′ → T be acontinuous map. Then the following map Ω : V × S ′ × T ′ → U ′ , Ω( x, σ, τ ) = u ◦ Ω( x, µ ( σ ) , ρ ( τ )) is an ( S ′ ; T ′ , k ) -deformation. (cid:3) Shift map
Local flows.
Let F be a vector field on M tangent to ∂M . Thenthe orbit of a point x ∈ M is a unique map F x : R ⊃ ( a x , b x ) → M such that F x (0) = x and ddt F x = F ( F x ), where ( a x , b x ) ⊂ R is themaximal interval on which a map with the previous two properties canbe defined. If x is either zero or periodic point for F , then ( a x , b x ) = R .By standard theorems in ODE the following subset of M × R dom ( F ) = ∪ x ∈ M x × ( a x , b x ) , is an open, connected neighbourhood of M × M × R .Then the local flow of F is the following map, being in fact C ∞ , F : M × R ⊃ dom ( F ) −→ M, F ( x, t ) = F x ( t ) . If F has compact support, then dom ( F ) = M × R , so F is a global flow,e.g. [38].It is well-known that if F is not global, then we can find a smoothstrictly positive function µ : M → (0 , + ∞ ) such that the flow G of G = µF is global, e.g. [12, Cor. 2].3.2. Shift map.
For open V ⊂ M let func ( F , V ) be the subset of C ∞ ( V, R ) consisting of functions α whose graph { ( x, α ( x )) | x ∈ V } is contained in dom ( F ). It F either has no non-periodic orbits, orgenerates a global flow, then func ( F , V ) = C ∞ ( V, R ).Then we can define the following map ϕ V : C ∞ ( V, R ) ⊃ func ( F , V ) −→ C ∞ ( V, M )by ϕ V ( α )( x ) = F ( x, α ( x )). We will call ϕ V the shift map along theorbits of F and denote its image in C ∞ ( M, M ) by Sh ( F , V ).Denote by E ( F , V ) the subset of C ∞ ( V, M ) consisting of all mappings h : V → M such that(1) h ( o ∩ V ) ⊂ o for every orbit o of F , and(2) h is a local diffeomorphism at every singular point z ∈ Σ ∩ V . SERGIY MAKSYMENKO
Let also E id ( F , V ) k , (0 ≤ k ≤ ∞ ), be the path-component of the iden-tity inclusion i V : V ⊂ M in E ( F , V ) with respect to the topology W k .It consists of all h ∈ E ( F , V ) which are k -homotopic to i V in E ( F , V ).If V = M , then we will omit V and simply write E ( F ) := E ( F , M ), func ( F , M ) := func ( F ), Sh ( F ) := Sh ( F , M ), and so on.It can be shown similarly to Lemma 1.2 that Sh ( F , V ) ⊂ E id ( F , V ) ∞ ⊂ · · · ⊂ E id ( F , V ) ⊂ E id ( F , V ) . We study the problem whether Sh ( F , V ) = E id ( F , V ) k for some k .Lemma 3.9 below implies that E ( F , V ) and Sh ( F , V ) do not changeunder reparametrizations, that is when we replace F with µF for somestrictly positive, C ∞ function µ : M → (0 , + ∞ ). Therefore we canalways assume that F generates a global flow. This simplifies manyarguments.3.3. The kernel of shift map.
The following subset ker( ϕ V ) := ϕ − V ( i V )of C ∞ ( V, R ) will be called the kernel of ϕ V . Thus ker( ϕ V ) consists ofall µ ∈ C ∞ ( V, R ) such that F ( x, µ ( x )) ≡ x for every x ∈ V . Lemma 3.7. [22, Lm. 5 & 7]
Let α, β ∈ func ( F , V ) . Then ϕ V ( α ) = ϕ V ( β ) iff α − β ∈ ker( ϕ V ) . Hence if func ( F ) = C ∞ ( V, R ) , then ker( ϕ V ) is a group with respect to the point-wise addition and ϕ V yields abijection between the factor group C ∞ ( V, R ) / ker( ϕ V ) and the image Sh ( F , V ) .Every θ ∈ ker( ϕ V ) is locally constant on orbits of F . If x ∈ V isnon-periodic, then θ ( x ) = 0 . If x is periodic, then θ ( x ) = n Per( x ) forsome n ∈ Z . Theorem 3.8 (Description of ker( ϕ V )) . [22, Th. 12], [29] Let V ⊂ M be a connected , open subset. If IntΣ ∩ V = ∅ , then ker( ϕ V ) = { µ ∈ func ( F , V ) : µ | V \ IntΣ = 0 } . Suppose Σ ∩ V is nowhere dense in V . Then one of the followingpossibilities for ker( ϕ V ) is realized: Non-periodic case: ker( ϕ V ) = { } , so ϕ V is injective. Periodic case: ker( ϕ V ) = { n θ } n ∈ Z for some θ ∈ C ∞ ( V, R ) calledthe positive generator of ker( ϕ V ) and having the following proper-ties: (1) θ > on all of V , so V \ Σ consists of periodic points only, andtherefore func ( F , V ) = C ∞ ( V, R ) ; (2) there exists an open and everywhere dense subset Q ⊂ V suchthat θ ( x ) = Per( x ) for all x ∈ Q ; (3) for every orbit o of F the restriction θ | o ∩ V is constant; In [22] ker( ϕ V ) was denoted by Z id . MAGE OF A SHIFT MAP ALONG THE ORBITS OF A FLOW 9 (4) θ extends to a C ∞ function on the F -invariant set U = F ( V × R ) and the vector field G = θF generates a circle action on U : G : U × S → U, G ( x, t ) = F ( x, θ ( x ) · t ) , where x ∈ U , t ∈ S = R / Z . Reparametrization of time.
The following lemma describes thebehavior of Sh ( F ) and E ( F ) under reparametrizations of time. Lemma 3.9. c.f.[30]
Let µ : M → R be a C ∞ function. Consider thevector field G = µF . Denote by Σ F and Σ G the sets of singular pointsof F and G respectively. Then the following statements hold true. E ( G ) ⊂ E ( F ) , and E ( G ) = E ( F ) iff µ = 0 on M \ Σ F . Sh ( G ) ⊂ Sh ( F ) , and Sh ( G ) = Sh ( F ) iff µ = 0 on M \ IntΣ F .Proof.
1) Evidently, every orbit of G is contained in some orbit of F ,whence E ( G ) ⊂ E ( F ). Suppose µ = 0 on M \ Σ F . Then the foliationsby orbits of F and G coincide, whence E ( G ) = E ( F ). Conversely, if µ ( x ) = 0 for some x ∈ M \ Σ F , then x is a fixed point for G , so h ( x ) = x for every h ∈ E ( G ), while E ( F ) contains maps g such that g ( x ) = x . Hence E ( G ) ( E ( F ).2) Define the following function α : dom ( G ) → R by(3.4) γ ( x, s ) = Z s µ ( G ( x, t )) dt. Then it is well known that G ( x, t ) = F ( x, γ ( x, t )), e.g. [4, Prop. 1.28].It follows that(3.5) G ( x, α ( x )) = F (cid:0) x, γ ( x, α ( x )) (cid:1) , α ∈ func ( G ) . Hence Sh ( µF ) ⊂ Sh ( F ) . Suppose µ = 0 on M \ IntΣ F . Then µ = 0 on some neighbourhood of M \ IntΣ F , so we can find another function ν : M → (0 , + ∞ ) which isstrictly positive and coincides with µ on a neighbourhood of M \ IntΣ F .Hence G = µF = νF and F = ν G . Therefore Sh ( F ) ⊂ Sh ( G ), andthus Sh ( G ) = Sh ( F ).Conversely, suppose µ ( x ) = 0 for some x ∈ M \ IntΣ F . Let β ∈ func ( F ) be a function such that β ( x ) = 0, and g = ϕ ( β ). If x isperiodic, we will assume that 0 < β ( x ) < Per( x ). We claim that g ∈ Sh ( F ) \ Sh ( G ).Suppose g ( x ) = G ( x, α ( x )) for some α ∈ func ( G ). Put β ′ ( x ) = γ ( x, α ( x )). Then by (3.5) g ( x ) = F ( x, β ′ ( x )), whence ν = β − β ′ ∈ ker( ϕ ). On the other hand by (3.4) γ ( x, s ) = 0 for all s . In particular, β ′ ( x ) = 0. We will show that β ( x ) = 0 which contradicts to theassumption. Consider three cases.(a) IntΣ F = ∅ . Then by Theorem 3.8 β − β ′ vanishes on M \ IntΣ.In particular, β ( x ) = 0.(b) IntΣ F = ∅ and ker( ϕ ) = { } . Then β ≡ β ′ . (c) IntΣ F = ∅ and ker( ϕ ) = { nθ } n ∈ Z . Then β − β ′ = nθ for some n ∈ Z . But by assumption 0 < β ( x ) < Per( x ) ≤ θ ( x ) and β ′ ( x ) = 0,whence β ( x ) = 0 as well. (cid:3) Corollary 3.10.
Let µ : M → R be a C ∞ function such that µ ( x ) = 0 for some x ∈ Σ \ IntΣ , and G = µF . Then Sh ( G ) = E id ( G ) ∞ .Proof. By Lemma 3.9 Sh ( G ) ( Sh ( F ) ⊆ E id ( F ) ∞ = E id ( G ) ∞ . (cid:3) Property
GSF . In [27] shift map was used to describe a class ofvector fields F having the following property which will be called inthe present paper GSF : Definition 3.11. (c.f. [5], [27])
Say that a C ∞ vector field F on amanifold M has property GSF if for any C ∞ vector field G on M suchthat every orbit of G is contained in some orbit of F there exists a C ∞ function α such that G = αF . Evidently, α always exists on the set of non-singular points of F andthe problem is to prove that it can be smoothly extended to all of M .In particular, every non-singular vector field has GSF .It follows from [27] that if Sh ( F ) = E id ( F ) ∞ and the map ϕ of F satisfies a smooth variant of covering path axiom at each z ∈ Σ, then F has property GSF . As an application of Theorem 5.25 we will presenta class of vector fields having property
GSF , see Theorem 8.41.To explain the notation
GSF let us reformulate this definition inalgebraic terms. Notice that the space V ( M ) of C ∞ vector fields on M can be regarded as a C ∞ ( M, R )-module. For each F ∈ V ( M ) definethe principal submodule h F i of F as follows: h F i := { αF : α ∈ C ∞ ( M, R ) } . Thus the inclusion h G i ⊂ h F i means that G = αF for some α ∈C ∞ ( M, R ), so G is smoothly divided by F .On the other hand, we can introduce the following relation on V ( M ) being reflexive and transitive. Let F, G ∈ V ( M ). We say that G F if and only if each orbit of G is contained in some orbit of F .Evidently, h G i ⊂ h F i implies G F .Then F satisfies condition GSF iff G F implies h G i ⊂ h F i for any G ∈ V ( M ). In other words, h F i is the greatest principal submod-ule among all principal submodules h G i whose foliation by orbits isobtained by partitioning the corresponding foliation of F .In Theorem 8.41 we present a class of vector fields satisfying GSF .This will also extend [27, Th. 11.1]. In [27] I used the term parameter rigidity for this property. But, as the refereeof the present paper noted, usually the action of a group Γ is called parameter rigid if any other Γ-action with the same orbits is smoothly conjugate to the original one(up to a linear reparametrization of orbits). Therefore here another term
GSF isused.
MAGE OF A SHIFT MAP ALONG THE ORBITS OF A FLOW 11
Comparison of shift maps for open sets.Lemma 3.12.
Let W ⊂ V be two open connected subsets of M . If ϕ V is periodic, then so is ϕ W . Moreover, if v and w are the correspondingpositive generators of ker( ϕ V ) and ker( ϕ W ) respectively, then v | W = w. Hence if ϕ W is non-periodic, then so is ϕ V .Proof. We have that IntΣ ∩ V = ∅ and F ( x, v ( x )) = x for all x ∈ V .In particular, IntΣ ∩ W = ∅ as well and F ( x, v ( x )) = x for all in W . Hence v | W ∈ ker( ϕ W ) is a non-zero shift function for the identityinclusion i W : W ⊂ M , and therefore ϕ W is also periodic.Let w be the positive generator of ker( ϕ W ). Then v | W = nw forsome n ∈ N . We claim that n = 1.Indeed, by (4) of Theorem 3.8 one can assume that W and V are F -invariant. Define the following map h : V → V, by h ( x ) = F ( x, v ( x ) /n ) . Then it follows from (3) of Theorem 3.8 that h k ( x ) = F ( x, k v ( x ) /n )for all k = 1 , . . . , n . In particular, h n = id V and so h yields a Z n -actionon V . Evidently, this action is fixed on a non-empty open set W : if x ∈ W , then F ( x, v ( x ) /n ) = F ( x, w ( x )) = x. As V is connected , we get from the well-known theorem of M. Newmanthat h = id V , see e.g. [35, 6]. Thus n v ∈ ker( ϕ V ). But v is the leastpositive function which generates ker( ϕ V ), whence n = 1. (cid:3) Function of periods.
Suppose that all points of F are periodic.Consider the function Per : M → (0 , + ∞ )associating to each x ∈ M its period Per( x ) with respect to F . Thenby D. B. A. Epstein [8, §
5] Per is lower semicontinuous and the set B of its continuity points is open, see also D. Montgomery [32].We call the shift map ϕ periodic if, in accordance with (2) of Theo-rem 3.8, B is everywhere dense in M , Per is C ∞ on B , and even extendsto a C ∞ function on all of M . Otherwise, ϕ M is non-periodic . In thelatter case the zero function µ ≡ C ∞ function on M sat-isfying F ( x, µ ( x )) = x for every x ∈ M . In particular, Per can not beextended to a C ∞ function on all of M .We will now discuss obstructions for continuity of Per.It is possible that Per is unbounded near some points on M . The firstexamples of this sort seem to be constructed by G. Reeb [40]. Furtherexamples of flows with all orbits closed and with locally unboundedperiod function were obtained by D. B. A. Epstein [8] (a real analytic flow on a non-compact 3-manifold), D. Sullivan [44, 43] (a C ∞ flow ona compact S × S × S ), D. B. A. Epstein and E. Vogt [9](a flow on a compact 4-manifold defined by polynomial equations, with the vector field defining the flow given by polynomials ), E. Vogt [47],and others.On the other hand, Per continuously extends from B to all of M for the case of suspension flows (D. Montgomery [32]) and if M is acompact orientable 3-manifold (D. B. A. Epstein [8]). In these cases F can be reparametrized to a circle action. More general sufficientconditions for existence of such reparametrizations were obtained byR. Edwards, K. Millett, and D. Sullivan [7].It should also be noted that due to A. W. Wadsley [48] an exis-tence of a circle action with the orbits of F is equivalent to the ex-istence a Riemannian metric on M in which all the orbits are geodesic.Moreover, if G : M × S → M is a smooth circle action, then byR. Palais [37, Th. 4.3.1] M has an invariant Riemannian metric and byM. Kankaanrinta [17] this metric can be made complete. Also if M iscompact, then due to G. D. Mostow [33] and R. Palais [36] this actioncan be made orthogonal with respect to some embedding of M into acertain finite-dimensional Euclidean space.3.8. pn -points. Let V ⊂ M be an open connected subset such that b V := V \ Σ is also connected. Suppose that the shift map ϕ b V is periodicand let θ : b V → (0 , + ∞ ) be the positive generator of ker( ϕ b V ). Thendue to Lemma 3.12 the shift map ϕ V is also periodic if and only if θ extends to a C ∞ strictly positive function on all of V .Again one of the reasons for ϕ V to be non-periodic is unboundednessof θ at some points of Σ. These effects are reflected in the followingdefinition. Definition 3.13.
Say that z ∈ Σ is a pn -point if there exists an openneighbourhood V of z such that b V = V \ Σ is connected and the shift map ϕ b V is periodic, while for any open connected neighbourhood W ⊂ V of z the shift map ϕ W is non-periodic.Let θ : b V → (0 , + ∞ ) be the positive generator of ker( ϕ b V ) . Then z will be called a strong pn -point , if lim x → z θ ( x ) = + ∞ . The following lemma is a consequence of results obtained in [29].
Lemma 3.14. c.f. [29]
Let z ∈ Σ . Suppose that there exists an openconnected neighbourhood V of z such that the shift map ϕ b V is periodic,where b V = V \ Σ . Let B be the real Jordan normal form of the linearpart j F ( z ) of F at z . Then the following statements hold true. (1) ℜ ( λ ) = 0 for every eigen value λ of B , so B is similar to s ⊕ σ =1 J q σ ( ± ib σ ) ⊕ r ⊕ τ =1 J p τ (0) , for some q σ , p τ ∈ N and b σ ∈ R \ { } . MAGE OF A SHIFT MAP ALONG THE ORBITS OF A FLOW 13 (2) If ϕ V is periodic, then B = 0 and is similar to s ⊕ σ =1 J ( ± ib σ ) ⊕ r ⊕ τ =1 J (0) = b − b ··· b s − b s ··· , for some s ≥ and b , . . . , b s ∈ R \ . (3) If B = 0 , or if B contains either a block J q ( ± ib ) or J q (0) with q ≥ , then z is a pn -point. Moreover, in this case θ isunbounded at z .Proof. (1) Since ϕ b V is periodic, we have that b V consists of periodicpoints only. If B has an eigen value λ with ℜ ( λ ) = 0, then by theHadamard-Perron’s theorem, e.g. [13], there exists a non-periodic orbit o of F such that z ∈ o \ o . Hence b V ∩ o = ∅ . This is a contradiction.(3) It is shown in [29] that if B = 0 or if B contains either a block J q ( ± ib ) or J q (0) with q ≥
2, then there exists a sequence { x i } i ∈ N in b V converging to z such that lim i →∞ Per( x i ) = + ∞ . Since θ ( x i ) = n i Per( x i )for some n i ∈ N , we obtain lim i →∞ θ ( x i ) = + ∞ as well.(2) Suppose ϕ V is periodic, so we have a circle action G on F ( V × R )defined in (4) of Theorem 3.8. This action induces a linear circle ac-tion on the tangent space T z V , whence it follows from standard re-sults about representations of SO (2) that B is similar to the matrix ⊕ sσ =1 J ( ± ib σ ) ⊕ rτ =1 J (0). Notice that these arguments do not provethat B = 0, however this holds due to (3). (cid:3) Statement (3) of Lemma 3.14 does not claim that z is a strong pn -point, i.e. lim i →∞ θ ( x i ) = + ∞ for any sequence of periodic points { x i } i ∈ N converging to z . Example 3.15.
Let a ≤ b ∈ N and f : R → R be the polynomialdefined by f ( x, y ) = x a + y b . Then the Hamiltonian vector field F ( x, y ) = − f ′ y ∂∂x + f ′ x ∂∂y = − by b − ∂∂x + 2 ax a − ∂∂y of f has the following property: the origin ∈ R is a unique singularpoint of F , and all other orbits are concentric closed curves wrappedonce around , see Figure 1. It follows from smoothness Poincar´e’sreturn map for orbits of F that the period function θ : R \ → (0 , + ∞ )defined by θ ( z ) = Per( z ) is C ∞ . Also notice that, j F ( ) is given byone of the following matrices:( − ) ( ) ( )1) a = b = 1 , a = 1 , b ≥ , a, b ≥ . In the case 1) F is linear and its flows is given by F ( z, t ) = e it z .Hence θ ( z ) = Per( z ) = π for all z ∈ R \ , so θ extends to a C ∞ function on all of R if we put θ ( ) = π . In the case 2) j F ( ) is nilpotent and in the case 3) j F ( ) = 0.Then by (3) of Lemma 3.14 θ is unbounded at , so is a pn -point.Moreover, it easily follows from the structure of orbits of F that in factlim x → θ ( z ) = + ∞ , i.e. is a strong pn -point for F .1) 2) 3) Figure 1. Shift functions on the set of regular points
Throughout the paper we will assume that S is a connected smoothmanifold, T is a path connected and locally path connected topologicalspace, σ ∈ S , and τ ∈ T .Let V ⊂ M be an open subset and Ω : V × S × T → M be an( S ; T, k )-deformation in E ( F , V ), and A ⊂ V × S × T be a subset.Then a function Λ : A → R will be called a shift function for Ω ifΩ( x, σ, τ ) = F ( x, Λ( x, σ, τ )) , ∀ ( x, σ, τ ) ∈ A. Theorem 4.16. c.f. [22, Th. 25].
Suppose V ∩ Σ = ∅ . (1) Let ( x , σ , τ ) ∈ V × S × T and a ∈ R be such that (4.6) Ω( x , σ , τ ) = F ( x, a ) . Then there exist connected neighbourhoods W x ⊂ V of x , W σ ⊂ S of σ , W τ ⊂ T of τ , and a unique continuous shift function ∆ : W x × W σ × W τ → R for Ω such that ∆( x , σ , τ ) = a . Thus Ω( x, σ, τ ) = F ( x, ∆( x, σ, τ )) for all ( x, σ, τ ) ∈ W x × W σ × W τ . Moreover, ∆ is a ( W σ ; W τ , k ) -deformation in C ∞ ( W x , R ) . (2) Any continuous shift function
Λ : V × S × T → R for Ω is an ( S ; T, k ) -deformation. (3) Denote P = S × T . Suppose that (a) for each x ∈ V the map Ω x : P → o x defined by Ω x ( ρ ) = Ω( x, ρ ) is null-homotopic, (this holds e.g. when P is simply connectedi.e. π P = 0 , or when F has no closed orbits), (b) for some ρ ∈ P the map Ω ρ has a continuous shift function α : V → R , i.e. Ω ρ ( x ) = F ( x, α ( x )) .Then there exists a unique continuous shift function Λ : V × P → R for Ω such that Λ ρ = α . MAGE OF A SHIFT MAP ALONG THE ORBITS OF A FLOW 15
Proof.
Let x ∈ V . By assumption V ∩ Σ = ∅ , so x is a regularpoint of F . Hence there exist ε >
0, a neighbourhood U of x , and adiffeomorphism η : U → R n − × ( − ε, ε ) such that in the coordinates( y, s ) on U induced by η we have that(4.7) F (( y, s ) , t ) = ( y, s + t ) , whenever s, s + t ∈ ( − ε, ε ) . We will call U a ε -flow-box at x . Notice that for any periodic point z ∈ U its period Per( z ) ≥ ε . We will always assume below that for any periodic point z ∈ U (4.8) Per( z ) > ε. To achieve this it suffices to replace ε -flow-box U e.g. with the “central” ε -flow-box U ′ = η − (cid:0) R m − × ( − ε , ε ) (cid:1) . Then Per( z ) ≥ ε = 12 · ε > · ε for each periodic point z ∈ U ′ .Let also p : U → ( − ε, ε ) be the projection to the last coordinate, i.e. p ( y, s ) = s .(1) Let U be an ε -flow box at x satisfying (4.8). Then by (4.6) F − a ◦ Ω( x , σ , τ ) = x , hence there exists a neighbourhood W of( x , σ , τ ) in V × S × T such that F − a ( W ) ⊂ U . We can also assumethat W has the desired form W x × W σ × W τ with connected multiples.Define the function ∆ : W → R by(4.9) ∆( x, σ, τ ) = p ◦ F − a ◦ Ω( x, σ, τ ) − p ( x ) + a. Then it easily follows from (4.7) that ∆ is a shift function for Ω satis-fying statement (1). In particular, it follows from Lemma 2.6 that ∆is a ( W σ ; W τ , k )-deformation as well as Ω. The following statementimplies uniqueness of ∆. In fact it proves much more. Claim 4.17.
Let ( x, σ, τ ) ∈ W and b ∈ R be such that Ω( x, σ, τ ) = F ( x, b ) and | ∆( x, σ, τ ) − b | ≤ ε. Then ∆( x, σ, τ ) = b .Proof. For simplicity denote ξ = ( x, σ, τ ). Notice that Ω( ξ ) ∈ o x .If x is non-periodic, then there can exist a unique c ∈ R such thatΩ( ξ ) = F ( x, c ), whence b = c = ∆( ξ ).Suppose x is periodic. Then ∆( ξ ) − b = k · Per( x ) for some k ∈ Z .On the other hand by (4.8) Per(Ω( ξ )) = Per( x ) > ε . Therefore | k · Per( x ) | = | ∆( ξ ) − b | ≤ ε < Per( x ) , whence k = 0. (cid:3) Corollary 4.18.
Let A ⊂ W be a connected subset, and ∆ ′ : A → R bea continuous shift function for Ω such that for some ξ = ( x, σ, τ ) ∈ A we have that | ∆( ξ ) − ∆ ′ ( ξ ) | < ε . Then ∆ = ∆ ′ on A . Proof.
Put e B = { η ∈ A | | ∆( η ) − ∆ ′ ( η ) | < ε } , B = { η ∈ A | ∆( η ) = ∆ ′ ( η ) } . We have to show that B = A .Evidently, B is closed in A while e B is open. Moreover, by Claim 4.17 B = e B and ξ ∈ B = ∅ . Hence B = e B = A . (cid:3) Statement (2) is a direct consequence of (1).(3) For x ∈ V let F x : R −→ o x ⊂ M, F x ( t ) = F ( x, t )be the map representing the orbit o x of x . Since Ω is a deformationin E ( F , V ), we have that Ω( x × P ) ⊂ o x . By assumption this mapis null-homotopic. Then by the path covering property for coveringmaps there exists a continuous function Λ x : x × P → R such thatΛ x ( ρ ) = α ( x ) and the following diagram is commutative: R F x (cid:15) (cid:15) x × P Λ x ; ; Ω / / o x i.e Ω( x, ρ ) = F ( x, Λ x ( ρ )) , Define the following function Λ : V × P → R by Λ( x, ρ ) = Λ x ( ρ ).Then Λ( x, ρ ) = α ( x ) and Ω( x, ρ ) = F ( x, Λ( x, ρ )) for ( x, ρ ) ∈ V × P .It remains to show that Λ is continuous.By (1) for each ( x, ρ ) ∈ V × P there are connected neighbourhoods W ( x,ρ ) x ⊂ V of x and W ( x,ρ ) ρ ⊂ P of ρ , and a continuous shift function∆ ( x,ρ ) : W ( x,ρ ) = W ( x,ρ ) x × W ( x,ρ ) ρ −→ R for Ω such that(4.10) ∆ ( x,ρ ) ( x, ρ ) = Λ x ( ρ ) . Our aim is to show that ∆ ( x,ρ ) = ∆ ( x ′ ,ρ ′ ) on W ( x,ρ ) ∩ W ( x ′ ,ρ ′ ) for all( x, ρ ) , ( x ′ , ρ ′ ) ∈ V × P . Hence the functions { ∆ ( x,ρ ) } ( x,ρ ) ∈ V × P will definea unique continuous function on all of V × P .Notice that the projection p takes the values in ( − ε, ε ). Thereforewe get from (4.9) that(4.11) | ∆ ( x,ρ ) ( ξ ) − ∆ ( x,ρ ) ( η ) | < ε, for all ξ, η ∈ W ( x,ρ ) . Claim 4.19.
Suppose that | ∆ ( x,ρ ) ( ξ ) − ∆ ( x ′ ,ρ ′ ) ( ξ ) | < ε for some ξ ∈ W ( x,ρ ) ∩ W ( x ′ ,ρ ′ ) . Then ∆ ( x,ρ ) = ∆ ( x ′ ,ρ ′ ) on W ( x,ρ ) ∩ W ( x ′ ,ρ ′ ) . MAGE OF A SHIFT MAP ALONG THE ORBITS OF A FLOW 17
Proof. If η ∈ W ( x,ρ ) ∩ W ( x ′ ,ρ ′ ) , then | ∆ ( x,ρ ) ( η ) − ∆ ( x ′ ,ρ ′ ) ( η ) | ≤ | ∆ ( x,ρ ) ( η ) − ∆ ( x,ρ ) ( ξ ) | ++ | ∆ ( x,ρ ) ( ξ ) − ∆ ( x ′ ,ρ ′ ) ( ξ ) | + | ∆ ( x ′ ,ρ ′ ) ( ξ ) − ∆ ( x ′ ,ρ ′ ) ( η ) | << ε + ε + 4 ε = 10 ε. Hence by Claim 4.17 ∆ ( x,ρ ) ( η ) = ∆ ( x ′ ,ρ ′ ) ( η ). (cid:3) Claim 4.20. ∆ ( x,ρ ) = ∆ ( x,ρ ′ ) on W ( x,ρ ) ∩ W ( x,ρ ′ ) .Proof. Since ∆ ( x,ρ ) and Λ x are continuous shift functions for Ω on theconnected set x × W ( x,ρ ) ρ , it follows from (4.10) and Corollary 4.18 that∆ ( x,ρ ) = Λ x on x × W ( x,ρ ) ρ . Hence∆ ( x,ρ ) ( x, ρ ′ ) = Λ x ( ρ ′ ) (4.10) === ∆ ( x,ρ ′ ) ( x, ρ ′ ) for all ρ ′ ∈ W ( x,ρ ) ρ . Then by Claim 4.19 ∆ ( x,ρ ) = ∆ ( x,ρ ′ ) on W ( x,ρ ) ∩ W ( x,ρ ′ ) . (cid:3) Thus for each x ∈ V the functions { ∆ ( x,ρ ) } ρ ∈ P give rise to a contin-uous shift function ∆ x : W x → R for Ω on the open neighbourhood W x := ∪ ρ ∈ P W ( x,ρ ) of x × P in V × P . Since P and each W ( x,ρ ) is con-nected, we see that so is W x . Then it easily follows from Claim 4.17and Corollary 4.18 that ∆ x is a unique shift function for Ω on W x suchthat ∆ x ( ρ ) = Λ x ( ρ ) for at least one ρ ∈ P . Claim 4.21. ∆ ( x,ρ ) = ∆ ( x ′ ,ρ ) on W ( x,ρ ) ∩ W ( x ′ ,ρ ) , whence ∆ x = ∆ x ′ on W x ∩ W x ′ .Proof. Notice that ∆ ( x,ρ ) and α are continuous shift functions for Ωon the connected set W ( x,ρ ) x × ρ and by assumption∆ ( x,ρ ) ( x, ρ ) = Λ x ( ρ ) = α ( x ) . Then by Corollary 4.18∆ ( x,ρ ) ( x ′ , ρ ) = α ( x ′ ) = ∆ ( x,ρ ) ( x ′ , ρ ) for all x ′ ∈ W ( x,ρ ) x . Hence by Claim 4.19 ∆ ( x,ρ ) = ∆ ( x ′ ,ρ ) on W ( x,ρ ) ∩ W ( x,ρ ) . (cid:3) Thus the functions { ∆ x } x ∈ V define a continuous function on all of V × P which coincides with Λ. Theorem 4.16 is completed. (cid:3) Corollary 4.22.
Let f ∈ E ( F , V ) , z ∈ V \ Σ , and a ∈ R be such that f ( z ) = F ( z, a ) . Then there exists a neighbourhood W of z and a uniquecontinuous shift function α : W → R for f such that α ( z ) = a . In fact α is C ∞ . Corollary 4.23.
Let V ⊂ M be an open subset, b V = V \ Σ , α ∈C ∞ ( b V \ Σ , R ) , and Ω : V × S × T → M be an ( S ; T, k ) -deformation in E ( F , V ) such that Ω( x, σ , τ ) = F ( x, α ( x )) for all x ∈ b V .If π ( S × T ) = 0 , then there exists a unique ( S ; T, k ) -deformation Λ : b V × S × T → R such that Λ( x, σ , τ ) = α ( x ) for x ∈ b V , and Ω( x, σ, τ ) = F ( x, Λ( x, σ, τ )) for ( x, σ, τ ) ∈ b V × S × T. Main result
Let V ⊂ M be an open connected set and Ω : V × I → M be a k -homotopy in E ( F , V ) such that Ω = ϕ V ( α ) for some C ∞ function α : V → R . Then by Corollary 4.23 α extends to a continuous shiftfunction Λ : ( V \ Σ) × I → R for Ω such that Λ = α . But in generalΛ τ for τ > C ∞ function on V .Such examples were constructed in [26]. There Ω is a 0-homotopybeing not a 1-homotopy, Ω = id, Λ = 0, and lim z → Λ ( z ) = + ∞ . Ourmain result shows that this situation is typical: non-extendability ofΛ τ to a C ∞ function almost always happens only for 0-homotopies thatare not 1-homotopies and for special types of singularities.Let T be a compact, simply connected and locally path connectedtopological space and τ ∈ T . Theorem 5.24.
Suppose Σ is nowhere dense in V and there exists z ∈ Σ such that either of the following conditions holds true: (a) z is not a pn -point; (b) z is a pn -point, j F ( z ) = 0 , and k ≥ ; (c) z is a strong pn -point and k ≥ .Let Ω : V × T → M be a ( T, k ) -deformation in Sh ( F , V ) , α ∈C ∞ ( V, R ) be any shift function for Ω τ , and Λ : ( V \ Σ) × T → R be a unique continuous shift function for Ω such that Λ τ = α , seeTheorem 4.16. Then there exists a neighbourhood W of z such that foreach τ ∈ T the function Λ τ also extends to a C ∞ function on W , so Ω τ | W = ϕ W (Λ τ | W ) . Notice that the assumptions (a)-(c) do not include the case when z is a pn -point being not strong and such that j F ( z ) = 0. Moreover, itis not claimed that Λ becomes continuous on V × T . Theorem 5.25.
Let F be a vector field on M such that Σ is nowheredense and let k = 0 , . . . , ∞ . Suppose that for each z ∈ Σ there existsan open, connected neighbourhood V such that Sh ( F , V ) = E id ( F , V ) k and either of the conditions (a)-(c) of Theorem 5.24 holds true. Then Sh ( F ) = E id ( F ) k . Proof.
Let f ∈ E id ( F ) k , so there exists a k -homotopy Ω : V × I → M in E ( F , V ) between the identity map Ω = id M and Ω = f . ByTheorem 4.16 there exists a unique k -homotopy Λ : ( V \ Σ) × I → R being a shift function for Ω and such that Λ ≡
0. We will show thatΛ τ extends to a C ∞ function on all of M , whence Ω τ = ϕ (Λ τ ) for all τ ∈ I . This will imply Sh ( F ) = E id ( F ) k . Let z ∈ Σ and V be a neighbourhood of z such that Sh ( F , V ) = E id ( F , V ) k . Then Ω τ | V ∈ E id ( F , V ) k , so Ω | V × I is a k -homotopy in Sh ( F , V ). Moreover, Λ | ( V \ Σ) × I is a unique extension of Λ | V = 0. Byassumption either of the conditions (a)-(c) of Theorem 5.24 holds true, MAGE OF A SHIFT MAP ALONG THE ORBITS OF A FLOW 19 whence for each τ ∈ I the function Λ τ extends to a C ∞ function onsome neighbourhood of z . Since Σ is nowhere dense such an extensionis unique. As z is arbitrary, we obtain that Λ τ smoothly extends to allof M . (cid:3) For the proof of Theorem 5.24 we need two preliminary results.5.1.
Reduction of -jets at singular points. Let Ω : V × T → M be a ( T, k ) deformation in Sh ( F , V ) and Λ : ( V \ Σ) × T → R be ashift function for Ω. Let also z ∈ Σ ∩ V . Choose local coordinates( x , . . . , x m ) at z in which z = ∈ R m . Let A be an ( m × m )-matrixbeing the linear part of F at z . Then we have the exponential map e : R → GL( m, R ) , e ( t ) = exp( At ) . Denote its image by R A . If A = 0, then e is an immersion, i.e. a localhomeomorphism onto its image.For each τ ∈ T denote by J τ the Jacobi matrix of Ω τ at z . Byassumption Ω τ = ϕ V ( α τ ) for some α τ ∈ C ∞ ( V, R ). Then it is easy toshow, [27, Th. 5.1], that J τ = exp( A α τ ( z )) , so we have a well-defined map η : T → R A , η ( τ ) = J τ . If k ≥
1, then η is continuous. Lemma 5.26.
Suppose that η is continuous and lifts to a continuousmap e η : T → R making the following diagram commutative: R e (cid:15) (cid:15) T e η > > η / / R A The latter holds for instance when T is compact, path connected, andsimply connected, e.g. T = I d .Let W ⊂ V be an arbitrary small neighbourhood of z . Then thereexists a function ν : V × T → R being a ( T, ∞ ) deformation in C ∞ ( V, R ) and vanishing outside W × T such that the ( T, k ) -deformation Ω ′ : V × T → M, Ω ′ ( x, τ ) = F (Ω( x, τ ) , − ν ( x, τ )) has the following properties: for each τ ∈ T (1) j Ω ′ τ ( z ) is the identity, and (2) the function Λ ′ τ = Λ τ − ν τ is a shift function for Ω ′ .In particular, Λ τ extends to a C ∞ function near z iff so does Λ ′ τ .Proof. Let µ : V → [0 ,
1] be a C ∞ function such that µ = 1 on someneighbourhood of z and µ = 0 on V \ W . Define ν ( x, τ ) = µ ( x ) b η ( τ ).Then it is easy to verify that ν satisfies the statement of our lemma. (cid:3) Smoothness of shift functions.
In this section we will assumethat V ⊂ M is an open connected subset such that b V = V \ Σ isconnected and the shift map ϕ b V is periodic. Let θ : b V → (0 , + ∞ ) bethe positive generator of ker( ϕ b V ).Define the map h : V → M by(5.12) h ( x ) = ( F ( x, θ ( x ) / , x ∈ b V = V \ Σ ,x, x ∈ Σ ∩ V. Evidently, h is continuous on b V but in general is discontinuous on Σ ∩ V .If x ∈ b V , then θ ( x ) = n x Per( x ) for some n x ∈ Z . Hence h ( x ) = x iff n x is even, see Figure 2. Figure 2. θ ( x ) = n x Per( x ), n x is odd Theorem 5.27.
Let
Ω : V × T → M be a ( T, -deformation in Sh ( F , V ) , Λ τ ∈ C ∞ ( V, R ) be any shift function for Ω τ , and let also Λ : b V × T → R be a unique continuous extension of Λ τ being a shiftfunction for Ω . Suppose that there exists a point z ∈ Σ ∩ V satisfyingone of the following conditions: ( HD ) h is discontinuous at z ; ( JF0 ) j F ( z ) = 0 ; ( TI ) lim b V ∋ x → z θ ( x ) = ∞ .Then ( Z ) for each τ ∈ T the function Λ τ : b V → R extends to a C ∞ function on all of V as well, however the induced extension Λ : V × T → R of Λ is not claimed to be even continuous.Proof. First notice that if ν : V × T → R is any ( T, k )-deformation in C ∞ ( V, R ), then the following map(5.13) Ω ′ : V × T → M, Ω ′ ( x, t ) = F (Ω( x, τ ) , − ν ( x, τ ))is a ( T, k )-deformation as well as Ω. Moreover, the functionΛ ′ = Λ − ν : ( V \ Σ) × T → R is a shift function for Ω ′ on V \ Σ and Λ ′ τ = Λ τ − ν τ extends to a C ∞ function α τ − ν τ on all of V . Hence Λ ′ τ = Λ τ − ν τ extends to a C ∞ function on V iff so does Λ τ . Thus one may replace Ω with Ω ′ .In particular, we can reduce the problem to the case when Ω satisfiesthe following additional assumptions:( ΩΛ ) Ω τ = i V : V ⊂ M is the identity inclusion and Λ τ ≡ MAGE OF A SHIFT MAP ALONG THE ORBITS OF A FLOW 21 ( JΩid ) j Ω τ ( z ) = id for all τ ∈ T .To establish ( ΩΛ ) put ν ( x, τ ) = α τ ( x ). Then Λ ′ τ = 0, and Ω ′ τ = i V .Furthermore, suppose that ( ΩΛ ) holds for Ω. Since Ω( z, τ ) = z for all τ ∈ T , we obtain from Lemma 5.26 that there exists even a( T, ∞ )-deformation ν : V × T → R such that ν τ = 0 and the ( T, JΩid ).We will now introduce more conditions.( HC ) h is continuous on some neighbourhood of z .Let { x i } i ∈ N ⊂ b V be a sequence converging to z . Then we also con-sider the following conditions:( PU ) the sequence of periods { Per( x i ) } i ∈ N is unbounded;( TU ) the sequence { θ ( x i ) } i ∈ N is unbounded;( ΩH ) Ω τ ( x i ) = h ( x i ) for all τ ∈ T and all sufficiently large i ∈ N ;( D ) there exists L > k· , ·k on someneighbourhood of z such that k x i , z k < L · k x i , h ( x i ) k , i ∈ N . Lemma 5.28.
The following implications hold true:1) ( ΩΛ ) & ( TU ) & ( ΩH ) ⇒ ( Z ) ;2) ( HD ) ⇒ ∃{ x i } i ∈ N satisfying lim i →∞ x i = z , ( PU ) , and ( ΩH ) ;3) ( HC ) ⇒ ∃{ x i } i ∈ N satisfying lim i →∞ x i = z and ( D ) ;4) ( JF0 ) & ( D ) ⇒ ( PU ) ;5) ( HC ) & ( JΩid ) & ( D ) ⇒ ( ΩH ) ;6) ( TI ) ⇒ ( TU ) .7) ( PU ) ⇒ ( TU ) . Assuming that this lemma is proved let us complete Theorem 5.27.We have to show that either of the conditions ( HD ), ( JF0 ), or ( TI )together with ( ΩΛ ) and ( JΩid ) implies ( Z ).The implication ( HD ) & ( ΩΛ ) ⇒ ( Z ) follows from 2), 7) and 1) ofLemma 5.28.Further, if ( HD ) fail for any z ∈ V ∩ Σ, that is h is continuous ateach such z , then h is continuous on all of V , so ( HC ) holds true.Then ( HC ) & ( JF0 ) & ( ΩΛ ) & ( JΩid ) ⇒ ( Z ) by 4), 7), 5), and 1) ofLemma 5.28.Moreover, ( HC ) & ( TI ) & ( ΩΛ ) & ( JΩid ) ⇒ ( Z ) by 3), 6), and 1) ofLemma 5.28.This completes Theorem 5.27 modulo Lemma 5.28. (cid:3) Proof of Lemma 5.28.
The implications 6) and 7) are trivial.1) ( ΩΛ ) & ( TU ) & ( ΩH ) ⇒ ( Z ).Thus Ω τ = i V : V ⊂ M , Λ τ ≡
0, and there exists a sequence { x i } i ∈ N which converges to some z ∈ Σ ∩ V and satisfies( TU ) lim i →∞ θ ( x i ) = + ∞ , ( ΩH ) Ω τ ( x i ) = h ( x i ) , i ∈ N , τ ∈ T. We have to show that for each τ ∈ T the function Λ τ extends to a C ∞ function on all of V .Notice that ( TU ) implies that ϕ V is non-periodic. Indeed, suppose ϕ V is periodic and let ν be the positive generator of ker( ϕ V ). Then byLemma 3.12 ν = θ on b V , whence θ is bounded on b V which contradictsto ( TU ).Thus ϕ V is non-periodic, whence for each Ω τ ∈ Sh ( F , V ), ( τ ∈ T ),there exists a unique C ∞ shift function α τ ∈ C ∞ ( V, R ). In particular,Λ τ = α τ . We will show that Λ τ = α τ on b V for all τ ∈ T .Since α τ | b V and Λ τ are shift functions for Ω τ on b V , we obtain fromLemma 3.7 that α τ − Λ τ = k τ θ on b V for some k τ ∈ Z . It suffices to show that k τ = 0.Condition ( ΩH ) means that the image of the map ω i : T → o x i defined by ω i ( τ ) = F ( x i , Λ τ ( x i )) = Ω( x i , τ )does not contain the point h ( x i ). Moreover, since Ω τ ( x i ) = x i = h ( x i )and Λ τ = 0, it follows from the construction of Λ that(5.14) | Λ τ ( x i ) − Λ τ ( x i ) | = | Λ τ ( x i ) | < Per( x i ) ≤ θ ( x i )for all τ ∈ T and i ∈ N .On the other hand we get from ( TU ) and continuity of α τ that | α τ ( x i ) − α τ ( x i ) | = | α τ ( x i ) | < θ ( x i )for all sufficiently large i ∈ N .Hence k τ θ ( x i ) = | Λ τ ( x i ) − α τ ( x i ) | ≤ | Λ τ ( x i ) | + | α τ ( x i ) | << Per( x i ) + θ ( x i ) ≤ θ ( x i ) . This implies k τ = 0 and thus Λ τ = α τ on b V .2) ( HD ) ⇒ ∃{ x i } i ∈ N satisfying lim i →∞ x i = z , ( PU ), and ( ΩH ).Discontinuity of h (that is property ( HD )) at z means that thereexists a sequence { x i } i ∈ N converging to z such that { h ( x i ) } i ∈ N laysoutside some neighbourhood W of z .( ΩH ). Since Ω( z × T ) = z and T is compact, we can assume that { Ω τ ( x i ) } i ∈ N ⊂ W for all i ∈ N and τ ∈ T , therefore Ω τ ( x i ) = h ( x i ).( PU ). Suppose ( PU ) fails. Then there exists C > x i ) < C for all i . Since z is a fixed point of F , there exists anotherneighbourhood U of z such that F ( U × [0 , C ]) ⊂ W . Therefore (passingto a subsequence) we can assume that x i ∈ U for all i ∈ N , hence h ( x i ) = F ( x i , Per( x i ) / ∈ F ( U × [0 , C ]) ⊂ W which contradicts to the assumption h ( x i ) W . MAGE OF A SHIFT MAP ALONG THE ORBITS OF A FLOW 23
3) ( HC ) ⇒ ∃{ x i } i ∈ N satisfying lim i →∞ x i = z and ( D ).Thus we have that h is continuous on some neighbourhood W ⊂ V of z . Our aim is to show that there exist • a neighbourhood U ⊂ W of z such that h ( U ) = U , • a Riemannian metric d on U , • a sequence { x i } i ∈ N ⊂ U \ Σ converging to z , and • a number L > D ) k z, x i k < L · k x i , h ( x i ) k . First suppose that z ∈ Int M . Then we can assume that W is anopen subset of R n and d is the induced Euclidean metric on W . Since h is continuous on W and h ( z ) = z , we can find an open ball B s ⊂ W of some radius s > z such that h ( B s ) ⊂ W . Then U s := B s ∪ h ( B s ) ⊂ W. By assumption b V is connected, therefore by (3) of Theorem 3.8 therestriction of θ to o x ∩ b V is constant for any x ∈ b V . This easily impliesthat h ( x ) = F ( x, θ ( x )) = x, x ∈ B s , so h yields a Z -action on U s . Due to (2) of Theorem 3.8 this action isnon-trivial and therefore effective.By a well known theorem of M. Newman [35] all the orbits of a Z -action on a manifold U s can not be arbitrary small. D. Hoffman andL. N. Mann [15, Th. 1], using a result of A. Dress [6], obtained lowerbounds for diameters of such actions. Denote by ¯ r the radius of con-vexity of U s at z . Since B s ⊂ U s , it follows that s ≤ ¯ r . Then it wasshown in [15, Th. 1] that there exists an orbit or this action of diametergreater than s/
2, that is k x s , h ( x s ) k > s/ x s ∈ U s . Interchanging x s and h ( x s ), if necessary, we canassume that in fact x s ∈ B s . Hence(5.15) k z, x s k < s < L · k x s , h ( x s ) k , where L = 2.Decreasing s to 0 we will find a sequence { x s i } i ∈ N satisfying ( D ).A more deep analysis of the proofs of [15, Th. 1] and [6, Lm. 3] showsthat similar estimations but with another constant L hold in the casewhen z ∈ ∂M , see [29] for details.4) ( JF0 ) & ( D ) ⇒ ( PU ).Thus j F ( z ) = 0 and we have a sequence { x i } i ∈ N converging to z and satisfying ( D ). We will show that there is subsequence { x i k } k ∈ N such that lim k →∞ Per( x i k ) = + ∞ . Suppose that there exists
C > x i ) < C for all i .Since z is a fixed point of F , we can decrease U and assume that F ( U × [0 , C ]) ⊂ V . Moreover, as j F ( z ) = 0, we can also suppose thatthere exists A > | F ( x ) | ≤ A · k z, x k , for all x ∈ U .
Notice that the length l ( x i ) of the orbit of x i can be calculated bythe following formula: l ( x i ) = Z Per( x i )0 | F ( F ( x i , t )) | dt, whence l ( x i ) ≤ Per( x ) · sup t ∈ [0 , Per( x )] | F ( F ( x, t )) | ≤ C · A · k z, x i k . Therefore k z, x i k < L · k z, h ( x i ) k ≤ L · l ( x i ) ≤ L · C · A · k z, x i k , whence 0 < L · C · A ≤ k z, x i k , which contradicts to the assumption that { x i } i ∈ N converges to z .5) ( JΩid ) & ( D ) ⇒ ( ΩH );Thus j Ω τ ( z ) = id for all τ ∈ T and there is a sequence { x i } i ∈ N converging to z and satisfying k z, x i k < L · k x i , h ( x i ) k for some L > τ ( x i ) = h ( x i ) for all τ ∈ T and all sufficientlylarge i .Since Ω is a ( T, JΩid ) that thereexists a continuous function γ : U → R such that γ ( z ) = 0 and k x, Ω τ ( x ) k ≤ γ ( x ) · k z, x k for all ( x, τ ) ∈ U × T , see [29]. If Ω were a C map or at least a( T, k x, Ω τ ( x ) k ≤ C · k z, x k with some constant C > k h ( x i ) , Ω τ ( x i ) k ≥ k x i , h ( x i ) k − k x i , Ω τ ( x i ) k >> L k z, x i k − γ ( x i , t ) · k z, x i k = (cid:0) L − γ ( x i ) (cid:1) · k z, x i k . Notice that lim i →∞ γ ( x i ) = γ ( z ) = 0. Therefore (passing if necessary toa subsequence) we can assume that L − γ ( x i , τ ) > i , whence k h ( x i ) , Ω τ ( x i ) k >
0. This means that Ω τ ( x i ) = h ( x i ).Lemma 5.28 is completed. MAGE OF A SHIFT MAP ALONG THE ORBITS OF A FLOW 25
Proof of Theorem 5.24.
Let V be a neighbourhood of z suchthat Σ is nowhere dense in V , Ω : V × T → M be a ( T, k )-deformationin Sh ( F , V ), and Λ : b V × T → M be a continuous shift function for Ωsuch that Λ τ extends to a C ∞ function α on all of V . We have to provethat there exists a neighbourhood W of z such that for each τ ∈ T thefunction Λ τ extends to a C ∞ function on W .By assumption Ω τ ∈ Sh ( F , V ), so Ω τ = ϕ V ( α τ ) for some α τ ∈C ∞ ( V, R ). Hence Λ τ = α τ + µ τ on b V for some µ τ ∈ ker( ϕ b V ).(a) Suppose z is not a pn -point. Then we can find a neighbourhood W of z such that either(a ′ ) ϕ c W is non-periodic, or(a ′′ ) both ϕ W and ϕ c W are periodic,where c W = W \ Σ.In the case (a ′ ) we have ker( ϕ c W ) = { } . In particular, µ τ = 0, andtherefore Λ τ = α τ on c W . Hence Λ τ extends to a C ∞ function α τ on W .In the case (a ′′ ) due to Lemma 3.12 we have that ker( ϕ W ) = { nθ } n ∈ Z for some θ ∈ C ∞ ( W, R ) and ker( ϕ c W ) = { nθ | c W } n ∈ Z . Hence µ τ = k τ θ | c W for some k τ ∈ Z . Therefore again Λ τ extends to a C ∞ function α τ + k τ θ on all of W .Finally, in the cases (b) and (c) z is pn -point, so there exists a neigh-bourhood W of z such that ϕ W is non-periodic, while ϕ c W is periodic.Then in the case (b) (resp. (c)) the restriction Ω | W × T satisfies assump-tions of Theorem 5.27 and condition ( JF0 ) (resp. ( TI )). Hence for each τ ∈ T the function Λ τ also smoothly extends to all of W . (cid:3) Decreasing V The following lemma shows that the relation Sh ( F , V ) = E id ( F , V ) k is preserved if V is properly decreased. Lemma 6.29.
Let V ⊂ M be an open connected subset such that V ∩ Σ is nowhere dense in V and Sh ( F , V ) = E id ( F , V ) k for some k ≥ . Letalso W ⊂ V be an open connected subset such that W ∩ Σ is closed (and so open-closed ) in V ∩ Σ . Then Sh ( F , W ) = E id ( F , W ) k .Proof. Let Ω : W × I → M be a k -homotopy such that Ω = i W : W ⊂ M is the identity inclusion. We have to show that for every t ∈ I thereexists a C ∞ function α t : W → R such that h = ϕ W ( α ).Let Λ : ( W \ Σ) × I → R be a unique k -homotopy being a shiftfunction for Ω and such that Λ ≡
0. We will show that every Λ t eitherextends to a C ∞ function on all of W , or can be replaced with someother shift function of Ω which smoothly extends to W .First we extend Λ to ( V \ Σ) × I changing it outside ( W ∩ Σ) × I .Since W ∩ Σ is open-closed in V ∩ Σ, there exists a C ∞ function µ : V → [0 ,
1] such that µ = 1 on some open neighbourhood N ⊂ V of W ∩ Σ and µ = 0 on some open neighbourhood of V \ W , see Figure 3. Then it follows from Lemma 2.6 that a function Λ ′ : ( V \ Σ) × I → R Figure 3. defined by Λ ′ ( x, t ) = (cid:26) µ ( x )Λ( x, t ) , x ∈ W \ Σ , , x ∈ V \ W is a k -homotopy which coincides with Λ on ( N \ Σ) × I . Then thefollowing map Ω ′ : V × I → M defined byΩ ′ ( x, t ) = (cid:26) F ( x, Λ ′ ( x, t )) , x ∈ V \ ( W ∩ Σ) ,x, x ∈ W ∩ Σis a k -homotopy in E id ( F , V ) k which coincides with Ω on N × I .Since Sh ( F , V ) = E id ( F , V ) k , it follows that Ω ′ t has a C ∞ shift func-tion α ′ t on V . ThusΩ ′ t ( x ) = F ( x, µ ( x )Λ t ( x )) = F ( x, α ′ t ( x )) , x ∈ V \ Σ . Denote b V = V \ Σ. Then µ Λ t is a shift function for Ω ′ t on b V , while α ′ t is a shift function for Ω ′ t on V . Consider three cases.1) If ϕ V and ϕ b V are non-periodic, then it follows from Lemma 3.7that α ′ t = Λ ′ t = µ Λ t on b V . Since µ = 1 near W ∩ Σ and Λ t is smoothoutside this set, we put Λ t = α t on W ∩ Σ. Then Λ t becomes smoothon all of W .2) If both ϕ V and ϕ b V are periodic, then by Lemmas 3.7 and 3.12 µ Λ t = α ′ t + l t θ on b V , where θ : V → (0 , ∞ ) is the positive generator ofthe kernel of ϕ V and l t ∈ Z . Then similarly to the case 1) Λ t smoothlyextends to all of W .3) Suppose that ϕ b V is periodic while ϕ V is non-periodic, and let θ : b V → (0 , ∞ ) be the positive generator of the kernel of ϕ b V . Then α ′ t = µ Λ t − l t θ on b V for some l t ∈ Z . Again α t = Λ t − l t θ smoothlyextends to all of W .We claim that α t is a shift function for Ω t . Indeed, we have thatΩ t ( x ) = F t ( x, Λ t ( x )) and F ( x, θ ( x )) = F ( x, − l t θ ( x )) = x , whence F ( x, Λ t ( x ) − l t θ ( x )) = F (cid:0) F ( x, − l t θ ( x )) , Λ t ( x ) (cid:1) = F ( x, Λ t ( x )) = Ω t ( x ) . Thus Ω t = ϕ W (Λ t − l t θ ) ∈ Sh ( F , W ). (cid:3) Corollary 6.30.
Suppose z is an isolated singular point of F , and let V be a connected, open neighbourhood of z such that V ∩ Σ = { z } and Sh ( F , V ) = E id ( F , V ) k for some k ≥ . Then Sh ( F , W ) = E id ( F , W ) k for any other connected, open neighbourhood W ⊂ V of z . MAGE OF A SHIFT MAP ALONG THE ORBITS OF A FLOW 27 Regular extensions
For every m ∈ N let m ∈ R m be the origin. Then m + n = ( m , n ) ifwe regard R m + n as R m × R n . Denote by p m : R m + n → R m the naturalprojection to the first m coordinates.Let F be a germ of a vector field at m + n on R m + n , and G be a germof a vector field on R m ≡ R m × n ⊂ R m + n at m . Say that F is a aregular extension of G , if(7.16) F ( x, y ) = ( G ( x ) , H ( x, y ))for some C ∞ germ H : R m + n → R n at m + n . Thus the first m coordinatefunctions of F coincide with G and, in particular, they do not dependon the remaining n coordinates. In this case the local flows of F and G are related by the following identity: F ( x, y, t ) = ( G ( x, t ) , H ( x, y, t )) , for some C ∞ map H .Denote by Σ F and Σ G the sets of singular points of F and G respec-tively. Then it follows from (7.16)(7.17) Σ F ⊂ Σ G × V n . Indeed, if F ( x, y ) = ( G ( x ) , H ( x, y )) = 0, then G ( x ) = 0.Notice that if H is a germ of a vector field on R n at n , then the product F ( x, y ) = ( G ( x ) , H ( y )) of G and H is a regular extension ofeach of these vector fields. Also notice that a regular extension of aregular extension is a regular extension itself.It follows from the real version of Jordan’s theorem about normalforms of matrices that every linear vector field on R n is a regular ex-tension of a linear vector field defined by one of the following matrices:(7.18) ( λ ) , ( ) , (cid:0) a b − b a (cid:1) , λ, a, b ∈ R . Lemma 7.31.
Let V m ⊂ R m and V n ⊂ R n be open connected subsets, V m + n = V m × V n , S be a connected smooth manifold with dim S = d , T be a path-connected topological space, (7.19) Ω : V m + n × S × T → R m × R n be an ( S ; T, k ) -deformation in E ( F , V m + n ) , and Ψ = p m ◦ Ω : V m × V n × S × T → R m be the projection to the first m coordinates. Then the following state-ments hold true.
1) Ψ is a ( V n × S ; T, k ) -deformation in E ( G , V m ) . Suppose that Ω has a continuous shift function (7.20) Λ : V m + n × S × T ⊃ A −→ R , defined on some subset A of V m + n × S × T . Then Λ is a shift functionfor Ψ with respect to G . Proof.
1) The statement that Ψ is a ( V n × S ; T, k )-deformation in C ∞ ( V m , R m ) directly follows from Lemma 2.6.Denote P = S × T and ρ = ( σ, τ ). We have to show that Ψ ( y,ρ ) ∈E ( G , V m ) for each ( y, ρ ) ∈ V n × P , i.e.(i) Ψ ( y,ρ ) preserves orbits of G , so for each x ∈ V m , there exists t ∈ R such that Ψ ( y,ρ ) ( x ) = G ( x, t );(ii) Ψ ( y,ρ ) is a local diffeomorphism at each x ∈ Σ G ∩ V m .(i) Since Ω ρ preserves orbits of F , there exists t x,y ∈ R such thatΩ( x, y, ρ ) = F ( x, y, t x,y ) = (cid:0) G ( x, t x,y ) , H ( x, y, t x,y ) (cid:1) , whence Ψ ( y,ρ ) ( x ) = p m ◦ Ω( x, y, ρ ) = G ( x, t x,y ) . (ii) Let x ∈ Σ G ∩ V m , so G ( x ) = 0. Then F ( x, y ) = ( G ( x ) , H ( x, y )) = (0 , H ( x, y ))for every y ∈ R n . It follows that the orbit o ( x,y ) of F passing through( x, y ) is everywhere tangent to x × R n and therefore o ( x,y ) ⊂ x × R n .Since Ω ρ preserves orbits of F , we also get that(7.21) Ω ρ ( x × V n ) ⊂ x × R n . Consider two cases.(a) Suppose ( x, y ) is a singular point of F , so H ( x, y ) = 0 as well.Since Ω ρ ∈ E ( F , V m + n ), we have Ω ρ ( x, y ) = ( x, y ) and Ω ρ is a localdiffeomorphism at ( x, y ). Then we get from (7.21) that the tangentspace T y V n = T ( x,y ) ( x × V n ) is invariant with respect to the tangentmap T ( x,y ) Ω ρ and therefore the image T ( x,y ) Ω ρ ( T x V m ) of the transversalspace T x V m = T ( x,y ) ( V m × y ) is transversal to T y V n , see Figure 4. This Figure 4. implies that the tangent map T x Ψ ( y,ρ ) = T x ( p m ◦ Ω ρ ) is non-degenerate,i.e. Ψ ( y,ρ ) is a local diffeomorphism at x .(b) Suppose H ( x, y ) = 0, so ( x, y ) is a regular point of F . Then byCorollary 4.22 there exist neighbourhoods W m ⊂ V m of x and W n ⊂ V n of y , and a C ∞ shift function β : W m × W n → R for Ω ρ , soΩ ρ (¯ x, ¯ y ) = F (¯ x, ¯ y, β (¯ x, ¯ y )) = ( G (¯ x, β (¯ x, ¯ y )) , H (¯ x, ¯ y, β (¯ x, ¯ y )) , for all (¯ x, ¯ y ) ∈ W m × W n . In particular, Ψ ( y,ρ ) (¯ x ) = G (¯ x, β (¯ x, y )) for all¯ x ∈ W m . Thus the restriction Ψ ( y,ρ ) | W m belongs to the image of shiftmap Sh ( G , W m ) ⊂ E ( G , W m ), whence Ψ ( y,ρ ) is a local diffeomorphismat x . MAGE OF A SHIFT MAP ALONG THE ORBITS OF A FLOW 29
2) Suppose Ω has a shift function (7.20). Then the identityΩ( x, y, ρ ) = F ( x, y, Λ( x, y, ρ )) , ( x, y, ρ ) ∈ A, implies Ψ( x, y, ρ ) = p m ◦ Ω( x, y, ρ ) = G ( x, Λ( x, y, ρ )). (cid:3) Examples
In this section we consider examples of vector fields F for whichrelation between Sh ( F ) and E id ( F ) k is known and whose singular pointssatisfy assumptions (a)-(c) of Theorem 5.24.8.1. Vector fields on R . Let G ( x ) = β ( x ) ∂∂x be a vector field on R ,where β : R → R is a smooth function. Suppose Σ = { } , so G has onlythree orbits ( −∞ , { } , (0 , ∞ ). This is equivalent to the assumptionthat β − (0) = 0.By the Hadamard lemma β ( x ) = x µ ( x ), where µ ( x ) = R β ′ ( tx ) dt .Moreover, µ (0) = β ′ (0) and µ ( x ) = 0 for x = 0. Hence by Corol-lary 3.10 if β ′ (0) = µ (0) = 0, then Sh ( G ) = E id ( G ) ∞ .If β ′ (0) = 0, then µ = 0 on all of R and by Lemma 3.9 instead of G we can consider the linear vector field F ( x ) = x ∂∂x .It is shown in [22] that Sh ( F ) = E id ( F ) for any linear vector fieldon R n . We will now discuss these results.8.2. Let F ( x ) = vx ∂∂x be the linear vector field on R . Then it gener-ates the following flow F ( x, t ) = e vt x . Let V be an open neighbourhoodof 0 and Ω : V × S × T → R be an ( S ; T, k )-deformation in E ( F , V ).Then Ω ( σ,τ ) (0) = 0, and by the Hadamard lemmaΩ( x, σ, τ ) = x ∫ ∂ Ω ∂x ( tx, σ, τ ) dt. Since Ω ( σ,τ ) ∈ E ( F , V ), we have that ∂ Ω ∂x ( x, σ, τ ) >
0, i.e. Ω ( σ,τ ) is alocal diffeomorphism at 0. Hence we can define the following functionΛ : V × S × T → R byΛ( x, σ, τ ) = v ln ∫ ∂ Ω ∂x ( tx, σ, τ ) dt. Then Λ is a ( S ; T, k − x, σ, τ ) = e v · Λ( x,σ,τ ) , c.f. [22, Eq. (23)]. In particular, we get Sh ( F , V ) = E ( F , V ).8.3. Consider the following linear vector field on R : F ( x, y ) = ( ∂∂x , ∂∂y ) ( ) ( xy ) = y ∂∂x . It generates the flow F ( x, y, t ) = ( x + yt, y ). Let V be an open neigh-bourhood of , and Ω = (Ω , Ω ) : V × S × T → R be an ( S ; T, k )-deformation in E ( F , V ). Then Ω ( x, y, σ, τ ) ≡ y and Ω ( x, , σ, τ ) ≡ x . Hence by the Hadamard lemmaΩ ( x, y, σ, τ ) − x = y ∫ ∂ Ω ∂y ( x, ty, σ, τ ) dt. Define the function Λ : V × S × T → R byΛ( x, y, σ, τ ) = ∫ ∂ Ω ∂y ( x, ty, σ, τ ) dt. Then Λ is a ( S ; T, k − x, y, σ, τ ) = ( x + y · Λ( x, y, σ, τ ) , y ), c.f. [22, Eq. (26)]. Again weget Sh ( F , V ) = E ( F , V ).8.4. Let ω = a + bi = 0 be a complex number, F ( z, t ) = e ω t z be thecorresponding linear flow on C , and V ⊂ C be an open neighbourhoodof . By definition every Ω ∈ E ( F , V ) is a local diffeomorphism at .Let Ω : V × R d → M be a ( R d , C ∞ )-deformation in E ( F , V ) suchthat Ω σ preserves orientation at for each σ ∈ R d . Then by [22,Lm. 31] there exists a C ∞ function γ : V × R d → C \ such thatΩ( z, σ ) = z · γ ( z, σ ). Hence, [22, Eqs.(24),(25)], the following functionΛ : V × R d → R defined byΛ( z ) = ( a ln | γ ( z, σ ) | , a = 0 , b arg( γ ( z, σ )) , a = 0 , is a C ∞ shift function for Ω. In the second case Λ is defined up to aconstant summand 2 πk/n . Remark 8.32. In § § S ; T, k )-defor-mation Ω is expressed via partial derivatives of Ω. Therefore it loosesits “smoothness in τ ” by 1 though it remains C ∞ in σ . The main toolof constructing Λ is the Hadamard lemma.On the other hand in § all the partial derivatives of Ω for the proofof smoothness of γ . Hence if Ω were an ( S ; T, k )-deformation, then wecould not guarantee that the obtained function Λ is even continuous in τ though for each τ ∈ T the function Λ τ is C ∞ on V , c.f. Definition 9.42.Such effects are typical for divisibility by smooth functions, c.f.[31,p. 93. Eq. (2)] and [34].8.5. Properties ( E ) l . In [22, Defn. 24] for every l ≥ E ) l for a singular point z of a vector field F wasintroduced (see also Remark 1.4). We will now present a slight modifi-cation of this property which will be useful for further considerations. Definition 8.33.
Let z ∈ Σ be such that Σ is nowhere dense at z . Saythat F has property ( E ) l , ( l ≥ , at z if for any • open neighbourhood V of z , • a smooth manifold S l of dimension dim S l = l , MAGE OF A SHIFT MAP ALONG THE ORBITS OF A FLOW 31 • an ( S l , C ∞ ) deformation Ω : V × S l → M in E ( F , V ) , and • a C ∞ shift function Λ : ( V \ Σ) × S l → R for Ω , i.e. (8.22) Ω( x, σ ) = F ( x, Λ( x, σ )) , ( x, σ ) ∈ ( V \ Σ) × S l , there exists a neighbourhood W ⊂ V of z such that Λ extends to a C ∞ function Λ : W × S l → R . Thus in order to verify ( E ) l one should smoothly resolve (8.22) withrespect to Λ on a neighbourhood of z × S l using the assumption that Ω σ preserves orbits of F and is a local diffeomorphism at each x ∈ Σ ∩ V .In particular, vector fields of §§ E ) l for all l ≥ σ has a shift function on V \ Σ = V \ , then it shouldpreserve orientation at .Property ( E ) allows to prove that Sh ( F ) = E id ( F ) . Lemma 8.34. [22, Th. 25]
Suppose that Σ is nowhere dense and F hasproperty ( E ) at each z ∈ Σ . Then Sh ( F ) = E id ( F ) . As a direct consequence of Lemma 7.31, we obtain that properties( E ) l for l > E ) , and therefore to establish Sh ( F ) = E id ( F ) , for regular extensions. Lemma 8.35. [22, Lm. 29] , (see also Remark 1.4 and Lemma 7.31)Let G (resp. F ) be a germ of a vector field on R n at n (resp. on R m + n at m + n ). Suppose F is a regular extension of G . If G has property ( E ) l + n at m , then F has property ( E ) l at m + n . Definition 8.36.
Let F be a vector field on a manifold M and z ∈ Σ .Say that z is of type (L) for F , if the germ of F at z is a regularextension of some linear vector field on R n at n for some n > . Corollary 8.37. [22, Th. 27] . Every point of type (L) has property ( E ) l for each l ≥ .Proof. As noted above, every linear vector field is a regular extensionof some vector field considered in §§ (cid:3) The problem of linearization of a vector field in a neighbourhood ofa singular point was extensively studied, see e.g. [41, 42, 46, 19, 2].
Corollary 8.38. [22, Th. 1(A)] . Suppose every z ∈ Σ is of type (L) for F . Then Sh ( F ) = E id ( F ) . Reduced Hamiltonian vector fields.
In [27] the author pre-sented a class of examples of highly degenerate vector fields F on R n with singularity at for which Sh ( F ) coincides at with E id ( F ) for-mally . That is for each h ∈ E id ( F ) there exists α ∈ C ∞ ( R n , R ) suchthat j ∞ h ( ) = j ∞ ϕ ( α )( ). Every such vector field (and even its initial non-zero jet at ) turned out to be non-divisible by smooth functions,c.f. Corollary 3.10.Moreover, in the following special case it can be said more.Let g : R → R be a homogeneous polynomial of degree p ≥
2, so(8.23) g = L l · · · L l a a · Q q · · · Q q b b , where L i is a non-zero linear function, Q j is an irreducible over R (definite) quadratic form, l i , q j ≥ L i /L i ′ = const for i = i ′ , and Q j /Q j ′ = const for j = j ′ . Put D = L l − · · · L l a − a · Q q − · · · Q q b − b . Then g = L · · · L a · Q · · · Q b · D and it is easy to see that D is the great-est common divisor of the partial derivatives g ′ x and g ′ y . The following polynomial vector field on R : F ( x, y ) = − ( g ′ y /D ) ∂∂x + ( g ′ x /D ) ∂∂y will be called the reduced Hamiltonian vector field of g . In particular,if g has no multiple factors, i.e. l i = q j = 1 for all i, j , then D ≡ F is the usual Hamiltonian vector field of g .Notice that F ( g ) ≡ F are relativelyprime homogeneous polynomials of degree deg F = a + 2 b − F = 1, so F is linear, then by Corollary 8.38, Sh ( F ) = E id ( F ) .Suppose deg F ≥
2, so a + 2 b ≥
3. Then we will distinguish thefollowing two cases, see Figure 5.
Case (HE): a = 0 and b ≥
2. Thus g = Q · · · Q b is a product of atleast two distinct irreducible quadratic forms, and, in particular, theorigin ∈ R is a global extreme of g Case (HS): all other cases, so either a ≥ b = 0, or a ≥ b ≥
1. Then ∈ R is a saddle critical point of g .(HE) (HS) Figure 5.
It is shown in [26] that in the case (HE) Sh ( F ) = E id ( F ) . ByLemma 8.34 this implies that (HE)-singularities do not satisfy ( E ) .The last statement can also be verified by another arguments. Let θ : R \ → R be the function associating to every z = its periodwith respect to F . Then θ is a C ∞ shift function on R \ for theidentity map id R . But since j F ( ) = 0, we have that lim z → θ ( z ) = ∞ ,see Example 3.15. Hence θ can not be even continuously extended toall of R . Therefore the origin ∈ R does not satisfy ( E ) . MAGE OF A SHIFT MAP ALONG THE ORBITS OF A FLOW 33
Definition 8.39.
Let F be a vector field on a manifold M and z ∈ Σ .We say that z is of type (HE) , resp. (HS) , for F , if the germ of F at z is a regular extension of some reduced Hamiltonian vector field of ahomogeneous polynomial (8.23) of the case (HE) , resp. (HS) .In either the cases we say that z is of type (H) . Example 8.40.
A singular point of a vector field can belong to distincttypes. For instance, consider the following vector field on R : F ( a, b, c, d, e ) = (2 a, − c , b , − d − de, de + 2 e )Then F is a product of the linear vector field 2 a ∂∂a and reduced Hamil-tonian vector fields of polynomials b + c and d e + d e . Whence theorigin ∈ R belongs to each of the types (L), (HE), and (HS).As an application of [27] and the results of the present paper wewill prove in next section the following theorem which extends Corol-lary 8.38 to vector fields with singularities of type (H). This theoremis also a “global” variant of [27, Th. 11.1]. Theorem 8.41.
Let F be a vector field on a manifold M tangent to ∂M and such that every z ∈ Σ belongs to one of the types (L) or (H) .Then Sh ( F ) = E id ( F ) . Moreover, F satisfies condition GSF , i.e. if G is another vector field each of whose orbits is contained in some orbitof F , then there exists a C ∞ function α : M → R such that G = α F .Moreover, if, in addition, every z ∈ Σ belongs to one of the types (L) or (HS) , then Sh ( F ) = E id ( F ) . Properties
ESD ( l ; d, k )In this section we introduce another series of properties ESD ( l ; d, k )being weaker than ( E ) l , but similarly to them property ESD (0; 1 , k ) forsingular points of F guarantees Sh ( F ) = E id ( F ) k , while ESD ( l ; d, k ) im-plies ESD (0; d, k ) for regular extensions. We also describe
ESD ( l ; d, k )for singularities of types (H). This will allow to prove Theorem 8.41.For l ≥ R l + = { x l ≥ } be closed upper half space in R l . Definition 9.42.
Suppose that Σ is nowhere dense at z ∈ Σ and let ≤ l < ∞ , ≤ d < ∞ , and ≤ k ≤ ∞ . Say that F has extensionsof shift functions under k -deformations property ESD ( l ; d, k ) at z if the following holds true: let • V be a connected, open neighbourhood of z , • S l be a either R l or R l + , • Ω : V × S l × I d → M be an ( S l ; I d , k ) -deformation in E ( F , V ) ,such that for some ( σ , τ ) ∈ S l × I d the map Ω ( σ ,τ ) has a C ∞ function α on all of V , and • Λ : ( V \ Σ) × S l × I d → R be a unique continuous shift functionfor Ω such that Λ ( σ ,τ ) = α ; then there exists a neighbourhood W ⊂ V × S l of ( z, σ ) such that foreach τ ∈ I d the function Λ τ smoothly extends to W .For d = 0 the number k does not matter, therefore in this case wewill denote the corresponding property by ESD ( l ; 0 , − ) . Also notice that ESD (0; 0 , − ) is a tautology, therefore we usually assume that l + d ≥ . Remark 9.43. a) Definition 9.42 does not require for Λ to be contin-uous on W × S l × I d .b) Also notice that ESD ( l ; d, k ) is a property of the germ of F at z ,i.e. if U is an open neighbourhood of z , then F has ESD ( l ; d, k ) at z ifand only if the restriction F | U has this property at z .The following lemma is a direct consequence of definitions and weleft it for the reader. Lemma 9.44.
For ≤ l ≤ l ′ < ∞ , ≤ d ≤ d ′ < ∞ , and ≤ k ≤ k ′ ≤ ∞ the following implications hold true: (a) ( E ) l ′ ⇒ ( E ) l , (b) ( E ) l ⇒ ESD ( l ; d, k ) , (c) ESD ( l ′ ; d ′ , k ) ⇒ ESD ( l ; d, k ′ ) .Thus ESD (1; 0 , − ) and ESD (0; 1 , ∞ ) are the weakest properties. (cid:3) Evidently, Theorem 5.24 claims that either of its conditions (a)-(c)implies
ESD (0; d, k ) for z for all d ≥
0. Moreover, the proof of The-orem 5.25 only uses property
ESD (0; d, k ). Hence the following resultbeing also an analogue of [22, Th. 25] (see Lemma 8.34) holds true. Weleave the details for the reader.
Theorem 9.45.
Suppose Σ is nowhere dense in V and there exists k ∈{ , . . . , ∞} such that F has property ESD (0; 1 , k ) at every z ∈ Σ ∩ V .Then (9.24) Sh ( F , V ) = E id ( F , V ) k , In particular, if V = M , then Sh ( F ) = E id ( F ) k Lemma 9.46. c.f. [27, Th. 4.1]
Suppose F has the weakest properties ESD (1; 0 , − ) and ESD (0; 1 , ∞ ) at each of its singular points, then F has GSF as well, see § It is shown in [27, Th. 4.1] that if for each z ∈ Σ there existsa neighbourhood V such that Sh ( F , V ) = E id ( F , V ) ∞ and the shiftmap ϕ V satisfies the so-called smooth path-lifting condition , see [27,Defn. 4.2], then F has property GSF .The first assumption Sh ( F , V ) = E id ( F , V ) ∞ can be satisfied dueto property ESD (0; 1 , ∞ ) for singular points of F and Theorem 9.45.Moreover, smooth path-lifting condition is the same as ESD (1; 0 , − ). (cid:3) The next statement describes in such a way
ESD ( l ; d, k ) is inheritedwith respect to regular extensions, c.f. Lemma 8.35. MAGE OF A SHIFT MAP ALONG THE ORBITS OF A FLOW 35
Lemma 9.47.
Let G (resp. F ) be a germ of a vector field on R n at n (resp. on R m + n at m + n ). Suppose F is a regular extension of G , and F ( m + n ) = 0 , so G ( m ) = 0 as well. If G has property ESD ( l + n ; d, k ) at m , then F has property ESD ( l ; d, k ) at m + n .Proof. Let V m and V n be open connected neighbourhoods of m and n respectively. Then V m + n = V m × V n is a connected, open neigh-bourhood of m + n .Let S l = R l or R l + , ( σ , τ ) ∈ S × I d , Ω : V m + n × S l × I d → R m + n bean ( S l ; I d , k )-deformation in E ( F , V m + n ), such that Ω ( σ ,τ ) = ϕ V m + n ( α )for some α ∈ C ∞ ( V m + n , R ), and Λ : ( V m + n \ Σ F ) × S l × I d → R be aunique shift function for Ω such that Λ ( σ ,τ ) = α .By Lemma 7.31 the projectionΨ = p m ◦ Ω : V m × V n × S l × I d → R m is a ( V n × S l ; I d , k )-deformation in E ( G , V m ) and Λ is a shift functionfor Ψ. Then by ESD ( l + n ; d, k ) for m there exists a neighbourhood W of ( m , n , σ ) in V m × V n × S l such that for each τ ∈ I d the functionΛ τ smoothly extends to W . This implies ESD ( l ; d, k ) for m + n . (cid:3) Lemma 9.48.
Every point of type (HS) has property ( E ) d for all d ≥ .Every point of type (HE) has property ESD ( l ; d, k ) for all l, d ≥ and k ≥ , but in general it does not satisfy ESD ( l ; d, with l + d ≥ .Proof. Let g be a homogeneous polynomial (8.23), F be the correspond-ing reduced Hamiltonian vector field of g , and V be an open connectedneighbourhood of ∈ R . Suppose that deg F ≥
2. It is shownin [23, 27], see [27, Th. 11.1], that in this case1) E id ( F , V ) = { h ∈ E ( F , V ) : j h ( ) = id } , and2) there exists a map λ : E id ( F , V ) → C ∞ ( V, R ) with the followingproperties.(2a) ϕ V ◦ λ = id( E id ( F , V ) ), that is for each h ∈ E id ( F , V ) we havethat h ( z ) = F ( z, λ ( h )( z )). Thus λ is the inverse of ϕ V . Inparticular, we obtain that E id ( F , V ) = Sh ( F , V ).(2b) the map λ preserves smoothness in the following sense: for any( S, C ∞ )-deformation Ω : V × S → R in E id ( F , V ) the functionΛ : V × S → R defined by Λ( z, σ ) = λ (Ω σ )( z ) is also C ∞ .Now we can complete our lemma. Denote b V = V \ Σ. Let also S l be any smooth manifold of dimension l .(HS). Let Ω : V × S l → R be an ( S l , C ∞ )-deformation in E ( F , V )and Λ : b V × S l → R be any shift function for Ω. It is shown in [26]that E id ( F , V ) = E ( F , V ) for the case (HS), so Ω is a deformation in E id ( F , V ) = Sh ( F , V ). Put α σ = λ (Ω σ ). Then Λ σ and α | σ | b V are two shift functions for Ω σ on b V . Since is a “saddle” point, we obtain that b V contains non-closed orbits of F , see Figure 5. Hence ϕ b V is non-periodic. ThereforeΛ σ = α | σ | b V , so Λ σ smoothly extends to V .Thus we obtain a function Λ : V × S l → R which can be defined byΛ( z, σ ) = λ (Ω σ )( z ). Then by (2b) Λ is C ∞ . This proves ( E ) d for .(HE). Let Ω : V × S l × I d → R be an ( S l ; I d , E id ( F , V ) = ϕ V , ( σ , τ ) ∈ S l × I d , α ∈ C ∞ ( V, R ) be a C ∞ shift functionfor Ω ( σ ,τ ) and Λ : b V × S l × I d → R be a unique continuous shift functionfor Ω such that Λ ( σ ,τ ) = α .Notice that in the case (HE) the origin is a strong pn -point for F and also j F ( ) = 0, so it satisfies even both assumptions (b) and (c)of Theorem 5.24. Hence for each ( σ, τ ) ∈ S l × I d the function Λ ( σ,τ ) smoothly extends to all of V and coincides with a unique shift function α ( σ,τ ) = λ (Ω ( σ,τ ) ) for Ω ( σ,τ ) .Then for each τ ∈ I d the obtained function Λ τ : V × S l → R isdefined by Λ τ ( z, σ ) = λ (Ω ( σ,τ ) )( z ). Hence by (2b) Λ τ is C ∞ on V × S l .This proves ESD ( l ; d,
1) for .To show that does not satisfy ESD ( l ; d,
0) consider the mapping h : R → R defined by h ( z ) = − z . Since g is a product of definitequadratic forms, it follows that g ◦ h = g , whence h ∈ E ( F , V ). It isshown in [26] that h ∈ E id ( F , V ) \ E id ( F , V ) , therefore Sh ( F , V ) = E id ( F , V ) .Let us briefly recall the idea of proof. First it is not hard to showthat the set of 1-jets at of elements of E ( F , V ) constitute a finite(actually either dihedral or cyclic) subgroup of GL(2 , R ). Then, usinganalogue of a Alexander trick, it was constructed a 0-homotopy h t between h = id R and h = h in E ( F , V ). Neither of such homotopiescan be a 1-homotopy. Indeed, j h ( ) = id, while j h ( ) = − id. Butas just noted, j h t ( ) can take only finitely many values, whence j h t is not continuous in t , so ( h t ) is not a 1-homotopy. (cid:3) Proof of Theorem 8.41.
By Corollary 8.37 and Lemmas 9.44,9.47, and 9.48 every point of type (L) and (HS) has property ( E ) , whileevery point of type (HE) has property ESD (0; 1 , Sh ( F , V ).Moreover, again by Lemma 9.44 every singular point of F has prop-erties ESD (1; 0 , − ) and ESD (0; 1 , ∞ ), whence by Lemma 9.46 F hasproperty GSF . (cid:3) Acknowledgment
I would like to thank the anonymous referee for very useful commentswhich allow to improve the exposition of the paper: in particular, forthe information about the papers [32, 8, 43] and for pointing out to thestandard definition of parameter rigidity.
MAGE OF A SHIFT MAP ALONG THE ORBITS OF A FLOW 37
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