∞ -jets of difeomorphisms preserving orbits of vector fields
aa r X i v : . [ m a t h . D S ] D ec ∞ -JETS OF DIFFEOMORPHISMS PRESERVINGORBITS OF VECTOR FIELDS SERGIY MAKSYMENKO
Abstract.
Let F be a C ∞ vector field defined near the origin O ∈ R n , F ( O ) = 0, and ( F t ) be its local flow. Denote by ˆ E ( F ) theset of germs of orbit preserving diffeomorphisms h : R n → R n at O ,and let ˆ E id ( F ) r , ( r ≥ E ( F ) withrespect to C r topology. Then ˆ E id ( F ) ∞ contains a subset ˆ Sh ( F )consisting of maps of the form F α ( x ) ( x ), where α : R n → R runsover the space of all smooth germs at O . It was proved earlier bythe author that if F is a linear vector field, then ˆ Sh ( F ) = ˆ E id ( F ) . In this paper we present a class of examples of vector fields withdegenerate singularities at O for which ˆ Sh ( F ) formally coincideswith ˆ E id ( F ) , i.e. on the level of ∞ -jets at O .We also establish parameter rigidity of linear vector fields and“reduced” Hamiltonian vector fields of real homogeneous polyno-mials in two variables. Keywords: orbit preserving diffeomorphism, parameter rigidity, Bo-rel’s theorem.
AMSClass:
Introduction
Let F be a smooth ( C ∞ ) vector field on a smooth manifold M , ( F t )be the local flow generated by F , and Σ F be the set of singular pointsof F . In this paper we consider smooth maps h : M → M preservingthe (singular) foliation on M by orbits of F , i.e. h ( M ∩ γ ) ⊂ γ for everyorbit γ of F .The groups of leaf preserving diffeomorphisms and homeomorphismsof foliations are intensively studied. Most of the results concern withregular foliations, see e.g. [B77, Ryb1, Ryb2, AF03] and referencesin these papers. For singular foliations the situation is much moredifficult. Therefore usually foliations by orbits of actions of finite-dimensional Lie groups are considered, e.g. [Sch75, Ma77, AF01, Ryb3].Homeomorphisms preserving foliations of vector fields are studied e.g.in [CN77, GM77].The approach used in this paper is specific for the case of flows. Bydefinition for every x ∈ M its image h ( x ) belongs to the orbit γ x of x .Therefore we want to associate to x the time α h ( x ) between x and h ( x ) along γ x , so that(1.1) h ( x ) = F α h ( x ) ( x ) . -JETS OF ORBIT PRESERVING DIFFEOMORPHISMS 2 We will call α h a shift function for h , which in turn will be called the shift along orbits of F via α h .Such ideas were used e.g. in [Hopf37, Ch66, To66, Ko72, Pa72,Koc73] and others for reparametrizations of measure preserving flowsand study their mixing properties. In these papers α h is required to bemeasurable, so it can even be discontinuous and its values on subsets ofmeasure 0 can be ignored. In [OS78] continuity of shift functions wasinvestigated. In contrast, we work in C ∞ category and require that α h is C ∞ whenever so is h . One of the main problems here is to define α h near a singular point of a vector field, see 2.1-2.5.Smooth shift functions were used in authors papers [M1, M2, M3] forcalculations of homotopy types of certain infinite-dimensional spaces.The present paper brings another application of shift functions tosmooth reparametrization of flows and in particular to parameter rigid-ity.In [M1] the problem of finding representation (1.1) was solved forlinear flows. It was shown that if F is a linear vector field on R n ,then for every diffeomorphism h : R n → R n preserving orbits of F andbeing isotopic to the identity map id R n via an orbit preserving isotopythere exists a C ∞ shift function α h . Moreover, if a family h s of orbitpreserving diffeomorphisms smoothly depends on some k -dimensionalparameter s , then so does the family α h s of their shift functions † .Our first result claims that the last two properties easily imply pa-rameter rigidity of a vector field, see Theorem 4.4. In particular, asa consequence of [M1], we obtain parameter rigidity of linear vectorfields and their regular extensions. Notice that this statement togetherwith the result of S. Sternberg [St57] implies parameter rigidity of alarge class of “hyperbolic” flows, which agrees with discovered aboutthirty years ago rigidity of locally free hyperbolic actions of certainLie groups [KS94]. Though we consider actions of the one-dimensionalgroup R only, Theorem 4.4 is nevertheless stronger in the part that weadmit fixed points, i.e. non-locally free actions. † I must warn the reader that my paper [M1] contains mistakes in the estimationsof continuity of the correspondence h α h regarded as a map between certainfunctional spaces. In particular in [M1, Defn. 15] it should be additionally requiredthat the image ϕ V ( M ) is at least C ∞ W -open in the image of the map ϕ V . Withoutthis assumption [M1, Th. 17] is not true. Moreover in [M1, Lm. 31] the mapping Z − is C r +1 ,rW,W -continuous in the real case and only C ∞ , ∞ W,W -continuous in the complexcase. As a result the formulations of [M1, Th 1, Th.27 (part concerning (S)-points)& Lm. 28] should be changed.Unfortunately [M1, Th. 1] was essentially used in [M2] for the calculations of thehomotopy types of stabilizers and orbits of Morse functions on surfaces. We willrepair the mistakes of [M1] in another paper and show that the part of results [M1]used in [M2] remains true.Also notice that the formula [M1, Eq. (10)] for the shift functions at regularpoints is misprinted. It must be read as follows: α ( x ) = p ◦ f ( x ) − p ◦ Φ( x, a ) + a. -JETS OF ORBIT PRESERVING DIFFEOMORPHISMS 3 Further we deal with the situation when F is a vector field defined onsome neighbourhood V of the origin O ∈ R n being its singular point.Denote by ˆ Sh ( F ) the group of germs at O of smooth shifts, i.e. maps ofthe form (1.1). In § ∞ -jets of elements ofˆ Sh ( F ) and establish a necessary and sufficient condition for a certaingroup G of germs of diffeomorphisms of ( R n , O ) to coincide with ˆ Sh ( F )on the level of ∞ -jets at O , see Theorem 5.1.Let J ( F , V ) be the space of all smooth maps h : V → R n whose ∞ -jet j ∞ ( h ) at O coincides with the ∞ -jet of some shift F α h , where α h ∈ C ∞ ( V, R ). Though in general such a function α h is not unique,we show in §§ h α h becomes a continuous (and in a certain sense smooth) mapΛ : J ( F , V ) ⊃ X → C ∞ ( V, R ) defined on some subset X of J ( F , V )(Theorem 7.6). Actually § § § D ( F ) the group of germs of orbit preserving diffeomor-phisms for F , and let ˆ D id ( F ) be its path component with respect toweak C W topology. In § ∗ ) on F guaranteeing that ˆ Sh ( F ) coincides with ˆ D id ( F ) on the (formal) level of ∞ -jets, see Theorem 8.5. The proof of this theorem is given in §§ §
11 we present a class of vector fields on R satisfyingcondition ( ∗ ) and explain that for these vector fields ˆ Sh ( F ) = ˆ D id ( F ) .This improves results of [M4], which were based on a previous (unpub-lished) version of this paper (Theorem 11.1).In another paper the last theorem will be used to extend calculationsof [M2] to a large class of functions with degenerate singularities onsurfaces.1.1. Preliminaries.
Let A and B be smooth manifolds. Then for ev-ery r = 0 , , . . . , ∞ we can define the weak C rW topology on C ∞ ( A, B ),see e.g. [GG, Hi]. We will assume that the reader is familiar with thesetopologies. It easily follows from definition that topology C W coincideswith the compact open one. Moreover, let J r ( A, B ) be the manifold of r -jets of C r maps A → B . Associating to every h ∈ C ∞ ( A, B ) its r -jetextension being an element of J r ( A, B ), we obtain a natural inclusion C ∞ ( A, B ) ⊂ C r ( A, J r ( A, B )). Then C rW topology on C ∞ ( A, B ) can bedefined as the topology induced by C W topology of C r ( A, J r ( A, B )).We say that a subset
X ⊂ C ∞ ( A, B ) is C kW -open if it is open withrespect to the induced C kW -topology of C ∞ ( A, B ).1.2.
Definition.
Let H : A × I → B be a homotopy such that for every t ∈ I the mapping H t : A → B is C r . We will call H an r -homotopy -JETS OF ORBIT PRESERVING DIFFEOMORPHISMS 4 if the following map j r H : A × I → J r ( A, B ) , ( a, t ) j r ( H t )( a ) , associating to every ( a, t ) ∈ A × I the r -th jet of H t at a , is continuous.In local coordinates this means that H t and all its partial derivatives“along A ” are continuous in ( a, t ) . In particular, every C r -homotopyis an r -homotopy as well.Equivalently, regarding a homotopy H as a path ˆ H : I → C ∞ ( A, B ) defined by ˆ H ( t )( a ) = H ( a, t ) , we see that H is an r -homotopy if andonly if ˆ H is a continuous path into C rW -topology of C ∞ ( A, B ) .If H is an r -homotopy consisting of embeddings, it will be called an r -isotopy . Let C and D be some other smooth manifolds and Y ⊂ C ∞ ( C, D )be a subset. A map u : X → Y will be called C s,rW,W -continuous ifit is continuous from C sW -topology of X to C rW -topology of Y , ( r, s =0 , , . . . , ∞ ).1.3. Definition.
We will say that u : X → Y preserves smoothness if for any C ∞ map H : A × R n → B such that H t = H ( · , t ) ∈ X for all t ∈ R n the following mapping u ( H ) : C × R n → D, u ( H )( c, t ) = u ( H t )( c ) is C ∞ as well. Obstructions for shift functions
In this section we briefly discuss obstructions for smooth resolvabilityof (1.1).2.1.
Evidently, the value α h ( z ) is uniquely defined only if z is regularand non-periodic for F . If z is a periodic point of period θ , then α h ( z ) is defined only up to a constant summand nθ , ( n ∈ Z ) , while if z isfixed, we can set α h ( z ) to arbitrary number. In applications to ergodic flows this problem usually does not appear:the union of periodic and fixed points is an invariant subset, thereforeit can be assumed to have measure 0. Hence the values of α h on this setmay be ignored. Sometimes it is also assumed that the set of periodicpoints is empty, e.g. [Ko72, p.357].2.2. If z is a regular (even periodic) point of F , then α h can besmoothly defined on some neighbourhood of z , see [M1, § Representation (1.1) with smooth α h implies that h is homotopicto the identity id via a smooth orbit preserving homotopy . For instancewe can take the following one: h t ( x ) = F ( x, t α h ( x )). -JETS OF ORBIT PRESERVING DIFFEOMORPHISMS 5 if h is homotopic to id via some smooth orbit pre-serving homotopy, then (using this homotopy and 2.2) we can smoothlydefine α h on the set of regular points of F , see [M1, Th. 25] , though α h can even be discontinuous at singular points of F , In general, α h depends on a particular homotopy.2.5. If α h can be defined at some singular point z of F so that it be-comes smooth near z , then h must be an embedding at z , [M1, Cor. 21].3. Shift map
Observations of the previous section lead to the following construc-tion of shift map used in [M1].Let M be a smooth manifold and F be a vector field on M tangentto ∂M . Then for every x ∈ M its orbit with respect to F is a uniquemapping γ x : R ⊃ ( a x , b x ) → M such that γ x (0) = x and ˙ γ x = F ( γ x ),where ( a x , b x ) ⊂ R is the maximal interval on which a map with theprevious two properties can be defined. Then dom ( F ) = ∪ x ∈ M x × ( a x , b x ) , is an open neighbourhood of M × M × R , and by definition the local flow of F is the following map F : M × R ⊃ dom ( F ) −→ M, F ( x, t ) = γ x ( t ) . If M is compact, or more generally if F has compact support, then dom ( F ) = M × R and thus F is global , i.e. is defined on all of M × R ,see e.g. [PM].For every open V ⊂ M denote by func ( F , V ) the subset of C ∞ ( V, R )consisting of functions α whose graph Γ α = { ( x, α ( x )) : x ∈ V } iscontained in dom ( F ). Then we can define the following map(3.1) ϕ V : C ∞ ( V, R ) ⊃ func ( F , V ) −→ C ∞ ( V, M ) ,ϕ V ( α )( x ) = F ( x, α ( x )) , which will be called the shift map of F on V . Its image in C ∞ ( V, M ) willbe denoted by Sh ( F , V ). If F is global, then func ( F , V ) = C ∞ ( V, R ).It is easy to see that ϕ V is C r,rW,W -continuous for all r = 0 , , . . . , ∞ ,[M1, Lemma 2]. Moreover, if the set Σ F of singular points of F isnowhere dense in V , then ϕ V is locally injective with respect to any C rW topology of func ( F , V ), [M1, Prop. 14].Denote by E ( F , V ) ⊂ C ∞ ( V, M ) the subset consisting of all smoothmaps h : V → M such that • h ( ω ∩ V ) ⊂ ω for every orbit ω of F , in particular h is fixed onΣ F ∩ V , and • h is a local diffeomorphism at every z ∈ Σ F ∩ V . -JETS OF ORBIT PRESERVING DIFFEOMORPHISMS 6 Let E id ( F , V ) r , (0 ≤ r ≤ ∞ ), be the path component of the iden-tity embedding i V : V ⊂ M in E ( F , V ) with respect to C rW -topology,i.e. E id ( F , V ) r consists of all smooth maps h : V → M which are r -homotopic to i V in E ( F , V ).Then we have the following inclusions:(3.2) Sh ( F , V ) ⊂ E id ( F , V ) ∞ ⊂ · · · ⊂ E id ( F , V ) ⊂ E id ( F , V ) . The first one follows from 2.3 and 2.5, and all others are evident.For V = M we denote ϕ = ϕ M , func ( F ) = func ( F , M ), Sh ( F ) = Sh ( F , M ), and E id ( F ) r = E id ( F , M ) r .3.1. Local shift map.
For z ∈ M let ˆ F z ( M ) be the algebra of germsat z of smooth functions M → R , ˆ E ( M, z ; M ) be the space of germs at z of all smooth maps f : M → M , ˆ E ( M, z ) ⊂ ˆ E ( M, z ; M ) be the subsetconsisting of germs f such that f ( z ) = z , and ˆ D ( M, z ) ⊂ ˆ E ( M, z ) bethe subset consisting of germs of all diffeomorphisms.We want to define the following local shift map analogous to (3.1):ˆ ϕ : ˆ F z ( M ) → ˆ E ( M, z ; M ) , ˆ ϕ ( α )( x ) = F ( x, α ( x )) . If F is not global, then as well as in the definition of shift map ϕ ,see (3.1), ˆ ϕ is defined only on a certain subset of ˆ F z ( M ). Nevertheless,the following lemma shows that if z is a singular point of F , then ˆ ϕ iswell-defined and its image is contained in ˆ D ( M, z ).3.2.
Lemma.
Suppose that F ( z ) = 0 . For h ∈ ˆ E ( M, z ) and α, β ∈ ˆ F z ( M ) define the following two maps F α , F h,β by F α ( x ) = F ( x, α ( x )) , F h,β ( x ) = F ( h ( x ) , β ( x )) . Then the following statements hold true. (a)
The germs at z of F α and F h,β are well-defined. (b) F α , F β ∈ ˆ Sh ( F ) are germs of a diffeomorphisms at z and (3.3) F − α = F − α ◦ F − α , F α ◦ F β = F α ◦ F β + β . (c) If h is a germ of a diffeomorphism, then so is F h,β and (3.4) F h,β = F β ◦ h − ◦ h. (d) The following conditions are equivalent: (3.5) h = F α ⇔ F h, − α = id . Proof. (a) Since F ( z ) = 0, it follows from the standard result on de-pendence of solutions of ODE on initial values that for arbitrary large A ≥ W A of z such that W A × [ − A, A ] ⊂ dom ( F ). Hence if A > | α ( z ) | then F α is defined on some neighbourhoodof z contained in W A . The proof for F h,β is similar.(b) is proved in [M1, Eqs.(8),(9) and Corollary 21], see also 2.5.(c) Eq. (3.4) just means that F h,β ( x ) = F ( h ( x ) , β ◦ h − ◦ h ( x )).(d) Finally, the verification of (3.5) is direct. (cid:3) -JETS OF ORBIT PRESERVING DIFFEOMORPHISMS 7 Let F ( z ) = 0. Then we have a well-defined local shift map at z ˆ ϕ : ˆ F z ( M ) → ˆ D ( M, z ) , ˆ ϕ ( α ) = F α . Denote its image ˆ ϕ ( ˆ F z ( M )) in ˆ D ( M, z ) by ˆ Sh ( F , z ).Let also ˆ D ( F , z ) be the subset of ˆ D ( M, z ) consisting of orbit preserv-ing germs, i.e. h ∈ ˆ D ( M, z ) provided there exists a neighbourhood V of z such that h ( γ ∩ V ) ⊂ γ for every orbit γ of F .For every r = 0 , , . . . , ∞ let ˆ D id ( F , z ) r be the “identity component”of ˆ D ( F , z ) with respect to C rW -topology, i.e. ˆ D id ( F , z ) r consists of all h ∈ ˆ D ( F , z ) for which there exists a neighbourhood V ⊂ M of z and an r -isotopy H : V × I → M such that H = i V : V ⊂ M , H t ∈ ˆ D ( F , z )for all t ∈ I , and H = h .Then similarly to (3.2) we have the following inclusions:ˆ Sh ( F , z ) ⊂ ˆ D id ( F , z ) ∞ ⊂ · · · ⊂ ˆ D id ( F , z ) ⊂ ˆ D id ( F , z ) . Parameter rigidity
In recent years there were obtained many results concerning rigidityof hyperbolic and locally free actions of certain Lie groups and theirlattices, see e.g. [KS94, Hur94, Ka96, KS97, MM03, D07, EF07] andreferences there. Roughly speaking a rigidity of an action T meansthat every action T ′ which is sufficiently close in a proper sense to T isconjugate to T .For instance, in a recent paper [Sa07] by N. dos Santos parameterrigidity of locally free actions of contractible Lie groups on closed ma-nifolds are considered. In the case of vector fields, i.e. actions of R ,local freeness means regularity of orbits. In contrast we will considercertain classes of vector fields with singular points, i.e. not locally free R -actions, and prove their parameter rigidity, see 4.4 and 4.6.4.1. Definition. (c.f. [Sa07])
We say that a vector field F on a manifold M is parameter rigid if for any vector field G on M such that everyorbit of G is contained in some orbit of F there exists a C ∞ -function α such that G = αF . Let Σ F and Σ G be the sets of singular points of F and G respectively.The assumption that orbits of G are contained in orbits of F impliesthat Σ F ⊂ Σ G and that F and G are parallel on M \ Σ G . Thereforethere exists a C ∞ function µ : M \ Σ F → R \ { } such that G = µF on M \ Σ F . Then Definition 4.1 requires that µ smoothly extends toall of M for any such G .4.2. Lemma (Extensions of shift functions under homotopies) . Let V ⊂ M be an open subset, α ∈ func ( F , V ) , and H : V × I → M bea C ∞ -homotopy such that H = ϕ ( α ) and H t ∈ Sh ( F , V ) for every -JETS OF ORBIT PRESERVING DIFFEOMORPHISMS 8 t ∈ I . Then there exists a unique C ∞ function Λ : ( V \ Σ F ) × I → R such that (4.1) Λ( x,
0) = α ( x ) , H ( x, t ) = F ( x, Λ( x, t )) , for all ( x, t ) ∈ ( V \ Σ F ) × I . Thus Λ is a shift function for H on ( V \ Σ F ) × I which extends α .Proof. This statement was actually established during the proof of [M1,Th. 25] for the case α ≡
0. But the same arguments show that theproof holds for any α ∈ func ( F , V ). We leave the details for thereader. (cid:3) Definition.
Let V ⊂ M be an open subset such that Σ F is nowheredense in V . We will say that the shift map ϕ V of F satisfies smoothpath-lifting condition if for every C ∞ -homotopy H : V × I → M and α ∈ func ( F , V ) such that H = ϕ ( α ) and H t ∈ Sh ( F , V ) , ( t ∈ I ) the shift function Λ : ( V \ Σ F ) × I → R of H satisfying (4.1) smoothlyextends to all of V × I . See also [Sch80], where the problem of lifting smooth homotopies oforbit spaces of Lie groups is considered.4.4.
Theorem.
Let F be a vector field on a manifold M . Suppose thatfor every singular point z ∈ Σ F there exists an open neighbourhood V such that Sh ( F , V ) = E id ( F , V ) ∞ and the corresponding shift map ϕ V satisfies smooth path-lifting condition. Then F is parameter rigid.Proof. Let G be a vector field on M such that every orbit of G iscontained in some orbit of F . Then there exists a smooth function µ : M \ Σ F → R such that G = µF . We have to show that µ smoothlyextends to all of M .We can assume that G generates a global flow G : M × R → M .Otherwise, there exists a smooth function β : M → (0 , ∞ ) such thatthe vector field G ′ = βG generates a global flow, see e.g. [Hr83, Corol-lary 2]. Then G and G ′ have the same orbit foliation. Moreover, if G ′ = γF for some smooth function γ : M → R , then G = γβ F , where γβ is smooth on all of M as well.Let z ∈ Σ F and V be a neighbourhood of z such that ϕ V satisfiessmooth path-lifting condition. Then for each t ∈ R we have a well-defined embedding G t | V : V → M belonging to E ( F , V ). Moreover,since G = id M = ϕ (0) and G is C ∞ , it follows that G t | V ∈ E id ( F , V ) ∞ = Sh ( F , V ) . Then by smooth path-lifting condition for ϕ V there exists a smoothfunction ¯ µ : V × R such that(4.2) G ( x, t ) = F ( x, ¯ µ ( x, t )) -JETS OF ORBIT PRESERVING DIFFEOMORPHISMS 9 and ¯ µ ( x,
0) = 0 for all x ∈ V . Let us differentiate both parts of (4.2)in t and set t = 0. Then we will get G ( x ) = ∂ G ∂t ( x,
0) = ∂ F ∂t ( x, ¯ µ ( x, · ¯ µ ′ t ( x,
0) = F ( x ) · ¯ µ ′ t ( x, . Hence µ ≡ ¯ µ ′ t ( x, F is nowhere dense in V , we obtain that µ smoothly extends to all of V . Applying these arguments to all z ∈ Σ F we will get that µ is smooth on all of M . (cid:3) As an application of this theorem and results of [M1] we will nowobtain parameter rigidity of linear vector fields and their regular ex-tensions.4.5.
Definition.
Let
M, N be two manifolds, G : M → T M be a vectorfield of M and F : M × N → T ( M × N ) = T M × T N be a vector field of M × N regarded as sections of the correspondingtangent bundles. Say that F is a regular extension of G providedthat F ( x, y ) = ( G ( x ) , H ( x, y )) , ( x, y ) ∈ M × N, for some smooth map H : M × N → T N such that H ( x, y ) ∈ T y N .In other words the “first” coordinate function of F “coincides with G ”and does not depend on y ∈ N . For instance if G i is a vector field on a manifold M i , ( i = 1 , F ( x, y ) = ( G ( x ) , G ( y )) on M × M is a regularextension of either of G i . Every linear vector field F ( x ) = Ax on R n is a product of linear vector fields generated by Jordan cells of realJordan form of A . Moreover, every Jordan cell vector field is a regularextension of a linear vector fields defined by the one of the followingmatrices: ( λ ), ( ), (cid:0) a − bb a (cid:1) , where either λ = 0 or b = 0, see also [V66].4.6. Corollary.
Let F be a vector field on a manifold M and V ⊂ M be an open subset. Suppose that the restriction of F to V is aregular extension of some non-zero linear vector field . Thismeans that there exist a non-zero ( m × m ) -matrix A , a smooth manifold N , and a diffeomorphism η : V → R m × N such that the induced vectorfield η ∗ F on R k × N is a regular extension of the linear vector field G ( y ) = Ay on R m . Then Sh ( F , V ) = E id ( F , V ) and the shift map ϕ V satisfies smooth path-lifting condition.Hence if every z ∈ Σ F has a neighbourhood V with the above property,then F is parameter rigid.Proof. The proof follows from [M1, Theorem 25 & Theorem 27, state-ment about (E)-point]. (cid:3)
In [Si52, St57, V66] and others there were obtained sufficient con-ditions for a vector field F defined in a neighbourhood of O ∈ R n to -JETS OF ORBIT PRESERVING DIFFEOMORPHISMS 10 be linear in some local coordinates at O . These results together withCorollary 4.6 imply parameter rigidity for a large class of “hyperbolic”flows which is in the spirit of mentioned above results of [KS94, Hur94,Ka96, KS97, MM03, D07, EF07, Sa07] concerning rigidity of hyper-bolic locally free actions of Lie groups. A new feature of Corollary 4.6is that F has singularity, and therefore the corresponding R -action (i.e.the flow of F ) is not locally free.The following lemma gives sufficient condition for a vector field tosatisfy smooth path-lifting condition.4.7. Lemma.
Suppose that Σ F is nowhere dense in V and for every h ∈ Sh ( F , V ) there exists a C ∞ W -neighbourhood N in Sh ( F , V ) and apreserving smoothness map (not necessarily continuous in any sense) σ : Sh ( F , V ) ⊃ N −→ func ( F , V ) ⊂ C ∞ ( V, R ) , such that τ ( x ) = F ( x, σ ( τ )( x )) for all τ ∈ N . In other words, σ isa section of ϕ V , i.e. ϕ V ◦ σ = id( N ) . Then ϕ V satisfies smoothpath-lifting condition.Proof. Before proving this lemma let us make two remarks.
R1.
Let h ∈ Sh ( F , V ) and σ , σ : N → func ( F , V ) be two preservingsmoothness sections of ϕ V defined on some C ∞ W -neighbourhood N of h , and H : V × I → M be a C ∞ map such that H = h and H t ∈ N for all t ∈ I . Since σ i preserves smoothness we have that the followingfunction Λ i : V × I → R given byΛ i ( x, t ) = σ i ( H t )( x )is C ∞ as well as H . If σ ( h ) = σ ( h ) , i.e. Λ ( · ,
0) = Λ ( · , , then Λ ≡ Λ on all of V × I . Indeed, since Λ ( · ,
0) = Λ ( · , ( x, t ) = Λ ( x, t ) for all ( x, t ) ∈ ( V \ Σ F ) × I . But Σ F is nowheredense in V and each Λ i is continuous. Therefore Λ = Λ on all of V × I . R2.
Let σ : N → func ( F , V ) be a preserving smoothness section of ϕ V defined on some neighbourhood N of h and α ∈ ϕ − V ( h ) be any shiftfunction for h . Notice that σ ( h ) ∈ ϕ − V ( h ) as well. Then there exists(possibly) another preserving smoothness section σ ′ : N → func ( F , V ) such that σ ′ ( h ) = α . Suppose that σ ( h ) = α . Then ϕ V is not injective map. Since Σ F is nowhere dense, it follows from [M1, Lm. 5 & Th. 12(2)] that thereexists a smooth function ν : V → (0 , + ∞ ) such that F ( x, ν ( x )) ≡ x for all x ∈ V , and α = σ ( h ) + nν for some n ∈ Z . Define the followingmap σ ′ : N → func ( F , V ) by σ ′ ( τ ) = σ ( τ ) + nν for τ ∈ N . Then σ ′ isalso a preserving smoothness section of ϕ V and σ ′ ( h ) = σ ( h ) + nν = α .Now we are ready to complete Lemma 4.7. Let α ∈ func ( F , V ) and H : V × I → M be a C ∞ map such that H = ϕ ( α ) and H t ∈ Sh ( F , V ) -JETS OF ORBIT PRESERVING DIFFEOMORPHISMS 11 for all t ∈ I , i.e. H t = ϕ V ( α t ) for some (not necessarily unique) α t ∈ func ( F , V ).We will show that under assumptions of lemma it is possible tochoose α t so that the correspondence ( x, t ) α t ( x ) becomes a C ∞ shift function Λ ′ : V × I → R for H such that Λ ′ ( x,
0) = α ( x ). Letalso Λ : ( V \ Σ F ) × I → R be a unique C ∞ shift function for H suchthat Λ( x,
0) = α ( x ) = Λ ′ ( x, ′ on ( V \ Σ F ) × I .Since Σ F is nowhere dense in V , we will get Λ ≡ Λ ′ on all of V × I .This will imply smooth path-lifting condition for ϕ V .Notice that H can be regarded as a continuous path into C ∞ W topologyof Sh ( F , V ). Since the image of this path is compact, there exist finitelymany points 0 = t < t < · · · < t n = 1 and for each k = 0 , , . . . , n − C ∞ neighbourhood N k of H t k in Sh ( F , V ) such that • H t ∈ N k for all t ∈ [ t k , t k +1 ], • and for every α ∈ func ( F , V ) such that ϕ V ( α ) = H t k there existsa preserving smoothness section σ α : N k → func ( F , V ) of ϕ V such that σ α ( H t k ) = α , (this follows from R2 ).Since H = ϕ V ( α ), the map σ α is well defined and we putΛ( x, t ) = σ α ( H t )( x ) , ( x, t ) ∈ V × [0 , t ] . Then Λ is smooth on V × [0 , t ].Denote α t = σ α ( H t ). Then H t = ϕ V ( α t ) and σ α is well defined.Therefore we also putΛ( x, t ) = σ α ( H t )( x ) , ( x, t ) ∈ V × [ t , t ] . Then Λ is smooth on V × [ t , t ]. Moreover σ α and σ α are two sectionsof ϕ V such that σ α ( H t ) = σ α ( H t ) = α t . Therefore by R1 σ α ( H t ) = σ α ( H t ) = Λ( · , t ) for all t sufficiently close to t . Since σ α and σ α preserve smoothness, it follows that Λ is smooth on all of [0 , t ].Using induction on n we can smoothly extend Λ on all of V × I . (cid:3) ∞ -jets of shifts Let F be a smooth vector field near the origin O ∈ R n such that F ( O ) = 0 and ( F t ) be the local flow of F . Define the following map j ∞ : ˆ D ( R n ) → ( R [[ x , . . . , x n ]]) n associating to every h ∈ ˆ D ( R n ) its ∞ -jet at O . Let also(5.1) ˆ J ( F ) = ( j ∞ ) − h j ∞ (cid:16) ˆ Sh ( F ) (cid:17)i . Thus ˆ J ( F ) is the subgroup of ˆ D ( R n ) consisting of germs h for whichthere exists a smooth function α h ∈ ˆ F ( R n ) such that j ∞ ( h ) = j ∞ ( F α h ). -JETS OF ORBIT PRESERVING DIFFEOMORPHISMS 12 Evidently, ˆ Sh ( F ) ⊂ ˆ J ( F ). Our first result gives necessary and suf-ficient conditions for a subgroup G ⊂ ˆ D ( R n ) containing ˆ Sh ( F ) to beincluded into ˆ J ( F ).5.1. Theorem.
Suppose that F is not flat at O , i.e. there exists p ≥ such that j p − ( F ) = 0 and P = j p ( F ) : R n → R n is a non-zerohomogeneous map of degree p . For p = 1 we will write P ( x ) = L · x ,where L is a certain non-zero ( n × n ) -matrix.Let G be a subgroup of ˆ D ( R n ) having the following properties: (A1) ˆ Sh ( F ) ⊂ G . (A2) For every h ∈ G there exists ω ∈ R such that j p ( h )( x ) = j p ( F ω )( x ) = ( e L ω · x, p = 1 ,x + P ( x ) · ω , p ≥ . (A3) Moreover, if j k − ( h ) = j k − (id) for some k ≥ p , then thereexists a unique homogeneous polynomial ω l of degree l = k − p such that j k ( h )( x ) = j k ( F ω l )( x ) = x + P ( x ) · ω l ( x ) . Then j ∞ (cid:0) ˆ Sh ( F ) (cid:1) = j ∞ ( G ) . In other words, ˆ Sh ( F ) ⊂ G ⊂ ˆ J ( F ) . The rest of this section is devoted to the proof of Theorem 5.1 whichwill be completed in § Spaces of jets.
Let ˆ E ( R n , O ; R m ) be the space of germs at theorigin O ∈ R n of smooth maps h : R n → R m and ˆ E ( R n , O ; R m , O ) beits subset consisting of all germs such that h ( O ) = O . For n = m , wewill write ˆ E ( R n ) instead of ˆ E ( R n , O ; R n , O ). Let also ˆ D ( R n ) ⊂ ˆ E ( R n )be the subset consisting of germs of diffeomorphisms at O . The spaceˆ E ( R n , O ; R ) of germs of smooth functions will be denoted by ˆ F ( R n ).For h ∈ ˆ E ( R n , O ; R m ) denote by j k ( h ) its k -jet at O ∈ R n . It will beconvenient to formally assume that ( − h is identically zero:(5.2) j ( − ( h ) ≡ . We will say that h ∈ ˆ E ( R n , O ; R m ) is k -small , ( k ≥ O provided j k − ( h ) = 0. In particular, by assumption (5.2) every h ∈ ˆ E ( R n , O ; R m )is 0-small and h is 1-small iff h ∈ ˆ E ( R n , O ; R m , O ), i.e. j ( h ) = h ( O ) = O . If j ∞ ( h ) = 0 then h is called flat .Let α, β ∈ ˆ F ( R n ) be such that α is a -small, and β is b -small for some a, b ≥
0. Then their product αβ is ( a + b )-small. In other words(5.3) j a − ( α ) = j b − ( β ) = 0 ⇒ j a + b − ( αβ ) = 0 . -JETS OF ORBIT PRESERVING DIFFEOMORPHISMS 13 We also say that h ∈ ˆ E ( R n , O ; R m , O ) is homogeneous of degree k ifits coordinate functions are homogeneous polynomials of degree k . Letˆ J k ( R n ), (0 ≤ k < ∞ ), be the space of all polynomial maps h : R n → R n of degree ≤ k such that h ( O ) = O . Similarly, putˆ J ∞ ( R n ) = (cid:8) τ ∈ (cid:0) R [[ x , . . . , x n ]] (cid:1) n : τ ( O ) = 0 (cid:9) . Define the following linear map j k : ˆ E ( R n ) → ˆ J k ( R n ) associating toevery h ∈ ˆ E ( R n ) its k -jet j k ( h ) at O . Then it is easy to verify that:(5.4) j k ( f ◦ g ) = j k ( j k ( f ) ◦ j k ( g )) , f, g ∈ ˆ E ( R n ) , see e.g. [St57, § Lemma.
Let f, g ∈ ˆ D ( R n ) . Then for every k = 0 , . . . , ∞ thefollowing conditions are equivalent: j k ( f ) = j k ( g ) ⇔ j k ( f − ◦ g ) = j k (id) ⇔ j k ( f − ) = j k ( g − ) . Moreover if α, β ∈ ˆ F ( R n ) , then j k ( α ) = j k ( β ) ⇐⇒ j k ( α ◦ f ) = j k ( β ◦ f ) . (cid:3) Jets of a flow.
First we introduce the following notation. For asmooth mapping G = ( G , . . . , G n ) : R n → R n denote ∇ G = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∂G ∂x ∂G ∂x · · · ∂G ∂x n ∂G ∂x ∂G ∂x · · · ∂G ∂x n · · · · · · · · · · · · ∂G n ∂x ∂G n ∂x · · · ∂G n ∂x n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) . Thus ∇ G is an ( n × n )-matrix whose rows are the gradients of thecorresponding coordinate functions of G .Now let F = ( F , . . . , F n ) be a smooth vector field defined on someneighbourhood V of O ∈ R n and F : V × R ⊃ dom ( F ) −→ R n be the local flow of F , so(5.5) ∂ F ∂t ( x, t ) = F ( F ( x, t )) and F ( x,
0) = x. Hence the Taylor expansion of F in t at x = O is given by(5.6) F ( x, t ) = x + v ( x ) t + v ( x ) t . . . + v n ( x ) t n n ! + · · · , where v i ( x ) = ∂ i F ∂t i ( x, t ) | t =0 . It follows from (5.5) that v = F . -JETS OF ORBIT PRESERVING DIFFEOMORPHISMS 14 Lemma.
For every i ≥ we have that v i +1 = ∇ v i · F = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∂v i ∂x ∂v i ∂x · · · ∂v i ∂x n ∂v i ∂x ∂v i ∂x · · · ∂v i ∂x n · · · · · · · · · · · · ∂v ni ∂x ∂v ni ∂x · · · ∂v ni ∂x n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) · (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) F F · · · F n (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) h∇ v i , F ih∇ v i , F i· · ·h∇ v ni , F i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) , where v ji is the j -th coordinate function of v i . Moreover, ∂ i F ∂t i ( x, t ) = v i ( F ( x, t )) . If j p − ( F ) = 0 , then (5.7) j i ( p − ( v i ) = 0 , ∀ i ≥ . Proof.
We will now calculate v . Let F j be the j -th coordinate functionof F . Then ∂ F j ∂t ( x, t ) = ∂∂t ∂ F j ∂t ( x, t ) (5.5) === ∂∂t F j ( F ( x, t )) (5.5) ==== n X k =1 ∂F j ∂x k ( F ( x, t )) · F k ( F ( x, t )) == (cid:16) n X j =1 ∂F j ∂x k · F k (cid:17) ◦ F ( x, t ) = h ▽ F j , F i ◦ F ( x, t ) . Therefore ∂ F ∂t ( x, t ) = ( ∇ F · F ) ◦ F ( x, t ) = v ◦ F ( x, t ) . Calculations for other v i are similar and are left to the reader. Proof of (5.7) . We have j · ( p − ( v ) = j p − ( F ) = 0. Suppose byinduction that j i ( p − ( v i ) = 0 for some i .Then j i ( p − − ( ∇ v i ) = 0. Since j p − ( F ) = 0, it follows from (5.3)that j i ( p − p − ( ∇ v i · F ) = j ( i +1)( p − ( v i +1 ) = 0. (cid:3) Initial non-zero jets of smooth shifts for non-flat vectorfields.
Suppose now that there exists p ≥ j p − ( F ) = 0 and P = j p ( F ) : R n → R n is a non-zero homogeneous map of degree p . For p = 1 we will write P ( x ) = L · x, where L is a certain non-zero n × n -matrix.5.7. Lemma.
Let α : R n → R be an l -small germ at O for some l ≥ , so j l ( α ) = ω is a homogeneous polynomial of degree l . Put F α ( x ) = F ( x, α ( x )) . Then (5.8) j p + l ( F α )( x ) = ( e L ω · x, p = 1 and l = 0 ,x + P ( x ) · ω ( x ) , otherwise . -JETS OF ORBIT PRESERVING DIFFEOMORPHISMS 15 Thus the ( p + l ) -jet of F α depends only on the l -jet of α . Moreover, (5.9) j p + l ( F − α ) = j p + l ( F − α ) . Proof. (5.8). Substituting α into (5.6) instead of t we get F α ( x ) = F ( x, α ( x )) = x + F ( x ) α ( x ) + · · · + v i ( x ) α ( x ) i i ! + · · · Though this series converges near z , the infinitesimal orders of its sum-mands at O do not necessarily increase when n → ∞ . Therefore inorder to find the initial non-zero jet of F α we should investigate eachof v i α i . In fact we will get only one exceptional case p = 1 and l = 0.First we calculate j ( F α ). Suppose that j ( F ) = L · x for somepossibly zero matrix L . Then j ( ∇ F ) = ∇ F ( O ) = L , j ( v ) = j ( ∇ F · F ) = j ( ∇ F ) · j ( F ) + j ( ∇ F ) · j ( F ) == L · L · x + j ( ∇ F ) · L · x, and by induction on i we obtain j ( v i ) = j (cid:0) ∇ v i − · F (cid:1) = L i · x. Therefore j ( v i α i ) = j ( v i ) · j ( α i ) + j ( v i ) · j ( α i ) == L i · x · α ( O ) + 0 · j ( α i ) = α ( O ) · L i · x, whence j ( F α )( x ) = x + ∞ X i =1 α ( O ) i i ! · L i · x = e L · α ( O ) · x. Thus if p = 1 and l = 0, i.e L = 0 and j ( α ) = α ( O ) = ω = 0, weobtain that j p + l ( F α )( x ) = j ( F α )( x ) = e L ω · x .Otherwise p + l ≥
2. We claim in this case j p + l ( v i · α i ) = 0 for i ≥ j p + l ( F α )( x ) = j p + l ( x + F ( x ) α ( x )) = x + P ( x ) · ω ( x ) . To calculate j p + l ( v i · α i ) notice that by (5.7) j i ( p − ( v i ) = 0 and byassumption j l − ( α ) = 0 as well. Then it follows from (5.3) that j il − ( α i ) = 0 and j i ( p − il ( v i · α i ) = j i ( p + l − ( v i · α i ) = 0 . It remains to note that p + l < i ( p + l −
1) if i ≥ p + l ≥
2. Hence j p + l ( v i α i ) = 0 for i ≥ Proof of (5.9). Recall that by (3.3) F − α = F − α ◦ F − α . We will showthat j l ( α ◦ F − α ) = j l ( α ). Then it will follow from (5.8) that j p + l ( F − α ) = j p + l ( F − α ◦ F − α ) (5.8) === j p + l ( F − α ) . Suppose that l = 0. Since F t ( O ) = O for all t ∈ R , we obtain that j ( α ◦ F − α ) = j ( α ) = α ( O ). -JETS OF ORBIT PRESERVING DIFFEOMORPHISMS 16 If l ≥
1, then it follows from (5.8) that j ( F α )( x ) = x , whence j l ( α ◦ F − α ) = j l ( α ). (cid:3) Corollary.
Let α, β ∈ ˆ F ( R n ) , h ∈ ˆ D ( R n ) , l, k = 0 , , . . . , ∞ . (1) The following conditions (A)-(C) are equivalent: (A) j l ( α ) = j l ( β ) , (B) j p + l ( F α ) = j p + l ( F β ) , (C) j p + l ( F h,α ) = j p + l ( F h,β ) . (2) The following conditions (D) and (E) are equivalent: (D) j k ( h ) = j k ( F α ) , (E) j k ( F h, − α ) = j k (id) .Proof. (1) (A) ⇔ (B). Notice that(5.10) F α ◦ F − β (3.3) === F α ◦ F − β ◦ F − β (3.3) === F α ◦ F − β − β ◦ F − β = F ( α − β ) ◦ F − β . Then the following statements are equivalent:(A) j l ( α ) = j l ( β ) , (c) j p + l ( F ( α − β ) ◦ F − β ) = j p + l (id) , (a) j l ( α − β ) = 0 , (d) j p + l ( F α ◦ F − β ) = j p + l (id) , (b) j l (( α − β ) ◦ F − β ) = 0 , (B) j p + l ( F α ) = j p + l ( F β ) . The equivalence of (A), (a), and (b) is trivial, (b) ⇔ (c) holds by (5.8),(c) ⇔ (d) by (5.10), and (d) ⇔ (B) by Lemma 5.3.(A) ⇔ (C) Recall that by (3.4) F h,α = F α ◦ h − ◦ h . Then the followingconditions are equivalent:(C) j p + l ( F h,α ) = j p + l ( F h,β ) , (f) j l ( α ◦ h − ) = j l ( β ◦ h − ) , (e) j p + l ( F α ◦ h − ) = j p + l ( F β ◦ h − ) , (A) j l ( α ) = j l ( β ) . The equivalence (C) ⇔ (e) holds by (3.4), (e) ⇔ (f) by the equivalence(B) ⇔ (A) which is already proved, and (f) ⇔ (A) by Lemma 5.3.(2) (D) ⇒ (E). Suppose that j k ( h ) = j k ( F α ). Then j k ( F − α ◦ h ) = j k (id) , (5.11) j k ( α ) = j k ( α ◦ F − α ◦ h ) . (5.12)Notice also that(5.13) F − α ◦ h (3.3) === F − α ◦ F − α ◦ h = F h, − α ◦ F − α ◦ h . Hence j k ( F h, − α ) (5.12) === j k ( F h, − α ◦ F − α ◦ h ) (5.13) === j k ( F − α ◦ h ) (5.11) === j k (id) . (E) ⇒ (D). It is easy to verify that(5.14) F α ◦ F − h, − α ◦ F h, − α = h. This identity simply means that F (cid:0) F ( h ( x ) , − α ( x )) , α ( x ) (cid:1) = h ( x ). Sup-pose that j k ( F h, − α ) = j k (id). Then(5.15) j k ( α ◦ F h, − α ) = j k ( α ) , -JETS OF ORBIT PRESERVING DIFFEOMORPHISMS 17 whence j k ( h ) (5.14) & ( E ) ======= j k ( F α ◦ F − h, − α ) (5.15) & (5.8) ======= j k ( F α ) . Corollary 5.8 is proved. (cid:3)
Proof of Theorem 5.1.
Let F be a vector field defined on someneighbourhood of the origin in R n and G be a subgroup of ˆ D ( R n )satisfying (A1)-(A3). We have to show that j ∞ (cid:0) ˆ Sh ( F ) (cid:1) = j ∞ ( G ). Dueto (A1) it remains to verify that j ∞ (cid:0) ˆ Sh ( F ) (cid:1) ⊃ j ∞ ( G ).Let h ∈ G . We will find a germ of a smooth function α ∈ ˆ F ( R n ) suchthat j ∞ ( F h, − α ) = j ∞ (id). Then it will follow from (2) of Corollary 5.8that j ∞ ( F α ) = j ∞ ( h ), i.e. j ∞ ( h ) ∈ j ∞ ( ˆ Sh ( F )).Since G is a group, it follows from (A1) and (3.4) that(5.16) F h,α = F α ◦ h − ◦ h ∈ G , ∀ α ∈ ˆ F ( R n ) . Put h = h . Then by (A2) there exists ω ∈ R such that j p ( h ) = j p ( F ω ). Denote h ( x ) = F h, − ω ( x ) = F ( h ( x ) , − ω ) = F − ω ◦ h ( x ) . Then h ∈ G and by (2) of Corollary 5.8 j p ( h ) = j p (id R n ) . Therefore by (A3) there exists a homogeneous polynomial ω of de-gree 1 such that j p +1 ( h ) = j p +1 ( F ω ), where F ω ( x ) = F ( x, ω ( x )).Denote h ( x ) = F h , − ω ( x ) = F ( h ( x ) , − ω ( x )) == F ( F ( h ( x ) , − ω ) , − ω ( x )) == F ( h ( x ) , − ω − ω ( x )) = F h, − ω − ω ( x ) . Then by (5.16) h ∈ G and by (2) of Corollary 5.8 j p +1 ( h ) = j p +1 (id).Therefore we can again apply (A3) to h and so on. Using induc-tion we will construct a sequence of homogeneous polynomials { ω l } ∞ l =0 ,(deg ω l = l ), and a sequence { h l } in G such that for every l ≥
0, wehave that j p + l − ( h l ) = j p + l − (id) , j p + l ( h l ) = j p + l ( F ω l ) , (5.17) h l +1 ( x ) = F h l , − ω l ( x ) = F (cid:0) h ( x ) , − l X i =0 ω i ( x ) (cid:1) . (5.18)Put τ = ∞ P l =0 ω l ( x ). Then by a well-known theorem of E. Borel, seealso §
6, there exists a smooth function α : R n → R whose ∞ -jet at O coincides with τ .We claim that j ∞ ( F h, − α ) = j ∞ (id). Evidently, it suffices to showthat j p + l ( F h, − α ) = j p + l (id) for arbitrary large l . -JETS OF ORBIT PRESERVING DIFFEOMORPHISMS 18 For l ≥ α l = l P i =0 ω i and α >l = α − α l . Then F h, − α ( x ) = F ( h ( x ) , − α l ( x ) − α >l ( x )) == F (cid:0) F ( h ( x ) , − α l ( x )) , − α >l ( x ) (cid:1) == F ( h l +1 ( x ) , − α >l ( x )) = F h l +1 , − α >l . Notice that by (5.17) j p + l ( h l +1 ) = j p + l (id). Moreover, since j l ( α >l ) =0, it follows from Lemma 5.7 that j p + l ( F α >l ) = j p + l (id) for l ≥ j p + l ( h l +1 ) = j p + l ( F α >l ) = j p + l (id) . Then by (2) of Corollary 5.8 j p + l ( F h, − α ) = j p + l ( F h l +1 , − α >l ) = j p + l (id). (cid:3) Borel’s theorem
In this section we present a variant of a well-known theorem ofE. Borel. It will be used in the next section for the construction of j ∞ -sections of a shift map.Let V be an open subset of R n and f = ( f , . . . , f m ) : V → R m be a smooth mapping. For every compact K ⊂ V and r ≥ r -norm of f on K by k f k rK = m P j =1 P | i |≤ r sup x ∈ K | D i f j ( x ) | , where i =( i , . . . , i n ), | i | = i + · · · + i n , and D i = ∂ | i | ∂x i ··· ∂x inn . For a fixed r the norms k f k rK , where K runs over all compact subsets of V , define the weakWhitney C rW topology on C ∞ ( V, R m ). A C ∞ W -topology on C ∞ ( V, R m ) isgenerated by C rW -topologies for all finite r ≥ i ≥ P i the space of real homogeneous polyno-mials in n variables x , . . . , x n of degree i . Associating to every ω ∈ P i its coefficients we can identify P i with R C i − n + i − , where C i − n + i − = ( n + i − n ! ( i − .On the other hand P i ⊂ C ∞ ( R n , R ). It is easy to show that every of C rW -topologies on P i induced from C ∞ ( R n , R ) coincides with the Eu-clidean one.Let A, B be smooth manifolds and
X ⊂ C ∞ ( A, B ) be a subset.We will say that a map λ : X → P i is C rW -continuous provided it iscontinuous from C rW -topology of X to the Euclidean topology of P i .6.1. Theorem.
Let
A, B be smooth manifolds,
X ⊂ C ∞ ( A, B ) a subset,and V ⊂ R n an open neighbourhood of O ∈ R n . Suppose that for every i ≥ we are given a number s i ≥ and a C s i W -continuous map λ i : X → P i . Thus we can define the following mapping λ : X → R [[ x , . . . , x n ]] , λ ( α ) = ∞ X i =0 λ i ( α ) . -JETS OF ORBIT PRESERVING DIFFEOMORPHISMS 19 Then there exists a C ∞ , ∞ W,W -continuous map
Λ :
X → C ∞ ( V, R ) suchthat ∞ -jet of Λ( α ) at O coincides with λ ( α ) for every α ∈ X .Moreover if every λ i preserves smoothness (in the sense of Defini-tion 1.3), then so does Λ .Proof. The proof uses a theorem of E. Borel claiming that there existsa map B : R [[ x , . . . , x n ]] → C ∞ ( V, R ) such that for every formal series τ ∈ R [[ x , . . . , x n ]] the Taylor series of the function B ( τ ) at O coincideswith τ , see e.g. [GG].Such a map B can be assumed to have the following properties (a)-(c)below. Let τ = ∞ P i =0 ω i be a formal series, where ω i ∈ P i . Then(a) for every i ≥ B| P i : P i → C ∞ ( V, R ) is con-tinuous from the Euclidean topology of P i to C ∞ W -topology of C ∞ ( V, R );(b) B ( ∞ P i =0 ω i ) = ∞ P i =0 B ( ω i );(c) for every compact K ⊂ V , ε >
0, and an integer number r ≥ s = s ( K, ε, r ) ≥ τ suchthat (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) B ∞ X i = s +1 ω i !(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) rK < ε. Indeed, for every i ≥ ρ i : R n → R be a smooth functionsupported in a sufficiently small neighbourhood V i ⊂ V of O and equalto 1 in a smaller neighbourhood of 0. Put B ( ω ) = ωρ i for all ω ∈ P i .Then (a) holds true.Property (b) is just the definition of B on all of R [[ x , . . . , x n ]].Finally, (c) can be reached if the supports of ρ i decrease sufficientlyfast when i tends to ∞ , see for details e.g. [GG].Assuming that B has properties (a)-(c) we will now show that thefollowing map(6.1) Λ = B ◦ λ : X λ −−→ R [[ x , . . . , x n ]] B −−→ C ∞ ( V, R ) , Λ( q )( x ) = ∞ P i =0 ρ i ( x ) λ i ( q )( x ) , q ∈ X , x ∈ V, satisfies the statement of our theorem.Indeed, for each α ∈ X the Taylor series of Λ( α ) = B ◦ λ ( α ) at O coincides with λ ( α ).Let us verify C ∞ , ∞ W,W -continuity of Λ at α . It suffices to prove thatfor every compact subset K ⊂ V , ε >
0, and r ≥ c = c ( K, ε, r ) ≥ α , and a C cW -neighbourhood U α of α in X such that k Λ( α ) − Λ( β ) k rK < ε for all β ∈ U α .For every s ≥ s , Λ >s : X → C ∞ ( V, R ) , -JETS OF ORBIT PRESERVING DIFFEOMORPHISMS 20 Λ s ( α ) = B ◦ s ⊕ i =0 λ i ( α ) , Λ >s = Λ − Λ s . Then(6.2) k Λ( α ) − Λ( β ) k rK ≤ k Λ s ( α ) − Λ s ( β ) k rK + k Λ >s ( α ) k rK + k Λ >s ( β ) k rK . For every s ≥ c ( s ) = max { s i } si =0 . Then it follows from (a)and assumptions about continuity of λ i that Λ s is C c ( s ) , ∞ W,W -continuous.Hence Λ s is C c ( s ) ,rW,W -continuous for every r ≥ ε > r ≥
0, and a compact subset K ⊂ V . Then by (c) thereexists s > α and β such that each of thelast two terms in (6.2) is less that ε . Moreover it follows from C c ( s ) ,rW,W -continuity of Λ s that there exists a C c ( s ) W -neighbourhood U α of α in X such that for every β ∈ U α the first term on the right side of (6.2) isless than ε as well. Then k Λ( α ) − Λ( β ) k rK < ε .Suppose now that every λ i preserves smoothness. Let q : A × R k → B be a C ∞ map such that q t ∈ X for every t ∈ R k . Then for every i ≥ λ i ( q ) : V × R k → R , λ i ( q )( x, t ) = λ i ( q t )( x )is also C ∞ . Therefore it follows from (6.1) that the following mapΛ( q ) : V × R k → R defined byΛ( q )( x, t ) = Λ( q t )( x ) = ∞ X i =0 ρ i ( x ) λ i ( q t )( x )is C ∞ as well. Hence Λ preserves smoothness. (cid:3) j ∞ -sections of the shift map Notation.
Let V be an open neighbourhood of O in R n , F be avector field on V such that F ( O ) = 0, F : V × R ⊃ dom ( F ) → R n bethe local flow of F , func ( F , V ) be the subset of C ∞ ( V, R ) consisting offunctions α whose graph is contained in dom ( F ), and ϕ : func ( F , V ) → C ∞ ( V, R n ) , ϕ ( α )( z ) = F α ( x ) = F ( z, α ( z ))be the corresponding shift map of F . Denote its image by Sh ( F , V ).For every k ≥ h ∈ C ∞ ( V, R n ) let j k ( h ) be the k -jet of h at O .Let also J ( F , V ) ⊂ C ∞ ( V, R n ) be the subset consisting of maps h for which there exists a smooth function α h ∈ C ∞ ( V, R ) such that j ∞ ( h ) = j ∞ ( F α h ), c.f. (5.1).In this section we will show how to choose α h so that the correspon-dence h α h becomes a continuous and preserving smoothness map.Such a map will be called a j ∞ -section of ϕ V . -JETS OF ORBIT PRESERVING DIFFEOMORPHISMS 21 Definition.
Let
X ⊂ J ( F , V ) be a subset. We will say that amapping Λ :
X → C ∞ ( V, R ) is a j ∞ -section of ϕ V on X provided Λ is C ∞ , ∞ W,W -continuous, preserves smoothness, and (7.1) j ∞ ( h ) = j ∞ ( ϕ V ◦ Λ( h )) , ∀ h ∈ X . If X = J ( F , V ) then Λ will be called a global j ∞ -section of ϕ V . Recall that we use the following notation ϕ V ◦ Λ( h )( x ) = F Λ( h ) ( x ) = F ( x, Λ( h )( x )) , F h, − Λ( h ) ( x ) = F ( h ( x ) , − Λ( h )( x )) . Then by (2) of Corollary 5.8 relation (7.1) is equivalent to each of thefollowing conditions(7.2) j ∞ ( h ) = j ∞ ( F Λ( h ) ) ⇔ j ∞ ( F h, − Λ( h ) ) = j ∞ (id) . Our main result proves existence of j ∞ -sections on certain subspacesof J ( F , V ), see Theorem 7.6. Before formulating this theorem, let usshow how j ∞ -sections can be used for constructing real sections of ϕ V .7.3. Applications of j ∞ -sections. Denote by E ∞ ( F , V ) ⊂ E ( F , V )the subset consisting of maps h such that j ∞ ( h ) = j ∞ (id). Since j ∞ ( h ) = j ∞ ( F ), we obtain that E ∞ ( F , V ) ⊂ E ( F , V ) ∩ J ( F , V ) . Let
X ⊂ E ( F , V ) ∩ J ( F , V ) be a subset. Thus every h ∈ X is anorbit preserving map V → R n being a diffeomorphism at every singularpoint z ∈ Σ F and such that j ∞ ( h ) = j ∞ ( F α h ) for some α h ∈ C ∞ ( V, R ).Suppose that there exists a j ∞ -section Λ : X → C ∞ ( V, R ) of ϕ V .Then by (7.2) F h, − Λ( h ) ∈ E ∞ ( F , V ) for every h ∈ X , whence the fol-lowing map is well defined:(7.3) H : X → E ∞ ( F , V ) , H ( h ) = F h, − Λ( h ) . Lemma. c.f. [M4, Pr. 3.4]
Suppose also that there exists a C ∞ , ∞ W,W -continuous and preserving smoothness section
Ψ : E ∞ ( F , V ) → C ∞ ( V, R ) of ϕ V , i.e. h ( x ) = F ( x, Ψ( h )( x )) for all h ∈ E ∞ ( F , V ) . Then thefollowing map σ : X → C ∞ ( V, R ) defined by σ ( h ) = Λ( h ) + Ψ ◦ H ( h ) = Λ( h ) + Ψ( F h, − Λ( h ) ) . is a section of ϕ V defined on all of X .Proof. This statement was actually established in [M4, Pr. 3.4]. Forthe convenience of the reader we recall the proof. It suffices to show -JETS OF ORBIT PRESERVING DIFFEOMORPHISMS 22 that F ( h ( x ) , − σ ( h )( x )) ≡ x for all x ∈ V and h ∈ X but this easilyfollows from definitions: F ( h ( x ) , − σ ( h )( x )) = F (cid:0) h ( x ) , − Λ( h )( x ) − Ψ ◦ H ( h )( x ) (cid:1) = F (cid:0) F ( h ( x ) , − Λ( h )( x )) , − Ψ ◦ H ( h )( x ) (cid:1) = F (cid:0) H ( h )( x ) , − Ψ ◦ H ( h )( x ) (cid:1) = x. (cid:3) Thus existence of j ∞ -sections reduces the problem of resolving (1.1)to the case when j ∞ ( h ) = j ∞ (id).7.5. Main result.
Suppose that F is not flat at O , i.e. there exists p ≥ j p − ( F ) = 0 and P = j p ( F ) is a non-zero homogeneousvector field. For p = 1 we will write P ( x ) = L · x , where L is a certainnon-zero ( n × n )-matrix. In this case we have the exponential mapexp L : R → GL( R , n ) , exp L ( t ) = e L t . Denote its image by E L = { e L t } t ∈ R . Then the following three cases of E L will be separated:(G1) E L ≈ R and is a closed subgroup of GL( R , n );(G2) E L ≈ SO (2);(G3) E L ≈ R and is a non-closed subset of GL( R , n ).For every k ≥ J k ( F , V ) = (cid:8) h ∈ J ( F , V ) | j k − ( h ) = j k − (id) (cid:9) , k ≥ . Since h ( O ) = O , i.e. j ( h ) = j (id), for all h ∈ J ( F , V ), we seethat J ( F , V ) = J ( F , V ). Moreover, it follows from Lemma 5.7 that j p − ( h ) = j p − (id) for p ≥
2. Hence for arbitrary p ≥ J ( F , V ) = J ( F , V ) = · · · = J p ( F , V ) ⊃ J p +1 ( F , V ) ⊃ · · · It will also be convenient to define local variants of the above con-structions. Recall that ˆ J ( F ) is the subgroup of ˆ D ( R n ) consisting ofspace of germs at O of maps from ˆ J ( F ), see (5.1). Therefore we defineˆ J k ( F ) to be the subspace of ˆ J ( F ) consisting of germs at O of mapsfrom J k ( F , V ). Then againˆ J ( F ) = ˆ J ( F ) = · · · = ˆ J p ( F ) ⊃ ˆ J p +1 ( F ) ⊃ · · · Theorem. (1)
There exists a j ∞ -section Λ of ϕ V on J ( F , V ) .Hence if p ≥ , then J ( F , V ) = J ( F , V ) , and thus Λ is defined on allof J ( F , V ) . (2) If p = 1 and E L satisfies (G1) , then there exists a global j ∞ -section Λ of ϕ V as well. (3) If p = 1 and E L satisfies (G2) , then for every g ∈ J ( F , V ) theshift map ϕ V has a j ∞ -section defined on some C W -neighbourhood of g in J ( F , V ) . -JETS OF ORBIT PRESERVING DIFFEOMORPHISMS 23 Remark. If p = 1 but E L satisfies (G3) then it seems that ϕ V hasno even local ( C ∞ , ∞ W,W -continuous) j ∞ -sections on J ( F , V ), though by(1) it has a j ∞ -section on J ( F , V ).The proof will be given in § Proposition.
Let p = 1 . In the case (G1) of E L there exists a C W -continuous map ∆ : J ( F , V ) → R such that for every h ∈ J ( F , V ) the germ at O of the mapping F h, − ∆ ( h ) ( x ) = F ( h ( x ) , − ∆ ( h )) belongs to ˆ J ( F ) .Suppose that E L satisfies (G2) . Then for every g ∈ J ( F , V ) thereexists a C W -neighbourhood N g in J ( F , V ) and a C W -continuous map-ping ∆ : N g → R such that for every h ∈ N g the germ at O of themapping F h, − ∆ ( h ) belongs to ˆ J ( F ) .In both cases ∆ preserves smoothness.Proof. Let j : J ( F , V ) → GL( R , n ) be the map associating to every h ∈ J ( F , V ) its Jacobi matrix J ( h, O ) at O . Then it follows fromLemma 5.7 that the image of j coincides with E L . Thus we have twomaps: J ( F , V ) j −−−→ E L exp L ←−−− R . Notice that in the cases (G1) and (G2) E L is a closed subgroup ofGL( R , n ). Moreover, in the case (G1) exp L is an embedding, so we canput ∆ : J ( F , V ) → R , ∆ = exp − L ◦ j . In the case (G2) exp L is a smooth Z -covering map, whence for every g ∈ J ( F , V ) there exists only a C W -neighbourhood N g in J ( F , V )such that the map ∆ = exp − L ◦ j : N g → R is well-defined.It is easy to see that in both cases ∆ has the desired properties. (cid:3) Proposition.
Let p + l ≥ . Then there exists a C p + lW -continuousand preserving smoothness map ∆ l : J p + l ( F , V ) → P l such that forevery h ∈ J p + l ( F , V ) the germ at O of the map F h, − ∆ l ( h ) ( x ) = F ( h ( x ) , − ∆ l ( h )( x )) belongs to ˆ J p + l +1 ( F ) .Proof. Let h ∈ J p + l ( F , V ). Since p + l ≥
2, we have that either p ≥ p = 1 but l ≥
1. It follows from Lemma 5.7 that in both casesthere exists a unique homogeneous polynomial ω l of degree l such that j p + l ( h )( x ) = x + P ( x ) · ω l ( x ). The correspondence h ω l is a well-defined map ∆ l : J p + l ( F , V ) → P l and by (2) of Corollary 5.8 the germat O of the mapping F h, − ∆ l ( h ) belongs to ˆ J p + l +1 ( F ). -JETS OF ORBIT PRESERVING DIFFEOMORPHISMS 24 Let us verify the continuity of ∆ l . Consider the following map j p + l : C ∞ ( V, R n ) → J p + l ( V, R n ) , h j p + l ( h ) , associating to each h ∈ C ∞ ( V, R n ) its ( p + l )-jet j p + l ( h ) at O . Evidently, j p + l is a C p + lW -continuous and preserving smoothness map Moreover,the image j p + l ( E p + l ( F , V, O )) is contained in the following set A l = { x + P ( x ) · ω ( x ) | ω ∈ P l } ⊂ J p + l ( V, R n ) . Further, it follows fromsmoothness of the Euclid algorithm of division of polynomials that thecorrespondence x + P ( x ) · ω ( x ) ω is a well-defined smooth map D : A l → P l . Therefore ∆ l = D ◦ j p + l is C p + lW -continuous and preservessmoothness. (cid:3) Proof of Theorem 7.6.
Let X be one of the spaces J ( F , V ), J ( F , V ), or N g with respect to the cases (1), (2), or (3) of our theo-rem, where N g is a C W -neighbourhood of g ∈ J ( F , V ) constructed inPropositions 7.8.Then similarly to the proof of Theorem 5.1 for every h ∈ X we canconstruct a sequence of homogeneous polynomials { ω i } ∞ i =0 , (deg ω i = i )via the following rule: ω = ∆ ( h ) , ω = ∆ ( F h, − ω ) , . . . , ω l = ∆ l (cid:0) F h, − P l − i =0 ω i (cid:1) . It follows from Propositions 7.8, 7.9, and formulae for ω i that for every i ≥ h ω i is a C p + lW -continuous and preservingsmoothness map λ i : X → P i , λ i ( h ) = ω i . Put λ ( h ) = P ∞ i =0 λ i ( h ). Then j ∞ ( F h, − λ ( h ) ) = j ∞ (id) as well as inTheorem 5.1.Now it follows from Borel’s Theorem 6.1 applied to X that thereexists a C ∞ , ∞ W,W -continuous and preserving smoothness mapΛ :
X → C ∞ ( V, R )such that j ∞ (Λ( h )) = λ ( h ) for h ∈ X . Hence j ∞ ( F h, − Λ( h ) ) = j ∞ (id).Then by (2) of Corollary 5.8 j ∞ ( h ) = j ∞ ( F Λ( h ) ) for all h ∈ X . (cid:3) Property ( ∗ )In this section we describe a class of vector fields F on R n for which j ∞ ˆ Sh ( F ) = j ∞ ˆ D id ( F ) , see Theorem 8.5. This class is rather spe-cial since it consists of completely integrable (i.e. having n − -JETS OF ORBIT PRESERVING DIFFEOMORPHISMS 25 Cross product.
Let ( x , . . . , x n ) be coordinates in R n . For everysmooth function f : R n → R define the following “gradient” vectorfield with respect to these coordinates: ▽ x f = ( f ′ x , . . . , f ′ x n ) . If f , . . . , f n − : R n → R is an ( n − cross-product vector field:(8.1) H = [ ∇ x f , . . . , ∇ x f n − ] = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂f ∂x ∂f ∂x · · · ∂f ∂x n · · · · · · · · · · · · ∂f n − ∂x ∂f n − ∂x · · · ∂f n − ∂x n ∂∂x ∂∂x · · · ∂∂x n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) being an analogue of the cross-product [ a, b ] of two vectors a, b in R .Notice that the first n − n × n )-matrix consist of smoothfunctions, while the n -th row is the standard basis h ∂∂x , . . . , ∂∂x n i of thespace of vector fields on R n . Therefore the corresponding determinantis a well-defined vector field.Equivalently, let us fix the standard Euclidean metric on R n . Thenwe have a Hodge isomorphism ∗ : Λ n − ( R n ) → Λ ( R n ) between thespaces of differential forms and the isomorphism φ : Λ ( R n ) → Γ( R n )between the space of 1-forms and the space of vector fields on R n . Thenit is easy to see that[ ∇ x f , . . . , ∇ x f n − ] = φ ◦ ∗ ( df ∧ · · · ∧ df n − ) . It is easy to see that every f i is constant along orbits of H , i.e. f i isan integral for H . Indeed, substituting ▽ x f i in (8.1) instead of last row,we will get df i ( H ) = 0. Thus H is completely integrable in the sensethat it has n − ▽ x f , . . . , ▽ x f n − are linearly dependent.8.2. Example.
Let f : R → R be a smooth function. Then H = [ ▽ x f ] = (cid:12)(cid:12)(cid:12)(cid:12) f ′ x f ′ y∂∂x ∂∂y (cid:12)(cid:12)(cid:12)(cid:12) = − f ′ y ∂∂x + f ′ x ∂∂y is the corresponding Hamiltonian vector field of f .8.3. Lemma.
Let x = ( x , . . . , x n ) and y = ( y , . . . , y n ) be two lo-cal coordinate systems at O related by a germ of diffeomorphism h =( h , . . . , h n ) of ( R n , O ) , i.e. x = h ( y ) . Let also H x and H y be vectorfields defined by (8.1) in the coordinates ( x i ) and ( y i ) respectively, and h ∗ H x be the vector field induced by h , i.e. this is H x in the coordinates ( y i ) . Then H y = | J ( h ) | · h ∗ H x . -JETS OF ORBIT PRESERVING DIFFEOMORPHISMS 26 Proof.
Notice that if H x ( x ) = n P i =1 T i ( x ) ∂∂x i , then in the coordinates ( y i )we can also write H x ( y ) = n X i =1 T i ( y ) ∂∂x i , where ∂∂x i = n P j =1 ∂y i ∂x j ∂∂y i . Hence(8.2) H x ( y ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂f ∂x ( y ) · · · ∂f ∂x n ( y ) · · · · · · · · · ∂f n − ∂x ( y ) · · · ∂f n − ∂x n ( y ) ∂∂x · · · ∂∂x n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) On the other hand, H y ( y ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂f ∂y ( y ) · · · ∂f ∂y n ( y ) · · · · · · · · · ∂f n − ∂y ( y ) · · · ∂f n − ∂y n ( y ) ∂∂y · · · ∂∂y n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n P i =1 ∂f ∂x i ( y ) · ∂x i ∂y · · · n P i =1 ∂f ∂x i ( y ) · ∂x i ∂y n · · · · · · · · · n P i =1 ∂f n − ∂x i ( y ) · ∂x i ∂y · · · n P i =1 ∂f n − ∂x i ( y ) · ∂x i ∂y n n P i =1 ∂∂x i · ∂x i ∂y · · · n P i =1 ∂∂x i · ∂x i ∂y n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂f ∂x ( y ) · · · ∂f ∂x n ( y ) · · · · · · · · · ∂f n − ∂x ( y ) · · · ∂f n − ∂x n ( y ) ∂∂x · · · ∂∂x n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) · (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂x ∂y · · · ∂x ∂y n · · · · · · · · · ∂x n ∂y · · · ∂x n ∂y n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (8.2) === H x ( y ) · | J ( h ) | . Lemma is proved. (cid:3)
Definition.
Let F be a vector field defined on some neighbourhood V of O ∈ R n . Say that F has property ( ∗ ) at O if there exist p ∈ N and n smooth non-flat at O functions η, f , . . . , f n − : V → R such that (a) j p − ( F ) = 0 , -JETS OF ORBIT PRESERVING DIFFEOMORPHISMS 27 (b) P = j p ( F ) is a non-zero homogeneous vector field being non-divisible by homogeneous polynomials , i.e. P can not berepresented as a product P = ωQ , where ω is a homogeneouspolynomial of degree deg ω ≥ and Q is a homogeneous vectorfield of degree deg Q ≥ , (c) vector fields ▽ x f , . . . , ▽ x f n − are linearly independent on aneverywhere dense subset of V and (8.3) η · F = [ ▽ x f , . . . , ▽ x f n − ] . Let us explain this definition.1) Since 1 ≤ p < ∞ , we have that F ( O ) = 0 and F is not flat at O .2) We allow η and therefore η · F vanish at some points of V whichcan be even non-singular for F . But due to (c) the set of zeros of η · F and therefore η − (0) are nowhere dense in V .3) Let k = ord( η, O ), p i = ord( f i , O ) < ∞ , ( i = 1 , . . . , n − γ = j k ( η ) , Γ i = j p i ( f i ) , ( i = 1 , . . . , n − k, p , . . . , p n − re-spectively. Then R = [ ▽ x Γ , . . . , ▽ x Γ n − ] is a homogeneous vectorfield of degree n − P i =1 ( p i −
1) and γ · P = R . Since P is non-divisibleby homogeneous polynomials, it follows that γ is the greatest commondivisor of coordinate functions of R in the ring R [ x , . . . , x n ].8.5. Theorem. If F has property ( ∗ ) then j ∞ ˆ Sh ( F ) = j ∞ ˆ D id ( F ) . Moreover, for every neighbourhood V of O and every g ∈ E id ( F , V ) there exist a C W -neighbourhood N g in E id ( F , V ) and a j ∞ -section ofthe shift map ϕ V on N g . Thus due to Lemma 7.4 in order to completely resolve (1.1) we haveto construct a section of ϕ V on E ∞ ( F , V ). This was done in [M4] forthe case of homogeneous polynomial vector fields on R satisfying ( ∗ ),see also §
11 for more general result.The proof of Theorem 8.5 will be given in §§
9, 10. The followinglemma presents a class of examples of vector fields with property ( ∗ ).8.6. Lemma.
Let f , . . . , f n − : R n → R be homogeneous polynomi-als such that ▽ x f , . . . , ▽ x f n − are linearly independent on everywheredense subset of R n , and let η be the greatest common divisor of the co-ordinate functions of H = [ ▽ x f , . . . , ▽ x f n − ] . Then the homogeneousvector field F = H/η has property ( ∗ ) . (cid:3) Example.
Let n = 2, f ( x, y ) = x y , and H = [ ▽ f ] = ( − f ′ y , f ′ x ) = ( − x y , x y ) = x y |{z} η ( − x, y ) | {z } F = ηF. Then F is non-divisible. Notice also that the singular set of H consistsof x - and y -axes while the singular set of F is the origin only. -JETS OF ORBIT PRESERVING DIFFEOMORPHISMS 28 Lemma.
Property ( ∗ ) does not depend on a particular choice oflocal coordinates at O .Proof. Suppose that in coordinates x = ( x , . . . , x n ) at O conditions(a)-(c) of Definition 8.4 are satisfied. Let y = ( y , . . . , y n ) be anothercoordinates at O related to ( x , . . . , x n ) by a germ of a diffeomorphism h = ( h , . . . , h n ) of ( R n , O ), i.e. x = h ( y ). We have to show thatconditions (a)-(c) of Definition 8.4 hold in the coordinates ( y , . . . , y n )for the induced vector field h ∗ F = T h − ◦ F ◦ h .Let A = J ( h, O ) be the Jacobi matrix of h at O . Then it easilyfollows from condition (a) for F that j p − ( h ∗ F ) = 0 , j p ( h ∗ F )( y ) = A − P ( Ay ) . The latter identity implies that the initial non-zero jet of h ∗ F is non-divisible by homogeneous polynomials iff so is P . This proves (a) and(b) for h ∗ F .To establish (c) apply h to both parts of (8.3). Then h ∗ ( η · F ) = η ◦ h · h ∗ F,h ∗ [ ▽ x f , . . . , ▽ x f n − ] Lemma 8.3 ======= 1 | J ( h ) | · [ ▽ y f , . . . , ▽ y f n − ] . Denote η ′ = η ◦ h · | J ( h ) | . Then η ′ is smooth and η ′ · h ∗ F = [ ▽ y f , . . . , ▽ y f n − ] . This proves (c). (cid:3) Stabilizers of functions and polynomials
In this section we present some statements which will be used in theproof of Theorem 8.5.9.1.
Lemma.
Let f ∈ ˆ F ( R n ) , h ∈ ˆ E ( R n ) , and δ = f ◦ h − f . Sup-pose that j p − ( f ) = 0 and j k − ( h ) = j k − (id) for some p, k ≥ . Inparticular Γ = j p ( f ) is a homogeneous polynomial of degree p .If k = 1 and j ( h )( x ) = Ax for some ( n × n ) -matrix, then j p ( δ )( x ) = Γ( A · x ) − Γ( x ) . If k ≥ and j k ( h )( x ) = x + v ( x ) for some homogeneous map v ofdegree k , then j p − k ( δ ) = h ▽ Γ , v i . The proof of this lemma is direct and we leave it for the reader.9.2.
Corollary.
Suppose that h preserves f , i.e. f ◦ h = f , and thus δ ≡ . If k = 1 , then Γ( A · x ) = Γ( x ) . If k ≥ , then h ▽ Γ , v i = 0 . -JETS OF ORBIT PRESERVING DIFFEOMORPHISMS 29 Stabilizers of polynomials.
Consider the right action of thegroup GL( R , n ) on the space of polynomials R [ x , . . . , x n ] by:(9.1) Φ : R [ x , . . . , x n ] × GL( R , n ) → R [ x , . . . , x n ]Φ(Γ , A ) = Γ ◦ A, i.e. Φ(Γ , A )( x ) = Γ( A · x ) , where (Γ , A ) ∈ R [ x , . . . , x n ] × GL( R , n ). For Γ ∈ R [ x , . . . , x n ] let S (Γ) = { A ∈ GL( R , n ) | Γ( A · x ) = Γ( x ) } be its stabilizer with respect to Φ. Then S (Γ) is a closed (and thereforea Lie) subgroup of GL( R , n ).9.4. Lemma.
For every Γ ∈ R [ x , . . . , x n ] the tangent space T E S (Γ) tothe stabilizer S (Γ) of Γ at E consists of matrices V ∈ M ( R , n ) suchthat h ▽ Γ( x ) , V x i = 0 for all x ∈ R n .Proof. Let V ∈ M ( R , n ) and A : R → GL( R , n ) be the followinghomomorphism A ( t ) = e V t . Evidently, A (0) = E and A ′ t (0) = V .Notice that ∂∂t Γ( A ( t ) x ) = h ▽ Γ( A ( t ) x ) , A ′ t ( t ) x i = h ▽ Γ( e V t x ) , V e V t x i . Then the following statements are equivalent:(i) V ∈ T E S (Γ);(ii) A ( t ) ∈ S (Γ), i.e. Γ( A ( t ) x ) = Γ( x ), for all t ∈ R and x ∈ R n ;(iii) ∂∂t Γ( A ( t ) x ) = h ▽ Γ( e V t x ) , V e V t x i = 0, for all x ∈ R n ;(iv) ∂∂t Γ( A ( t ) x ) | t =0 = h ▽ Γ( x ) , V x i = 0 for all x ∈ R n .The implications (i) ⇔ (ii) ⇔ (iii) ⇒ (iv) are evident and (iv) ⇒ (iii) can beobtained by substituting e V t x instead of x in (iv). It remains to notethat our lemma claims that (i) ⇔ (iv). (cid:3) Let Γ , . . . , Γ n − ∈ R [ x , . . . , x n ] and(9.2) S = n − ∩ i =1 S (Γ i )be the intersection of their stabilizers. Then S is a closed Lie subgroupof GL( R , n ). Denote by T E S the tangent space of S at the unity matrix E , and let S E be the unity component of S .9.5. Lemma.
Let H = [ ▽ Γ , . . . , ▽ Γ n − ] be the vector field on R n de-fined by (8.4) , η be the greatest common divisor of coordinate functionsof H , and P = H/η . Suppose that P . Then P is non-divisible bypolynomials, i.e. if P ( x ) = ω ( x ) U ( x ) , where ω is a polynomial and U is a polynomial vector field, then either deg ω = 0 , or deg U = 0 . (i) If deg P = 1 , i.e. P ( x ) = Lx for some non-zero matrix L ∈ M ( R , n ) , then T E S = { L t } t ∈ R and S E = { e L t } t ∈ R . (ii) If deg P ≥ then S E = { E } . -JETS OF ORBIT PRESERVING DIFFEOMORPHISMS 30 Proof.
Notice that T E S = n − ∩ i =1 T E S (Γ i ). Let U ∈ T E S . Then byLemma 9.4 h ▽ Γ i ( x ) , U x i = 0 for every i = 1 , . . . , n − . Therefore
U x is parallel to the cross product H ( x ) of gradients ▽ Γ i and there-fore to P ( x ) at every x ∈ R n . If U = 0, then there exists a non-zeropolynomial ω such that(9.3) P ( x ) = ω ( x ) · U x
Since P is non-divisible, this identity is possible only if ω is a constantand in this case deg P = 1.Hence if deg P ≥
2, then U is always zero, whence T E S = { } , and S E = { E } . This proves (ii).(i) Suppose that deg P = 1. Then it follows from Lemma 9.4 that { Lt } t ∈ R ⊂ T E S . On the other hand, as noted above for every U ∈ T E S there exists ω ∈ R such that (9.3) holds true, whence L = ωU .Therefore U ∈ { Lt } t ∈ R , and thus { Lt } t ∈ R = T E S . (cid:3) Proof of Theorem 8.5
Suppose that F has property ( ∗ ) at O . Thus η · F = [ ▽ x f , . . . , ▽ x f n − ] , where η, f , . . . , f n − : V → R are germs of smooth functions satisfyingassumptions of Definition 8.4. We have to show that(10.1) j ∞ ˆ Sh ( F ) = j ∞ ˆ D id ( F ) and for every open neighbourhood V of O and g ∈ E id ( F , V ) constructa local j ∞ -section of ϕ V defined on some C W -neighbourhood N g of g in E id ( F , V ) . Proof of (10.1) . Notice that G = ˆ D id ( F ) is a group which containsˆ Sh ( F ). Therefore it suffices to verify conditions (A2) and (A3) ofTheorem 5.1.Similarly to (8.4) set k = ord( η, O ), p i = ord( f i , O ), γ = j k ( η ), andΓ i = j p i ( f ), ( i = 1 , . . . , n − p = n − P i =1 ( p i − − k . Then(10.2) γ · P = [ ▽ x Γ , . . . , ▽ x Γ n − ] , where P = j p ( F ) is a homogeneous vector field of degree p . By as-sumption P is non-divisible by homogeneous polynomials. For p = 1we assume that P ( x ) = Lx for some non-zero matrix L ∈ M ( R , n ).Let h ∈ ˆ D ( F ). Then h leaves invariant every orbit of F and thereforepreserves every function f i , i.e. f i ◦ h = f i for all i = 1 , . . . , n .(A2) Let A be the Jacobi matrix of h at O , thus j ( h )( x ) = A · x .We have to show that A = e ω L for some ω ∈ R if p = 1, and A = E for p ≥
2. This is implied by the following lemma and Lemma 9.5. -JETS OF ORBIT PRESERVING DIFFEOMORPHISMS 31
Lemma.
Let S = n − ∩ i =1 S ( f i ) be the intersection of the stabilizersof f i with respect to the action of GL( R , n ) , see (9.2) , and S E be theunity component of S . Then for every h ∈ ˆ D id ( F ) its Jacobi matrix A at O belongs to S E .Proof. Since f i ◦ h = f i , ( i = 1 , . . . , n − i ( Ax ) = Γ i ( x ). In other words A belongs to the intersectionof the stabilizers S = n − ∩ i =1 S (Γ i ) . On the other hand the assumption h ∈ ˆ D id ( F ) means that there exists a 1-isotopy ( h t ) in ˆ D ( F ) between h = id R n and h = h . Let A t be the Jacobi matrix of h t at O . Since( h t ) is 1-isotopy, we have that ( A t ) continuously depend on t . Moreover, A = E , whence A = A belongs to the unity component S E of S . (cid:3) (A3) Suppose that j p + l ( h )( x ) = x + v ( x ), where v is a non-zerohomogeneous map of degree p + l ≥
2. Since f i ◦ h = f i , ( i = 1 , . . . , n − h ▽ Γ i , v i = 0, whence v is parallelto the cross-product of gradients ▽ Γ i and therefore to P . Since P isnon-divisible, it follows that k ≥ p and there exists a unique non-zerohomogeneous polynomial ω l ∈ P l such that v = P · ω l . This completes the proof of (10.1).It remains to construct j ∞ -sections of ϕ V . Let V be a neighbourhoodof O . Then (10.1) implies that E id ( F , V ) ⊂ J ( F , V ). If p ≥
2, thenby (1) of Theorem 7.6 there exists a j ∞ -section of ϕ V on all of J ( F , V )and therefore on E id ( F , V ) .Suppose that p = 1, so P ( x ) = Lx is a linear vector field. Notice thatthe corresponding one-parametric subgroup E L = { e Lt } t ∈ R is closed inGL( R , n ) as the unity component of the intersection of closed subgroupsof GL( R , n ) (stabilizers of f i ). Then by (2) and (3) of Theorem 7.6 forevery g ∈ J ( F , V ) there exists a local j ∞ -section of ϕ V . In particular,this holds for all g ∈ E id ( F , V ) ⊂ J ( F , V ). (cid:3) Reduced Hamiltonian vector fields
Let f : R → R be a real homogeneous polynomial in two variables,so we can write(11.1) f ( x, y ) = l Y i =1 L l i i ( x, y ) · q Y j =1 Q q j j ( x, y ) , where l, q ≥
1, every L i is a linear function, every Q j is a definite(i.e. irreducible over R ) quadratic form, L i /L i ′ = const for i = i ′ ,and Q j /Q j ′ = const for j = j ′ . Then it can easily be shown that thepolynomial D = l Y i =1 L l i − i · q Y j =1 Q q j − j -JETS OF ORBIT PRESERVING DIFFEOMORPHISMS 32 is the greatest common divisor of its partial derivatives f ′ x and f ′ y .Hence the following homogeneous vector field of degree p = l + 2 q − R F = [ ▽ f ] /D = − ( f ′ y /D ) ∂∂x + ( f ′ x /D ) ∂∂y is non-divisible by homogeneous polynomials. Thus F has property( ∗ ). We will call F the reduced Hamiltonian vector field of F . Noticethat O is a unique singular point of F .The following theorem improves [M4, Theorem 3.2] which was basedon the previous version of this paper.11.1. Theorem.
Let f be a real homogeneous polynomial in two vari-ables, F be its reduced Hamiltonian vector field, and V be an openneighbourhood of O . Then Sh ( F , V ) = E id ( F , V ) and for every g ∈ E id ( F , V ) there exist a C W -neighbourhood N g in E id ( F , V ) and a C ∞ , ∞ W,W -continuous and preserving smoothness section
Λ : N g → C ∞ ( V, R ) of ϕ V , i.e. for every h ∈ N g h ( x ) = ϕ V (Λ( h ))( x ) = F ( x, Λ( h )( x )) . In particular, it follows from Theorem 4.4 and Lemma 4.7 that F isparameter rigid.Proof. Since F has property ( ∗ ), it follows from Theorem 8.5 that forevery g ∈ E id ( F , V ) there exists a j ∞ -section defined on some C W -neighbourhood of g in E id ( F , V ) ⊂ E ( F , V ) ∩ J ( F , V ). Therefore byLemma 7.4 it suffices to construct a C ∞ , ∞ W,W -continuous and preservingsmoothness section Ψ : E ∞ ( F , V ) → C ∞ ( V, R ) of ϕ V .For the case D ≡
1, i.e., when f has no multiple factors, such asection was constructed in [M4, Theorem 3.2]. The detailed analysisof the proof shows that [M4, Theorem 3.2] uses only the assumptionthat coordinate functions of F are relatively prime in R [ x, y ], i.e. that F has property ( ∗ ), but not the assumption that f has no multiplefactors. This implies that the same arguments prove an existence of Ψfor arbitrary f . The details are left for the reader. (cid:3) Acknowledgments
I am sincerely grateful to V. V. Sharko, V. V. Lyubashenko, E. Polu-lyakh, A. Prishlyak, I. Vlasenko, I. Yurchuk, and O. Burylko for usefuldiscussions and interest to this work. I would like to thank anonymousreferee for constructive remarks which allow to clarify exposition of thepaper. -JETS OF ORBIT PRESERVING DIFFEOMORPHISMS 33
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