On the first integral conjecture of Rene Thom
aa r X i v : . [ m a t h . D S ] O c t On the First Integral Conjecture of Ren´e Thom
Jacky CRESSON, Aris DANIILIDIS & Masahiro SHIOTA
Abstract.
More than half a century ago R. Thom asserted in an unpublished manuscriptthat, generically, vector fields on compact connected smooth manifolds without boundary canadmit only trivial continuous first integrals. Though somehow unprecise for what concernsthe interpretation of the word “generically”, this statement is ostensibly true and is nowadayscommonly accepted. On the other hand, the (few) known formal proofs of Thom’s conjectureare all relying to the classical Sard theorem and are thus requiring the technical assumptionthat first integrals should be of class C k with k ≥ d, where d is the dimension of the manifold.In this work, using a recent nonsmooth extension of Sard theorem we establish the validity ofThom’s conjecture for locally Lipschitz first integrals, interpreting genericity in the C sense. Key words.
Structural stability, first integral, o-minimal structure, Sard theorem.
AMS Subject Classification.
Primary
Secondary
The purpose of this paper is to discuss the following conjecture attributed to Ren´e Thom (see [19],[8], [13], for example) which is part of the folklore in the dynamical systems community:
Thom conjecture : For ≤ r ≤ ∞ , C r -generically vector fields on d -dimensional compact,smooth, connected manifolds without boundary do not admit nontrivial continuous first integrals. Ren´e Thom [19] proposes a scheme for a formal proof relying on the assumption that a C r closing lemma ( r ≥
1) is true [8]. The C -closing lemma (case r = 1) has indeed been provedby Pugh ([14], [15]). Nevertheless, very little is known for a C r -closing lemma with r ≥ r = 1 assuming that first integrals are of class C k with k ≥ d , i.e. a relation between the regularity class of the first integrals to be considered andthe dimension of the underlying compact manifold. As pointed out by Peixoto, this conditionis only of technical nature and relates to the use of the classical Sard theorem in a crucial partof the proof. More precisely, denoting by X M the set of C -vector fields on the d -dimensionalcompact manifold M , Peixoto’s proof is divided in three steps:Let X ∈ X M and f : M → R be a first integral of class C d .- By Sard’s lemma, there exists in f ( M ) an interval ] a, b [ made up of regular values. Forany y ∈ ] a, b [, f − ( y ) is an ( d − X .- By Pugh’s general density theorem for each y ∈ ] a, b [, f − ( y ) does not contain singularitiesor closed orbits of X since they are generic. As a consequence, singularities and closed orbitsare all located at the critical levels of f . 1 Any trajectory γ in f − ( y ) is such that ω ( γ ) ⊂ f − ( y ) and cannot be contained in theclosure of the set of singularities or closed orbits, in contradiction to Pugh’s general densitytheorem.T. Bewley [2] extends Peixoto’s theorem for 1 ≤ r ≤ ∞ . The proof has then been simplifiedby R. Ma˜ne [11]. Ma˜ne’s proof seems to have been rediscovered by M. Hurley [10]. However,the technical assumption of Peixoto ( C d -regularity of the first integrals) stays behind all theseworks, because of the use of Sard’s theorem. Nevertheless, Thom conjecture seems to be true ingeneral.In this paper, we cancel the aforementioned regularity condition by interpreting the word“nontrivial” as “being essentially definable with respect to an o-minimal approximation” (seeDefinition 2.6 below). In this context we prove the validity of Thom’s conjecture for r = 1 andfor Lipschitz continuous first integrals.The technic of the proof follows the strategy used by M. Artin and B. Mazur ([1]) to provethat generically the number of isolated periodic points of a diffeomorphism grows at most ex-ponentially. Indeed, using a recently established nonsmooth version of Sard theorem (see [3,Theorem 7] or [4, Corollary 9]) and Peixoto’s scheme of proof ([13]) we first show that in ano-minimal manifold (that is, a manifold that is an o-minimal set), generically, o-minimal firstintegrals are constant. Then by approximating every compact differentiable manifold by a Nashmanifold we derive a general statement. • Preliminaries in dynamical systems.
Given a C manifold M we denote by X M the space of all C -vector fields on M equipped withthe C topology. Let φ t : M → M be the one-parameter group of diffeomorphisms generated bya vector field X on M . A point p ∈ M is called nonwandering , if given any neighborhood U of p , there are arbitrarily large values of t for which U ∩ φ t ( U ) = ∅ . Denoting by Ω the set of allnonwandering points we have the following genericity result due to C. Pugh ([14], [15]). General density theorem (GDT):
The set G M of vector fields X ∈ X M such that properties(G )–(G ) below hold is residual in X M .(G ) X has only a finite number of singularities, all generic ;(G ) Closed orbits of X are generic ;(G ) The stable and unstable manifolds associated to the singularities and the closed orbitsof X are transversal ;(G ) Ω = ¯Γ, where Γ stands for the union of all singular points and closed orbits of X .We use the following definition of a first integral : Definition 1.1 (First integral) . A first integral of a vector field X on a compact connectedmanifold M of dimension d is a continuous function f : M → R which is constant on the orbitsof the flow generated by X but it is not constant on any nonempty open set of M .As mentioned in the introduction, Peixoto [13] only considers C k -first integrals with k ≥ d .2 Preliminaries in o-minimal geometry.
Let us recall the definition of an o-minimal structure (see [7] for example).
Definition 1.2 (o-minimal structure) . An o-minimal structure on the ordered field R is asequence of Boolean algebras O = {O n } n ≥ such that for each n ∈ N (i) A ∈ O n = ⇒ A × R ∈ O n +1 and R × A ∈ O n +1 ;(ii) A ∈ O n +1 = ⇒ Π( A ) ∈ O n (Π : R n +1 → R n denotes the canonical projection onto R n ) ;(iii) O n contains the family of algebraic subsets of R n , that is, the sets of the form { x ∈ R n : p ( x ) = 0 } , where p : R n → R is a polynomial function ;(iv) O consists exactly of the finite unions of intervals and points.An important example of o-minimal structure is the collection of semialgebraic sets (see [6]for example), that is, sets that can be obtained by Boolean combinations of sets of the form { x ∈ R n : p ( x ) = 0 , q ( x ) < , . . . , q m ( x ) < } , where p, q , . . . , q m are polynomial functions in R n . Indeed, properties (i), (iii) and (iv) ofDefinition 1.2 are straightforward, while (ii) is a consequence of the Tarski-Seidenberg principle.A subset A of R n is called definable (in the o-minimal structure O ) if it belongs to O n . Givenany S ⊂ R n a mapping F : S → R is called definable in O (respectively, semialgebraic) if itsgraph is a definable (respectively, semialgebraic) subset of R n × R . • Preliminaries in variational analysis.
Let g : U → R be a Lipschitz continuous function where U is a nonempty open subset of R d .The generalized derivative of g at x in the direction v ∈ R n is defined as follows (see [5, Section 2]for example): g o ( x , e ) = lim sup x → x t ց + g ( x + te ) − g ( x ) t (1)where t ց + indicates the fact that t > t →
0. It turns out that the function v g o ( x , v )is positively homogeneous and convex, giving rise to the Clarke subdifferential of g at x definedas follows: ∂g ( x ) = { x ∗ ∈ R d : g o ( x , v ) ≥ h x ∗ , v i , ∀ v ∈ R d } . (2)In case that g is of class C (or more generally, strictly differentiable at x ) it follows that ∂g ( x ) = {∇ g ( x ) } . A point x ∈ U is called Clarke critical , if 0 ∈ ∂g ( x ) . We say that y ∈ g ( U ) is a Clarkecritical value if the level set g − ( y ) contains at least one Clarke critical point. Given a Lipschitzcontinuous function f : M → R defined on a C manifold M we give the following definition of(nonsmooth) critical value. 3 efinition 1.3 (Clarke critical value) . We say that y ∈ f ( M ) is a Clarke critical value of thefunction f : M → R , if there exists p ∈ f − ( y ) and a local chart ( ϕ, U ) around p such that0 ∈ ∂ ( f ◦ ϕ − )( ϕ ( p )). In this case, p ∈ M is a Clarke critical point for f .It can be easily shown (see [17, Exercise 10.7], for example) that the above definition does notdepend on the choice of the chart. Throughout this section M will be a C compact connected submanifold of R n (without bound-ary), T M its corresponding tangent bundle and X M the space of C vector fields on M equippedwith the C topology. Let us recall that submanifolds of R n admit ε -tubular neighborhoods forall ε > C -submanifold N of R n , a C -diffeomorphism F : M → N and ε > . Definition 2.1 (Approximation of a manifold) . ( i ) We say that ( N, F ) is a C -approximationof M (of precision ǫ ), if N belongs to an ǫ -tubular neighborhood U ǫ of M and F can be extendedto a C -diffeomorphism ˜ F : R n → R n , which is isotopic to the identity id, satisfies ˜ F | R n (cid:31) U r ≡ idand max x ∈ R n n || ˜ F ( x ) − x || + || d ˜ F ( x ) − id || o < ǫ. (We shall use the notation ˜ F ∼ ǫ id to indicate that ˜ F is ǫ - C -closed to the identity mapping.)( ii ) A C -approximation ( N, F ) of M is called semialgebraic (respectively, definable) if themanifold N is a semialgebraic subset of R n (respectively, a definable set in an o-minimal struc-ture).In the sequel, we shall need the following approximation result. Lemma 2.2 (Semialgebraic approximation) . Let M be a C compact submanifold of R n . Thenfor every ε > , there exists a semialgebraic ε -approximation of M . Proof.
Fix ε > U be an open ǫ -tubular neighborhood of M in R n for some ǫ ∈ (0 , ε ).Applying [18, Theorem I.3.6] (for A = R n and B = C ), we deduce the existence of a C -embedding F of M into U which is ǫ -close to the identity map id in the C topology such that F ( M ) = N is a Nash manifold (that is, N is a C ∞ -manifold and a semialgebraic set). Then F can be extended to a C diffeomorphism ˜ F of R n by a partition of unity of class C such that˜ F = id on R n \ U . Moreover there exists a C isotopy { F t } t ∈ [0 , , such that F t = id on R n \ U , F ≡ id and F = ˜ F and the map F t : R n × [0 , → R n is ǫ -close to the projection to R n in the C topology. (cid:3) Given a C -manifold M and a C -vector field X ∈ X M , the following result relates genericsingularities of hyperbolic type of X with Clarke critical values of Lipschitz continuous firstintegrals of X . Lemma 2.3 (Location of singularities) . Assume f : M → R is a Lipschitz continuous firstintegral for the vector field X ∈ X M . Then all generic singularities and all closed orbits ofhyperbolic type are located at the Clarke-critical level sets. roof. Let p be either a singular point of hyperbolic type or any point of a closed orbit ofhyperbolic type and let π : U → M be the exponential mapping around p = π (0) , where U isan open neighborhood of 0 ∈ T p M ∼ = R d ( d denoting the dimension of M ). It follows that thefunction g = f ◦ π is Lipschitz continuous. Since the stable and unstable manifolds of the flawof the field X at p are transversal and since f is a first integral, it follows that for some basis { e i } i ∈{ ,...,d } of T p M one has g o (0 , ± e i ) ≥ g ′ (0 , ± e i ) := lim t ց g ( ± te i ) − g (0) t = 0 , where g (0 , · ) is given by (1). In view of (2) we deduce that 0 ∈ ∂g (0) , hence f ( p ) is a criticalvalue of f . (cid:3) Lemma 2.4 (Density of critical values for GDT fields) . In the situation of Lemma 2.3, let usfurther assume that X ∈ G M . Then the Clarke critical values of f are dense in f ( M ) . Proof.
Let Γ denote the union of all singular points and closed orbits of the field X. Since X ∈ G M , it follows from Lemma 2.3 that the set f (Γ) is included to the Clarke critical values.Continuity of f and compactness of M yield f (Γ) = f (Γ) = f (Ω) . Since Ω contains all ω -limitsof orbits of X, taking any y ∈ f ( M ) and any x ∈ f − ( y ) (cid:31) Γ we denote by γ the orbit passingthrough x and by γ ∞ the set of ω -limits of γ . Then by continuity y = f ( γ ) = f ( γ ∞ ) ⊂ f (Ω).This proves the assertion. (cid:3) Corollary 2.5 (Thom conjecture – definable version) . Let X ∈ G M . Then X does not admitany Lipschitz continuous definable first integral. Proof.
Assume f is a Lipschitz continuous first integral of X on M and denote by S the setof its Clarke critical points. If f is o-minimal, then so is M (cf. property (ii) of Definition 1.2),the tangential mappings π : U ⊂ T p M → M (around any point p ∈ M ) and the compositefunctions of the form g = f ◦ π (notation according to the proof of Lemma 2.3). Note that p isa critical point of f if and only if π − ( p ) is a critical point of g = f ◦ π where π is any tangentialmapping with p ∈ π ( U ). Applying [4, Corollary 8] we deduce that the set of Clarke criticalvalues of each function g is of measure zero. Using a standard compactness argument we deducethat f ( S ) is of measure zero, thus in particular f ( M ) \ f ( S ) contains an interval ( y , y ) . Butthis contradicts the density result of Lemma 2.4. (cid:3)
If the manifold M is not a definable subset of R n the above result holds vacuously and givesno information. To deal with this case, the forthcoming notion of essential o-minimality withrespect to a given o-minimal approximation turns out to be a useful substitute for our purposes.Let us fix ǫ > ǫ -approximation ( N, F ) of M . Definition 2.6 (Essential o-minimality with respect to a definable approximation) . A mapping f : M → R is called essentially o-minimal with respect to a definable approximation ( N, F ) of M if the mapping f ◦ F − : N → R is o-minimal.Note that every o-minimal function on M is essentially o-minimal with respect to any approxi-mation ( N, F ) of M for which the diffeomorphism F is o-minimal. Setting M = { p ∈ R : p ∈ Graph( h ) } h ( t ) = t sin( x − ) , if t = 0 and 0 if t = 0, we obtain a nondefinable C -submanifoldof R . Thus the (projection) function f : M → R with f (( t, h ( t )) = t is not o-minimal. Itcan be easily seen that for every ǫ > ǫ -approximation ( N, F ) with respect towhich f is essentially o-minimal. On the other hand, if χ K is the characteristic function of theCantor set K of (0 , g : R → R defined by g ( x ) = R x χ K ( t ) dt for all x ∈ R is notessentially o-minimal with respect to any approximation. Roughly speaking, a function that isnot essentially o-minimal contains intrinsic irreparable oscillations.In view of Lemma 2.2, for every ǫ > C definable manifold N and a dif-feomorphism F : M → N such that F ∼ ǫ id . Fixing the approximation, we associate to everyvector field X : M → T M on M the conjugate C -vector field ˜ X : N → T N on N defined asfollows: ˜ X ( q ) = dF ( F − ( q ) , X ( F − ( q )) . Note that ˜ X is uniquely determined by X . Let us further denote by G N the vector fields of N that satisfy the generic GDT assumptions. We are ready to state our main result. Theorem 2.7 (Genericity of non-existence of first integrals) . Let M be a C compact subman-ifold of R n and ǫ > . For the C topology, the set of vector fields in M that do not admitLipschitz continuous first integrals which are essentially o-minimal with respect to a given de-finable ǫ -approximation of M is generic. Proof.
Let us fix any definable ǫ -approximation of M and let us denote by G N the vector fieldsof N that satisfy the generic GDT assumptions. By Pugh’s density theorem G N is a C -residualsubset of X N and by Corollary 2.5, if Y ∈ G N then Y does not possess any o-minimal Lipschitzcontinuous first integral. Let G denote the set of vector fields of X that conjugate inside G N ,that is, G = { X ∈ X M : ˜ X ∈ G N } . Then G is residual in X M . Pick any X ∈ G and assume that f : M → R is a Lipschitz continuousfirst integral of X. Since the trajectories of X are transported to the trajectories of ˜ X ∈ G N through the mapping F, it follows that ˜ f = f ◦ F is a first integral of ˜ X in N. This shows that f cannot be essentially o-minimal with respect to ( N, F ). (cid:3) ————————————– Acknowledgment.
Part of this work has been made during a research visit of the secondauthor to the University of Pau (June 2007). The second author wishes to thank his hosts forhospitality and Olivier Ruatta (University of Limoges) for useful discussions. The first authorthanks Jean-Pierre Bourguignon for his help with R. Thom’s archive and J-P. Fran¸coise foruseful references. J.C. has been supported by the French projet de l’Agence Nationale de laRecherche ”Int´egrabilit´e r´eelle et complexe en M´ecanique Hamiltonienne”, N.JC05-41465
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Unpublished manuscript . 7———————————–Jacky CRESSONLaboratoire de Math´ematiques Appliqu´ees, UMR 5142 du CNRSUniversit´e de Pau et des Pays de l’Adouravenue de l’universit´e, F-64000 Pau, France.andInstitut de M´ecanique C´eleste et de Calcul des ´Eph´em´erides (IMCCE)UMR 8028 du CNRS - Observatoire de Paris77 Av. Denfert Rochereau 75014 PARIS (FRANCE)E-mail: [email protected]://web.univ-pau.fr/~jcresson/
Aris DANIILIDISDepartament de Matem`atiques, C1/308Universitat Aut`onoma de BarcelonaE-08193 Bellaterra, Spain.E-mail: [email protected]://mat.uab.es/~arisd
Research supported by the MEC Grant No. MTM2005-08572-C03-03 (Spain).Masahiro SHIOTADepartment of MathematicsNagoya University (Furocho, Chikusa)Nagoya 464-8602, Japan.E-mail: [email protected]@math.nagoya-u.ac.jp