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Linearization of Cohomology-free Vector Fields

We study the cohomological equation for a smooth vector field on a compact manifold. We show that if the vector field is cohomology free, then it can be embedded continuously in a linear flow on an Abelian group.


INTRODUCTION
We consider a smooth flow (φ t ) t∈R generated by a vector field X on a smooth connected compact manifold M.
A major problem arising in many different contexts of the theory of dynamical systems is to solve the cohomological equation for the flow (φ t ) t∈R , the equation given by Here L X denotes the Lie derivative in the direction of the vector field X, f is a given function and h is the solution we seek.
To make sense of this problem it is of course necessary to impose some regularity conditions on the data f as well as on the solution h. In low regularity we shall interpret the equation (1) in a weak sense.
We endow the space C ∞ (M) with the C ∞ -topology and define the C ∞ -cohomology of the flow (φ t ) t∈R as the quotient vector space C ∞ (M)/L X (C ∞ (M)); the reduced C ∞ -cohomology is instead the topological vector space C ∞ (M)/L X (C ∞ (M)), where the closure is taken in the C ∞ -topology.
Restricting Katok's definition, ([Kat01], [Kat03]), to the C ∞ setting, we say that the flow (φ t ) t∈R is C ∞ -stable if its cohomology and reduced cohomology coincide, that is if the image of the Lie derivative operator L X is a closed subspace of C ∞ (M). By Hahn-Banach's Theorem this is equivalent to saying that, for every function f belonging to the kernel of all X-invariant Schwartz distributions, the equation (1) admits a solution h ∈ C ∞ .
We remark that since a continuous flow on a compact manifold admits always an invariant measure, the reduced cohomology of a flow is at least one-dimensional.
In all the cases studied so far, the flow cohomology is infinite dimensional, with one notable exception. In fact, linear flows on d-dimensional tori provide the classical and only known examples with one-dimensional reduced flow cohomology. It is well known that such a flow is C ∞ -stable if and only if the direction numbers α ∈ R d of the vector field X is a Diophantine vector, that is, if there exist positive constants C and τ such that When α is not Diophantine then there exists , [Kat03]) of even measurable ( [Her04]); thus fast approximation by periodic flows provides one of mechanisms through which C ∞ -stability fails to hold. A C ∞ -stable flow is called rigid or cohomology free if its flow cohomology is one dimensional; A. Katok ([Hur85], [Kat01], [Kat03]) suggested the following: Conjecture 1. Let {φ t } t∈R be a cohomology-free flow generated by a vector field X on a compact connected manifold M. Then, up to diffeomorphism, the manifold M is a torus and X is a Diophantine vector field.
Chen and Chi [CC00] have essentially proved that Katok's conjecture is equivalent to a conjecture formulated by Greenfield and Wallach in [GW73] and which states that a globally hypoelliptic vector field is a linear diophantine flow on the torus (see [For08] for a review of the relation betwwen the two conjectures).
Progress towards this conjecture has been limited. A major advance has been made by F. and J. Rodriguez-Hertz [RHRH06] who showed that a cohomology-free flow on a manifold M is up to semiconjugate to a Diophantine linear flow on a torus of dimension equal to the first Betti number of M (the Albanese variety of M); furthermore the semi-conjugacy is smooth. More substantial progress has made for low dimensional manifolds. In [LdS98] the analogous problem for diffeomorphisms is solved for tori of dimension four or less. Recently independent work of G. Forni [For08], A. Kocsard [Koc09] and S. Matsumoto [Mat09], have proved the Katok-Greenfield-Wallach conjecture when dim M ≤ 3, using Taubes' proof of Weinstein's Conjecture [Tau07].
More recently Avila and Kocsard have recently announced in [AK10] that the reduced cohomology of every minimal circle diffeomorphism -hence of very minimal flow on the two-torus -is one-dimensional; from this it follows easily that, up to a diffeomorphism, the only C ∞ -stable minimal flows on the two-torus are the diophantine linear flows.
Theorem 1.1. Let {φ t } t∈R be a cohomology-free flow generated by a vector field X on a compact connected manifold M. Then there are a (possibly non separated) topological Abelian group A, a continuous homomorphism t ∈ R → at ∈ A, and a continuous injection l : M → A, such that l(φ t (y)) = l(y) + at for every y ∈ M and t ∈ R. Furthermore there is a continuous projection π of A onto the Albanese torus of M such that π • l is F. and J. Rodriguez-Hertz' semi-conjugacy.

CURVES AND LOOPS
From now our standing hypothesis is that (φ t ) t∈R is a cohomologyfree flow generated by a vector field X on a compact connected manifold M.
Two curves γ 1 and γ 2 can be concatenated if ω(γ 1 ) = α(γ 2 ); with γ = γ 1 γ 2 we shall denote the usual concatenation For the sequel we restrict our consideration to the set Γ of curves which are finite concatenations of unoriented flow segments and geodesic segments (for some fixed Riemannian metric on M) transverse to the flow. We endow Γ with the topology of uniform convergence (say for the uniform speed parametrization of curves).
Curves in Γ have the following regularity property: Lemma 2.1. Two curves in Γ meet a finite collections of intervals (which may reduce to points). Furthermore there is a compact set K ⊂ Γ such that every two points of M can be joined by an element of K.
Proof. The first statement is an immediate consequence of the choice of Γ: if two geodesic segments intersect, they do so on an interval; the same is true for flow segments. For the set K we can choose the set paths which are concatenation of N geodesic segments of length less than some positive For x ∈ M, we denote by Γ x the curves γ ∈ Γ with α(γ) = x, that is the curves starting at x. Finally we let ∆ be the set of curves γ ∈ Γ with α(γ) = ω(γ), the set of closed loops in Γ.

CURRENTS
Let Ω 1 (M) denote the Fréchet space of C ∞ differential one-forms with the C ∞ -topology and let C 0 (M) and C 1 (M) be the Fréchet spaces of de Rham currents on M of degree zero and one, that is the dual space of Ω 0 (M) = C ∞ (M) and the dual space of Ω 1 (M), respectively. The spaces C 0 (M) and C 1 (M) are endowed with the vague topology.
To each parametrized curve γ it corresponds an integration onecurrent γ given by Clearly the current γ does not depend upon the choice of a parametrization of γ so that the map γ ∈ Γ → γ ∈ C 1 (M) is well defined and we set we also denote by Γ x and ∆ the sets of currents images of curves in Γ x and loops in ∆.
It is obvious from the definition that for any γ 1 and γ 2 in Γ for which the concatenation is defined we have The following elementary propositions are stated here for further reference. We Hence ∆ is a closed set both in Γ and in Γ x . Furthermore Γ and Γ x are both invariant by translation by ∆.
Proof. The first affirmation is obvious. Suppose that π Corollary 3.5. The map π x induces a homeomorphism Proof. The induced map p x is continuous for the quotient topology and injective by Proposition 3.4. Let K be as in Lemma 2.1. The set of currents images of elements in K ∩ Γ x is compact in the weak topology and surjects onto Γ x / ∆; hence Γ x / ∆ is compact. Since M is a Hausdorff space, the map p x is a homeomorphism.

TWISTING THE EMBEDDING
The hypothesis of Theorem 1.1 imply that for every η ∈ Ω 1 (M) there exist a C ∞ function h η : M → R and a constant c η ∈ R such that (4) L X h η = η(X) − c η . The function h η is only defined up to a constant, by the unique ergodicity of the flow (φ t ). We shall need the following observations. Remark 4.1. If η 1 , η 2 ∈ Ω 1 (M) and η 1 (X) − η 2 (X) = C is a constant function then h η 1 − h η 2 is a constant and c η 1 − c η 2 = C. Proposition 4.2. Let C ∞ 0 (M) be the quotient space C ∞ (M) modulo constants (which we can also identify to the space C ∞ functions of µ-average zero). Then the maps and Ω 1 (M) are endowed with the C ∞ Fréchet topology.
Proof. The Lie derivative L X : , is a continuous linear operator which, by hypothesis and by the unique ergodicity of µ, is also bijective. Since C ∞ 0 (M) is a Fréchet space, the open mapping theorem implies that this map is an isomorphism. The map Ω 1 (M) → C ∞ (M), η → η(X), being clearly continuous, our claim follows.
We shall use the family of functions h η to "twist" our homeomorphism M ≈ Γ x / ∆. The following lemma is obvious.

Lemma 4.4. The map L is a continuous linear operator (in fact a projection). The restriction of L to ∆ is the identity map of ∆.
Let L x be the restriction of L to Γ x .

INJECTIVITY OF L x
The aim of this section is to show the following proposition.
Clearly broad equivalence is an equivalence relation.
From a set of curves in Γ we can, in an iterative way, obtain a new set of curves, broadly equivalent to the given set and without retraced arcs.
In fact if {γ = arbr −1 c} we say that the set of curves {γ 1 = ac, γ 2 = b} is the set obtained from {γ} by simple excision of the retraced arc r. (Observe that γ 2 is a closed curve).
If a set of two curves is given, {γ 1 , γ 2 }, such that γ 1 = arb and γ 2 = cr −1 d we have two cases: if γ 2 is closed, the simple excision of the retraced arc r will yield a single curve set {γ 3 = adcb}; if γ 2 is open the simple excision of r will result in the set of two curves It is clear in the above procedure that after a simple excision the new set of curves is broadly equivalent to the original set.
The simple excision of a retraced arc from one or two arcs in a family of broken arcs γ 0 1 , γ 0 2 , . . . , γ 0 n yields a new family of broken arcs; successive simple excisions will lead to a family of broken arcs γ 1 , γ 2 , . . . , γ n ′ which we say obtained by maximal excision from γ 0 1 , γ 0 2 , . . . , γ 0 n if it does not contain any further retraced arcs. The proof of the following two elementary lemmata is obtained by induction on the number of excisions and by a direct application of the definition of excision.  The choice of Γ as a set of curves obtained as concatenations of geodesic segments and flow segments yields a simple proof the following lemma which holds true in greater generality (cf. [BS97]).
Lemma 5.6. Let G = {γ 1 , γ 2 , . . . , γ n } be a finite subset of Γ. Set Y = n i=1 γ i (I). Recall that a point y ∈ Y is regular if it satisfies the following two conditions: (1) every t ∈ n i=1 γ −1 i {y} is a regular point and (2) there exists an open neighborhood W of y such that Y ∩ W is a embedded arc.

The set of regular points y ∈ Y is an open and dense subset of Y.
Proof of Proposition 5.1.
The first condition means We have two cases. In the first case there exists t 0 ∈ I such that γ i (t 0 ) is regular and the velocityγ(t 0 ) is not collinear to X(γ i (t 0 )). By the previous Lemma, there exists an open neighborhood W of γ i (t 0 ), such that W ∩ ∪ j γ j (I) = W ∩ γ i (I) is an embedded arc. Since γ i (I) and X are transverse, there exists a one-form θ, supported in W, such that θ(X) vanishes identically and such that γ i θ = 0. Since there are no retraced arcs in the set G and by the choice of W and θ we have ∑ i γ i θ = 0. By the Remark 4.1 we have that h θ is identically constant. But this contradicts (6) since γ θ = ∑ i γ i θ = 0 and h θ (ω(γ)) − h θ (α(γ)) = 0. This case is impossible.
6. PROOF OF THEOREM 1.1 Let x ∈ M be a point fixed once for all. Consider the Abelian group A = C 1 (M)/ ∆. The map L : C 1 (M) → C 1 (M) of Definition 4.3 defines a quotient map of A into itself since, by Lemma 4.4, L| ∆ is the identity mapping of ∆. The restriction of L to Γ x , which we have denoted by L x , induces a mapping of Γ x / ∆ into A, which, by Proposition 5.1, is in fact a continuous bijection of Γ x / ∆ onto L x ( Γ x )/ ∆. Using the identification M ≈ Γ x / ∆ of Corollary 3.5 we then conclude that L x induces a continuous injection l : Fix y ∈ M and let γ be a curve starting at x and ending at y. For t ∈ R, let γ t be the arc of orbit s → φ ts (y). Integrating the equation (4) along the orbit γ t of y ∈ M we have Let c ∈ C 1 (M) be given by c(η) = c η . Then the equations (8), in view of the Definition 4.3, can be rewritten as the following equation in C 1 (M) for the currents γ and γ t associated to the arcs γ and γ t : As the endpoints of the arcs γ γ t and γ are respectively equal to φ t (y) and y, passing to the quotient by ∆, we obtain that l(φ t (y)) = l(y) + ta, where t → ta is the projection to A of the line subgroup t → ct.
Finally notice that, by restricting the space of currents C 1 (M) to the finite dimensional space H 1 (M) of harmonic one-forms on M, we obtain a projection q : C 1 (M) → H 1 (M) * . Taking a further quotient by the lattice P of periods of M, we obtain a projection q ′ of C 1 (M) onto the Albanese torus H 1 (M) * /P; the map q ′ factors through a map π : A → H 1 (M) * /P, as q sends the loop group ∆ onto P. Thus we have C 1 (M) → C 1 (M)/ ∆ → H 1 (M) * /P.
Since the maps L and q ′ are smooth, the composite map q ′ • L is also smooth. For any smooth local lift φ : U ⊂ M → Γ x of the projection mapΓ x → M, we have that q ′ • L • φ = π • l| U ; we conclude that π ′ = π • l is a smooth map of M into H 1 (M) * /P.
From the continuity of π ′ = π • l and the minimality of the flow it follows that the image of π ′ is a rational sub-torus of H 1 (M) * /P. However the map associating to a closed loop in M the period along this loop induces a surjection of ∆ onto P; it follows that π ′ is surjective in homology and thus (smooth and) surjective and indeed equal to F. and J. Rodriguez Hertz' semi-conjugacy.
Remark 6.1. The proof above shows that in fact there is, to some degree, a differential structure on the space L x ( Γ x )/ ∆, inherited from the linear structure of C 1 (M); in fact one could show that the map l is a morphism of differential (or diffeological) spaces in the sense of Chen and Iglesias ([Che77], [Igl]). As the major problem is to tie the topological structure with the differentiable one, we omit any discussion of this point.