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Cohomologically rigid vector fields: the Katok conjecture in dimension 3
Alejandro Kocsard
1Instituto Nacional de Matemática Pura e Aplicada, Estrada Dona Castorina, 110, Rio de Janeiro, Brasil Received 12 December 2007; received in revised form 10 July 2008; accepted 28 July 2008
Available online 9 September 2008
Abstract
A smooth vector fieldXon a closed orientabled-manifoldMis said to becohomologically rigidwhen given anyξ∈C∞(M,R), there existu∈C∞(M,R)andc∈Rsatisfying
LXu=ξ−c,
whereLXis the Lie derivative in theXdirection. In 1984, Anatole Katok conjectured that every cohomologically rigid vector field should be smoothly conjugated to a Diophantine vector field on thed-torusTd. In this work the validity of the Katok conjecture for 3-manifolds is proved.
©2008 Elsevier Masson SAS. All rights reserved.
Résumé
Un champ de vecteursXsur une variétéMcompacte orientable de dimensiondest ditcohomologiquement rigidesi pour toute fonctionξ∈C∞(M,R)il existeu∈C∞(M,R)etc∈Rtels que
LXu=ξ−c,
oùLXdésigne la dérivée de Lie dans la direction deX. En 1984, Anatole Katok a conjecturé que tout champ de vecteurs coho- mologiquement rigide devrait être conjugué par un difféomorphisme lisse à un champ linéaire diophantien sur le toreTd. Dans ce travail nous démontrons la conjecture de Katok pour les variétés de dimension trois.
©2008 Elsevier Masson SAS. All rights reserved.
Keywords:Vector fields; Cohomological equations; Katok conjecture
1. Introduction
WhenGis a Lie group acting on a manifoldMby a smooth actionΦ:G×M→M, many questions about the dynamics ofΦ can be answered studying the first cohomology group with real coefficientsH1(Φ), i.e. the quotient linear space of real cocycles overΦ(from now on, cocycles for short) by the subspace of coboundaries (see Section 2
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doi:10.1016/j.anihpc.2008.07.005
for definitions). The problem of analyzing the structure ofH1(Φ)leads to studycohomological equations(see [10,11]
for a great panoramic view of the subject).
WhenGis equal toZporRp, it is rather easy to verify that dimH1(Φ)p,
and very commonly H1(Φ)is infinite-dimensional, being its natural topology (induced by the Fréchet topology of C∞(M,R)) typically non-Hausdorff. Therefore, the “smallness” ofH1(Φ)is usually associated with some kind of
“rigidity” ofΦ, and so, it seems to be natural to say that aZq orRq-actionΦ iscohomologically rigidwhen
dimH1(Φ)=q. (1.1)
One of the simplest, although important, examples of Lie group actions is given by a flow (an R-action) ΦX:R×M→Minduced by a smooth vector fieldX∈X(M). In this case, the space of smooth cocycles overΦXis canonically identified withC∞(M,R), andξ ∈C∞(M,R)is a coboundary if and only if there existsu∈C∞(M,R), namedtransfer function, satisfying
LXu=ξ,
whereLXdenotes the Lie derivative in theXdirection.
Taking into account this identification,H1(ΦX)is naturally isomorphic toC∞(M,R)/LX(C∞(M,R)).
The prototypical example of cohomologically rigidR-action is given by a linear flow on a torus generated by a Diophantine vector field (see Definition 2.5). So far these are the only known examples, and Katok has conjectured [9–11] that, moduloC∞-conjugation, these are the only ones. More precisely, we have
Katok Conjecture. IfM is a closed orientable d-manifold andX is a cohomologically rigid smooth vector field on M, then there exist a C∞ diffeomorphismH:M→Td and a Diophantine vector α∈Rd (see Definition2.5) verifying
DH(X)≡α.
It is interesting to remark that an analogous statement for higher-rank actions is far from being true. In fact, Katok and Spatzier, in their seminal work [12], showed that all known AnosovZk-actions, withk2, are cohomologically rigid. A completely different kind of examples were constructed by Urzúa Luz in [28], who proved the existence of co- homologically rigid affine minimalZd-actions on tori, with some acting diffeomorphisms different from translations.
In all these cases the suspensions of these actions generate cohomologically rigidRd-actions on manifolds which are not tori.
The main result of this work is the following
Theorem A.The Katok conjecture is true for3-manifolds.
We would like to remark that, while this work was in progress, Forni [5] and Matsumoto [15] independently proved the same result.
2. Notations and preliminaries
For simplicity, in this article we will restrict ourselves to work on theC∞ category, and we shall use the word smoothas a synonymous ofC∞.
Thed-dimensional torus will be denoted byTdand the quotient Lie groupRd/Zd will be our favorite model for it. We will write prZd:Rd→Td for the quotient projection, and(θ1, θ2, . . . , θd)for the canonical coordinates inTd. The Haar probability measure onTd, also called the Lebesgue measure, will be denoted by Lebd. As usual, making some abuse of notation, we shall suppose that the elements of SL(d,Z)act onTdby Lie group automorphisms.
For usMwill always denote a smooth closed (i.e. compact and without boundary) orientabled-dimensional man- ifold. We will writeβk(M)for thek-th Betti number ofM, i.e.βk(M) .
=Hk(M,Q).
The group of smooth diffeomorphisms ofM, endowed with the WhitneyC∞topology, will denoted by Diff(M).
The subgroup of orientation-preserving diffeomorphisms will be denoted by Diff+(M).
Given a smooth fibrationp:N→M, thep-fiber over anyx∈Mwill be denoted byNx, and we shall writeΓ (N ) for the space of smooth sections ofp. The only exception for this notational convention is the tangent bundle overM:
in this caseπ:T M→Mdenotes the canonical projection, and we writeTxMforπ−1(x), andX(M)forΓ (T M).
The expressionΛk(M)will be used for the space of smoothk-forms onM.
Given anyX∈X(M), we write{ΦXt }for its induced flow,iX:Λk(M)→Λk−1(M)for the usual contraction byX, andLXfor the Lie derivative acting on any smooth tensor field onM.
2.1. Measures and distributions
The set of all finite signed Borel measures onM shall be denoted byM(M), and we will writeD(M)for the space of all real continuous linear functionals onC∞(M,R). Of course, we assume thatC∞(M,R)is equipped with its usual Fréchet topology.
In order to avoid confusions, we remark that along this work we use the termdistributionin the “sense of Schwartz”, i.e. for us a distribution is any element of D(M), and we reserve the term plane field to mean a section of the Grasmannian ofTM.
LetGbe an arbitrary Lie group andΦ:G×M→M be a smoothG-action. We define the space ofΦ-invariant distributions and measures by
D(Φ) .
=
T ∈D(M):
T , ψ
Φ(g,·)
= T , ψ, ∀g∈G, ∀ψ∈C∞(M,R)
, (2.1)
M(Φ) .
=D(Φ)∩M(M). (2.2)
WhenΦ is an R-action induced by X∈X(M), we shall use the notations D(X) andM(X) to meanD(Φ)and M(Φ), respectively. As usual, given any T ∈D(M)we defineLXT ∈D(M)byLXT , ψ .
= −T ,LXψ, for all ψ∈C∞(M,R).
We will supposeΛd(M)canonically embedded inM(M), andM(M)inD(M). On the other hand, since we are assuming thatMis orientable, any volume form induces an isomorphism betweenC∞(M,R)andΛd(M). However, notice that this identification is not canonical at all.
2.2. Cocycles and coboundaries
As above, letGbe an arbitrary Lie group andΦ:G×M→Mbe a smoothG-action.
For us, acocycleoverΦis a smooth real functionA:G×M→Rsatisfying A(g1g2, x)=A
g1, Φ(g2, x)
+A(g2, x), ∀x∈M, ∀g1, g2∈G. The vector space of all cocycles overΦwill be denoted byZ(Φ).
On the other hand, we say that a cocycleAis acoboundaryif there exists a smooth mapB:M→R, usually called transfer map, such that
A(g, x)=B
Φ(g, x)
−B(x), ∀x∈M, ∀g∈G. (2.3)
The set of coboundaries overΦ shall be denoted byB(Φ)and it is clearly a linear subspace ofZ(Φ). The quotient spaceZ(Φ)/B(Φ)is called thefirst cohomology group of Φand it is denoted byH1(Φ).
Very commonly,H1(Φ)is infinite-dimensional and its natural topology (induced by the usual Fréchet topology of Z(Φ)) is non-Hausdorff.
Notice that (2.3) implies that T , A(g,·)
=0, ∀A∈B(Φ),∀g∈G, ∀T ∈D(Φ). (2.4)
On the other hand, as we have already mentioned in the introduction of this work, ifKdenotes eitherZorR, and Φ is a smoothKq-action, we can easily verify that dimH1(Φ)q. In fact, the group ofK-linear homomorphisms HomK(Kq,R)naturally injects in Z(Φ)(i.e. eachK-linear homomorphism can be considered as a cocycle which does not depend on theM-coordinate), and taking into account (2.4), we easily show that the zero cocycle is the only coboundary contained in the image of this injection. Therefore,H1(Φ)contains a subspace algebraically isomorphic to HomK(Kq,R).
Taking into account these remarks, we have the following
Definition 2.1.LetGbe equal toZq orRq, andΦ:G×M→Mbe a smoothG-action. We say thatΦiscohomo- logically rigidiff
dimH1(Φ)=q.
Remark 2.2. It is important to notice that some authors [21,29,15] use the term “cohomology-free” or “param- eter rigid” instead of our terminology. The reader can find some examples of cohomologically rigid actions in [12,16,28,24].
2.3. Cohomologically rigid vector fields
We say that a vector field is cohomologically rigid when its induced flow is. The following is a very simple result that will be very useful in the future:
Proposition 2.3. A vector fieldX∈X(M)is cohomologically rigid if and only if given any ξ∈C∞(M,R), there existsc=c(ξ )∈Randu∈C∞(M,R)satisfying
LXu=ξ−c. (2.5)
From this it easily follows
Proposition 2.4.IfX∈X(M)is cohomologically rigid, then:
(1) dimD(X)=1. In particular,D(X)=M(X)and{ΦXt }is uniquely ergodic;
(2) There exists a smoothX-invariant volume formΩ∈Λd(M);
(3) {ΦXt }is minimal, i.e.{ΦXt (x): t∈R}is dense inM, for everyx∈M.
The reader can find a proof of this result in [13], which is a simple reformulation of the ideas presented in [11,10]
to prove the analogous statement for diffeomorphisms.
To introduce the first result of classification of cohomologically rigid vector fields, we need the following
Definition 2.5. We say that α=(α1, . . . , αd)∈Rd is a Diophantine vector if there exist real constantsC, τ >0 satisfying
d
i=1
αipi > C
1maxid|pi|−τ
, (2.6)
for everyp=(p1, . . . , pd)∈Zd\ {0}.
A vector fieldXαon thed-dimensional torusTdverifyingXα≡αwill be called aDiophantine vector field.
Now we can easily characterize the cohomologically rigid vector fields on tori:
Proposition 2.6.A smooth vector field onTd is cohomologically rigid if and only if it is smoothly conjugated to a Diophantine one.
Proof. The proof of Corollary 1.7 in [29] can easily be adapted to prove that every cohomologically rigid vector field onTdmust be smoothly conjugated to a constant one (see [13] for details).
Finally, using Fourier analysis it is not hard to verify that a constant vector field is cohomologically rigid if and only if it is Diophantine (see for instance [11]). 2
2.4. Topological obstructions
Taking into account Proposition 2.6, Katok conjecture essentially affirms that the only closed orientable manifolds that support cohomologically rigid vector fields are tori.
In Proposition 2.4 we saw that every cohomologically rigid vector field was minimal. In particular, it cannot exhibit any singularity, and hence, the Euler characteristic of the supporting manifold must vanish.
For a very long time this was the only known obstruction for the existence of cohomologically rigid vector fields, until F. and J. Rodríguez-Hertz produced a breakthrough in [21], finding additional restrictions for the topology of the supporting manifold.
To recall the main result of [21], we first need to introduce the following
Definition 2.7.Given anyX∈X(M), we say thatq:M→Tnis aDiophantine projectionforXwhenqis a smooth fibration and
Dq(X)≡Xα∈X(Tn), (2.7)
beingXα a Diophantine vector field (Definition 2.5).
Now we can precisely state
Theorem 2.8.(F. and J. Rodríguez-Hertz [21].) IfMis a closedd-manifold andX∈X(M)is cohomologically rigid, then there exists a Diophantine projectionp:M→Tβ1 forX, whereβ1=β1(M)is the first Betti number ofM.
This result is used as the fundamental tool in the proof of Theorem 3.1.
2.5. Tangent, normal and projective flows
In this short paragraph we introduce some terminology that we shall use in Section 4.
LetX∈Xr(M)(r2) be a singularity-free vector field inMand{ΦXt }be its induced flow. Then, thetangent flow ofXis nothing but the derivative of its induced flow and will be denoted by{DΦXt }.
Since X is singularity-free, we can define the normal bundle associated to X as the quotient bundle N X .
= T M/RX→M. Its natural bundle projection (induced by π:T M →M) will be denoted by πN:N X→M, and we shall write prX:T M→N Xfor the canonical quotient projection.
Since the line bundleRXis invariant under the action of the tangent flow{DΦXt }, it clearly induces a linear flow onN Xwhich will be called thenormal flowand will be denoted by{N ΦXt }.
Finally, we can projectivize each fiber ofπN:N X→M to get a new fiber bundleπP:P(N X)→M, called the projective bundle. The normal flow will induce a bundle flow onP(N X)which will be denoted by{P ΦXt }. We will write prP:N X\ {0} →P(N X)2for the canonical quotient projection given by prP:vˆ→(R\ {0})v.ˆ
2.6. Hyperbolic dynamics
In this paragraph we introduce some notations and known results on hyperbolic dynamics that will be useful in Section 4.
Given a vector bundleπ:E→Mand a singularity-free vector fieldY ∈Xr(M)(r2), we say thatA:R×E→E is alinear cocycle over{ΦYt}if, for everyt,ΦXt ◦π=π◦A(t,·), where the mapsA(t,·):π−1(p)→π−1(ΦXt (p)) are linear isomorphisms verifying
A(t0+t1, p)=A
t0, ΦXt1(p)
A(t1, p), ∀p∈M, ∀t0, t1∈R.
The cocycleAis said to beAnosovif there exist two continuous sub-bundlesEs, Eu⊂E, aC0Finsler structure · inE, and real constantsC >0 andρ∈(0,1)verifying
• Es⊕Eu=E,
• A(t, Epσ)=Eσ
ΦYt(p), for everyp∈M, everyt∈Randσ=s, u,
• A(t,·)|EsCρt, andA(−t,·)|EuCρt, for everyt >0.
2 In this context{0}means the zero section ofN X.
On the other hand, the cocycleAis said to bequasi Anosovif, given anyv∈E, it holds sup
t∈R
A(v, t )<∞ ⇒v=0. (2.8)
The following result appears in different forms, and in fact with different hypothesis, in the works of Mañé [14], Sacker and Sell [23], and Selgrade [25,26]:
Theorem 2.9.If the flow{ΦYt}does not have any wandering point(i.e. every non-empty open set U⊂M satisfies ΦYt(U )∩U= ∅, for somet >1), then a cocycleAover{ΦYt}is quasi Anosov if and only if it is Anosov.
On the other hand, we shall say thatY ∈Xr(M)isAnosovif there exists a codimension-oneDΦY-invariant sub- bundleF ⊂T MverifyingF⊕RX=T Mand such thatDΦY|F:R×F→F is an Anosov linear cocycle (over{ΦYt}.
Then we have the following result due to Doering:
Theorem 2.10.(Doering [3].) Let suppose that{ΦYt}does not have any wandering point. Then,Y is an Anosov vector field if and only if its normal flow{N ΦYt}(see Section2.5)is an Anosov linear cocycle(over{ΦYt}).
3. Non-vanishing first Betti number
In this section we begin the proof of Theorem A considering the case where the supporting manifold has non-trivial real first homology group. In fact, the main purpose of this section consists in proving the following
Theorem 3.1. LetM be a closed, orientable3-manifold such thatβ1(M)1, and let us suppose thatX∈X(M) is a cohomologically rigid vector field. Then,M is diffeomorphic toT3andXisC∞-conjugated to a Diophantine constant vector field onT3.
To simplify the exposition, we will subdivide the proof of Theorem 3.1 in the following partial results:
Lemma 3.2. Under the hypothesis of Theorem3.1, there existsn0∈Z such thatM is diffeomorphic to the torus bundleT3A .
=T2×R/(x, t )∼(Ax, t−1), where A .
=
1 0 n0 1
∈SL(2,Z). (3.1)
In particular, it holds
β1(M)2. (3.2)
Lemma 3.3.The flow{ΦXt}is smoothly conjugated to the suspension of the affine2-torus automorphismA+γ, for someγ∈T2andAgiven by(3.1).
Assuming Lemmas 3.2 and 3.3, now we can easily prove Theorem 3.1:
Proof of Theorem 3.1. Let us suppose thatn0=0 in (3.1). Then, applying a construction attributed to Katok [11], for eachm∈Z\ {0}we defineTm∈D(T2)by
Tm, ψ .
=
k∈Z
ψ (knˆ 0m, m)e−2π ikm(γ2+k−21n0γ1), ψ∈C∞(T2,R) (3.3)
whereγi are the coordinates moduloZofγ. Straight forward computations show that eachTmis(A+γ )-invariant and{Tm: m∈Z\ {0}}is linearly independent.
By Lemma 3.3, there exists aC∞diffeomorphismH:M→T3Aconjugating{ΦXt}and the suspension ofA+γ, denoted by{[A+γ]t}.
Finally, for eachm∈Z\ {0}, we defineT˜m∈D(M)by ˜Tm, ψ=.
1 0
Tm,
ψ◦H−1◦ [A+γ]−t
T2×{t}
dt, ∀ψ∈C∞(M,R). (3.4)
Notice that, by a minor abuse of notation, we are supposing that eachTmis a distribution onT2× {0} ⊂T3A.
Once again, simple computations let us show that the distributionsT˜mareX-invariant and linearly independent, which clearly contradicts Proposition 2.4.
Therefore, it must holdn0=0, andMis diffeomorphic toT3. By Proposition 2.6,Xis smoothly conjugated to a Diophantine vector field onT3. 2
Lemmas 3.2 and 3.3 will be proved in the following two paragraphs.
3.1. Torus bundle structure
In this paragraph we shall prove Lemma 3.2, so we will continue working under the hypothesis of Theorem 3.1.
Sinceβ1(M)1, we can apply Theorem 2.8 to affirm that there exists a Diophantine projection forXoverT1, i.e. a smooth fibrationp:M→T1such thatDp(X)≡α∈R\ {0}. This implies that, for allθ∈T1,Mθ=. p−1(θ )is a transverse section forX, and by the minimality of{ΦXt },Mθ must be a global section. So, the Poincaré first return map toMθis given by the diffeomorphism
Pθ .
=ΦXα−1
Mθ:Mθ→Mθ.
Clearly,Pθ cannot exhibit any periodic point. Then, the Euler characteristic ofMθ must vanish [6]. So we can conclude that any fiber ofpis diffeomorphic to a disjoint union ofkcopies ofT2. Let us notice that we do not lose any generality assuming thatk=1. In fact, ifEk:θ→kθ is the standardk-fold covering of T1, then it is easy to verify that we can find another fibrationp˜:M→T1satisfying
M
˜
p p
T1 Ek T1
From this commutative diagram it follows that eachp-fiber is a connected component of a˜ p-fiber, andp˜is clearly a Diophantine projection forXtoo.
To simplify our notation, we will suppose that our original fibrationphas connected fibers. Therefore,pis a 2-torus bundle overT1, and sinceM is orientable, there exist a matrixA∈SL(2,Z)and a diffeomorphismH:M→T3A verifying pr2◦H=p, whereT3A .
=T2×R/(x, t )∼(Ax, t−1)and pr2:T3A→T1is the projection on the second factor.
From this observation it easily follows thatL(Pθ), the Lefschetz number ofPθ, equals to det(A−id). In particular, by Lefschetz fixed point theorem, we conclude that the spectrum ofAis equal to{1}. Then, we can suppose thatA coincides with its Jordan form, i.e.
A=
1 0 n0 1
.
Finally, letΣA:T2×R→T3Abe the quotient projection induced by the suspension ofA∈SL(2,Z)⊂Diff(T2).
Then,ΣAis clearly a covering map and its deck group is generated by T2×R
(θ1, θ2), t
→
(θ1−n0θ2, θ2), t+1 .
This implies that the formsdθ2, dt∈Λ1(T2×R)are invariant under the action of the deck group ofΣA. Therefore, both forms can be pushed forward byΣAgetting two closed forms onT3Awhich clearly induced linear independent elements ofH1(T3A). In particular, this impliesβ1(M)2, as desired.
3.2. Smooth linearization
This paragraph is devoted to prove Lemma 3.3. The main ingredients of the proof are Theorem 2.8, Herman–Yoccoz linearization theorem for smooth circle diffeomorphisms [8,31], and the following “foliated” version of Moser isotopy theorem for volume forms [17], which is nothing but a two-dimensional reformulation of Theorem 6.1 in [29]:
Proposition 3.4.LetΩ1, Ω2∈Λ2(T2)be two volume forms and suppose they satisfy:
Ω1
pr−11(C)
=Ω2
pr−11(C) ,
for every Borel measurable setC⊂T1and where we are consideringΩ1andΩ2as elements ofM(T2). Then there existsH∈Diff(T2)isotopic to the identity verifying
H∗Ω1=Ω2, and H∗dθ1=dθ1, where(θ1, θ2)are the canonical coordinates ofT2.
By Lemma 3.2, we know that β1(M)2, and hence, by Theorem 2.8, there exists a Diophantine projection q:M→T2forX. Let us write
α=(α1, α2) .
=Dq(X)∈R2. (3.5)
If pr1:T2(θ1, θ2)→θ1∈T1 denotes the projection on the first coordinate, we can define the mapp .
=pr1◦ q:M→T1, which is a Diophantine projection forX, too.
Repeating an argument analogous to that used in Section 3.1 we can assume that the fibers ofpandqare connected, and therefore,p-fibers (q-fibers) are diffeomorphic toT2(T1, respectively).
LetΩ∈Λ3(M)be the normalized (i.e.
MΩ=1 when we have already fixed an orientation onM)X-invariant smooth volume form (see Proposition 2.4).
Ifθ¯is any point inT1, andP:p−1(θ )¯ →p−1(θ )¯ denotes the Poincaré first return map top−1(θ ), then¯ ω .
=iXΩ
p−1(θ )¯ ∈Λ2 p−1(θ )¯
,
is a smoothP-invariant area form onp−1(θ ).¯
On the other hand, every q-fiber is contained in a p-fiber, and then, on each p-fiber we have a 1-dimensional foliation, whose leaves are all diffeomorphic toT1, which is invariant under the action ofP.
It is well-known that any two smooth codimension-one foliations inT2with all their leaves compact are diffeo- morphic. Hence, we can find aC∞diffeomorphismH0:p−1(θ0)→T2verifying
DH0
T
q−1(θ , θ )¯
=kerdθ1, ∀θ∈T1. (3.6)
where(θ1, θ2)are the canonical coordinates inT2.
Notice that Eq. (3.6) lets us affirm thatH0◦P◦H0−1is a skew-product overT1, i.e. there existsh0∈Diff+(T1) andη∈C∞(T2,R)such that
H0
P
H0−1(θ1, θ2)
=
h0(θ1), θ2+n0θ1+
η(θ1, θ2)+Z
. (3.7)
On the other hand, the rotation number ofh0is equal toα2α−11+Z, whereα=(α1, α2)is given by (3.5). Taking into account thatαis Diophantine, it is easy to verify that there exist positive real constantsCandτ satisfying
m−nα2 α1
C
nτ, ∀m∈Z, ∀n∈N. (3.8)
Therefore, by Herman–Yoccoz linearization theorem [8,31] we know thath0is smoothly conjugated to the rotation θ→θ+(α2α−11+Z). So, we do not lose any generality supposing that in (3.7)h0is indeed equal to the rotation, and this is what we will do to simplify the notation.
In particular, this implies that Leb2 andH0−1∗ω satisfy the hypothesis of Proposition 3.4, so there existsH1∈ Diff+(T2)preserving the vertical foliation{{θ1} ×T1}θ1∈T1 ofT2and such that
H1∗Leb2=H0−1∗ω. (3.9)
From (3.7) and (3.9) we can conclude thatH1◦H0◦P◦H0−1◦H1−1is an area-preserving skew-product inT2, and therefore,
H1
H0
P
H0−1(H1−1(θ1, θ2))
=
θ1+ α2
α1+Z
, θ2+n0θ1+
ζ (θ1)+Z
, (3.10)
for someζ∈C∞(T1,R).
On the other hand, using Fourier series techniques analogous to those used to prove Proposition 2.6 we can easily show that, sinceα2α1−1satisfies estimate (3.8), the rotationθ→θ+(α2α1−1+Z)is cohomologically rigid. Therefore, we can findu∈C∞(T1,R)verifying
u θ+
α2α2−1+Z
−u(θ )= −ζ (θ )+
T1
ζdLeb1, ∀θ∈T1. (3.11)
Finally, conjugatingH1◦H0◦P◦H0−1◦H1−1with the diffeomorphism (θ1, θ2)→
θ1, θ2+u(θ1) ,
we get an affine 2-torus automorphism whose linear part is equal toA, as desired.
4. Vanishing first Betti number
In this section we complete the proof of Theorem A, proving the following
Theorem 4.1.IfM is a closed orientable3-manifold so thatH1(M,R)=0, thenMdoes not support any cohomo- logically rigid smooth vector field.
So, from now on we shall assume thatMsatisfies the hypothesis of Theorem 4.1, and by contradiction, we will suppose thatX∈X(M)is a cohomologically rigid vector field.
Our first step to prove Theorem 4.1 is the following
Proposition 4.2.There existsλ∈Λ1(M)such thatλ(p)=0, for everyp∈M, and LXλ≡0 and iXdλ≡0.
Proof. By Proposition 2.3 we know that there exists anX-invariant volume form Ω∈Λ3(M). Hence, if we write ω .
=iXΩ, Cartan’s formula lets us affirm 0=LXΩ=d(iXΩ)+iX(dΩ)=dω.
Notice that, by Poincaré duality,H2(M,R)=0. Then, there exists a 1-formλ˜ such thatω=dλ. Applying Cartan’s˜ formula once again we obtain
LXλ˜=d(iXλ)˜ +iX(dλ)˜ =d(iXλ)˜ +iX(iXΩ)=d(iXλ),˜
whereiXλ˜ is an element ofC∞(M,R). So, there exists a smooth functionu:M→Rverifying LXu= −iXλ˜+
M
(iXλ)Ω.˜ (4.1)
Therefore, if we defineλ .
= ˜λ+du, it still holdsdλ=ω, andiXdλ=iX(iXΩ)≡0. Moreover, LXλ=LXλ˜+LXdu=d(iXλ)˜ +d(iXdu)
=d
iXλ˜+LXu
=d
M
(iXλ)Ω˜
=0.
Then, taking into account the minimality of{ΦXt }, we easily see thatλexhibits a singularity if and only ifλ≡0.
On the other hand, sincedλ=iXΩ=0, we know thatλ≡0, and so,λdoes not have any singularity. 2
Notice that, since dimM=3, it holds 0=iX(λ∧Ω)=(iXλ)Ω−λ∧iXΩ
=(iXλ)Ω−λ∧ω=(iXλ)Ω−λ∧dλ.
That is
λ∧dλ=(iXλ)Ω, (4.2)
whereiXλis a real constant.
In the following paragraphs we will analyze separately the cases whereiXλ=0 andiXλ=0.
4.1. The contact case:iXλ=0
WheniXλ=0, Eq. (4.2) implies thatλ is a contact form. On the other hand, by Proposition 4.2 we know that iXdλ≡0, and therefore,(iXλ)−1Xis the Reeb vector field induced byλ. And then we easily get to a contradiction invoking the famous Weinstein conjecture [30], which was recently proven by Taubes:
Theorem 4.3. (Taubes [27].) Let N be a closed orientable3-manifold,η∈Λ1(N )be a smooth contact form and Y ∈X(N )be its Reeb vector field(i.e.iYη≡1andiYdη≡0). Then, the flow induced byY exhibits a periodic orbit.
Clearly, the existence of a periodic orbit is not compatible with the minimality of{ΦXt }, and we get the desired contradiction.
Nevertheless, at this point we must recognize that it would be very desirable to solve this case not invoking Taubes’
proof of Weinstein conjecture. In fact, in a certain way the Katok conjecture is just the first step toward the compre- hension of how the topology of the manifold generates obstructions when we try to solve cohomological equations.
And from this point of view the sophistication of the techniques used by Taubes in [27] does not let us understand what is really happening in this case.
4.2. The completely integrable case:iXλ=0
When iXλ=0, Eq. (4.2) implies that kerλ is a completely integrable plane field, i.e. there exists a smooth codimension-one foliation inMtangent to kerλ.
While this work was in progress, Forni communicated to the author that he had been able to solve this case using the foliation tangent to kerλto prove thatMshould be diffeomorphic to a nilmanifold and{ΦXt}smoothly conjugated to a homogeneous flow. On the other hand, Greenfield and Wallach had already proved in [7] thatT3 was the only 3-dimensional nilmanifold that supported cohomologically rigid homogeneous vector fields (this result was extended by Flaminio and Forni [4] to higher dimensional nilmanifolds).
Our techniques are completely different to those of Forni [5]. The main novelty of ours consists in using the integrability condition in a very indirect way, and this lets us believe that most of our proof could be reusable to solve the “contact case” independently of Taube’s proof of Weinstein conjecture [27].
Very roughly, our strategy consists in doing a very detailed analysis of the dynamics of the tangent flow {ΦXt}, proving that the normal flow as well as the tangent flow restricted to the kernel ofλexhibit a “parabolic behavior”.
Then, we shall see that this implies that our original flow{ΦXt }should be positively expansive (see Definition 4.12).
Finally, we show that there does not exist positively expansive flows on closed 3-manifolds.
4.2.1. Two simple properties of the normal flow
In this short paragraph we start the analysis of the dynamics of the normal flow{N ΦXt }, presenting two very simple lemmas.
Let us start introducing any smooth Riemannian structure·,·inT M. This naturally induces another Riemannian structure·,·N XinN Xdefining, for eachp∈M,
v1, v2N X
=. v1, v2
, ∀v1, v2∈N Xp,
wherevi is the only element of TpM verifying simultaneously X(p), vi(p) =0 and prX(vi)=vi. The Finsler structures induced by·,·and·,·N X will be denoted by · and · N X, respectively. As usual, we shall also use the Riemannian structures·,·and·,·N X to measure angles between non-null vectors of the same fiber. Making some abuse of notation, we shall use the symbol(·,·)for both.
Now, we can present our first result about the dynamics of our normal flow{N ΦXt }:
Lemma 4.4.There existsvˆ0∈N Xsuch thatvˆ0=0and sup
t∈R
N ΦXt (vˆ0)
N X<∞. (4.3)
Proof. Let us suppose estimate (4.3) is not satisfied by any non-vanishing vector inN X. In other words, let suppose thatN ΦX:R×N X→N Xis quasi-Anosov. By Theorem 2.9,{N ΦXt }is an Anosov cocycle. Then, Theorem 2.10 lets us affirm thatXis indeed Anosov.
Finally, it is a very well-known fact that any Anosov flow exhibits (infinitely many) periodic orbits [1], which clearly contradicts the minimality of{ΦXt }. 2
Our second result about the dynamics of the normal flow is the following
Lemma 4.5.The normal flow{N ΦXt }is conservative. More precisely, there exists a symplectic formκ on the vector bundleπN:N X→Mwhich is invariant under the action of{N ΦXt}.
Proof. Notice thatω=iXΩ=dλis a 2-form onT MverifyingiXω≡0. This implies that we may push-forward this form by prXonN X, i.e. we can find a smooth 2-formκ onN Xsuch that
κ
prX(v),prX(w)
=ω(v, w), ∀v, w∈TpM, ∀p∈M.
Now, it is very easy to verify thatκ is symplectic onN XandN ΦX-invariant. 2 4.2.2. Dynamics of the projective flow
This paragraph is devoted to prove that the dynamics of the projective flow is very simple. In fact, we shall get that the limit set of{P ΦXt }is a smooth submanifold ofP(N X)which happens to be a graph overM, being the dynamics on this set smoothly conjugated to{ΦXt}.
For this, first we need the following result due to Nakayama and Noda about the geometry and amount of minimal sets of the projective flow:
Theorem 4.6.(Nakayama and Noda [18].) LetV be a closed3-manifold and letY ∈X(V )be such that its induced flowΦY:R×V→V is minimal.
LetP ΦY:R×P(N Y )→P(N Y )be the projective flow ofY. Hence, we have:
(1) If{P ΦYt}exhibits more than two minimal sets, thenV is diffeomorphic toT3and{ΦXt }is continuously conjugate to an irrational translation.
(2) If {P ΦYt}exhibits exactly two minimal setsM1, M2⊂P(N Y ) and {ΦXt}is not C0-conjugate to an irrational translation onT3, then for anyz∈V it holds:M1∩πP−1(z)orM2∩πP−1(z)consists of a single point. Moreover, there exists a residual subsetB⊂V such that both setsM1∩πP−1(z)andM2∩πP−1(z)contain just a point, for everyz∈B.
Since we are assuming thatH1(M,R)=0, Theorem 4.6 lets us affirm that the flow{P ΦXt }exhibits at most two minimal sets. Then, observe that one of the minimal sets (in fact, we will prove this is the only one) is given by the prP-projection of the plane field
Σ .
=kerλ⊂T M.
Indeed, for anyp∈Mwe haveX(p)∈Σp, and hence, EΣ .
=prX Σ\ {0}
⊂N X, (4.4)
is a smooth one-dimensional linear sub-bundle ofN X. In this way,EΣ determines exactly one point on each fiber of πP:P(N X)→M. More precisely, we may defineθp=. prP(EΣ p\ {0})∈πP−1(p).
Notice that since the plane fieldΣis invariant under the action of{DΦXt }, we have the flow{N ΦXt }leaves invariant the line fieldEΣ, and therefore, it holdsP ΦXt (θp)=θΦt
X(p), for everyp∈Mand everyt∈R. So, summarizing we have
KΣ .
= {θp: p∈M} ⊂P(N X) (4.5)
is a minimal set for{P ΦXt }.
As it was already mentioned above, we aim to prove the following
Theorem 4.7.The only minimal set for the flow{P ΦXt}isKΣ⊂P(N X), defined in(4.5).
To prove Theorem 4.7 we shall suppose that there exists another minimal setK0⊂P(N X)(of course, different fromKΣ), and for the sake of clarity of the exposition we will divide the proof in several lemmas:
Lemma 4.8.The sub-bundleEΣ⊂N Xdefined in(4.4)is orientable, and therefore, it admits a non-vanishing section Yˆ0∈Γ (EΣ).
Proof. SinceMis orientable andΣwas defined as the kernel of a non-singular 1-form, the vector bundleπ|Σ:Σ→ Mis orientable. On the other hand, our vector fieldXcan be considered as a non-singular element ofX∈Γ (Σ ). This lets us affirm that Σ→Mis a globally trivial vector bundle. Therefore, we can find a smooth section Y0∈Γ (Σ ) verifyingΣp=span{X(p), Y0(p)}, for everyp∈M.
Finally, definingYˆ0=. prX(Y0)we get our desired section ofEΣ→M. 2
Lemma 4.9.Assuming that there exists another minimal setK0⊂P(N X)(different fromKΣ), we can find a non- vanishingYˆ∈Γ (EΣ)verifying
N ΦXtY (p)ˆ
= ˆY ΦXt (p)
, ∀p∈M, ∀t∈R. (4.6)
Proof. LetLΣ∈C∞(M,R)be defined by LΣ(p)Yˆ0(p)=lim
t→0
N ΦX−t(Yˆ0(ΦXt (p)))− ˆY0(p)
t , ∀p∈M, (4.7)
whereYˆ0is the smooth section ofEΣ given by Lemma 4.8.
Using the fact thatXis cohomologically rigid, we can find a functionu∈C∞(M,R)verifying LXu= −LΣ+
M
LΣΩ. (4.8)
Then, if we defineYˆ =. euYˆ0, we clearly get
tlim→0
N ΦX−t(Y (Φˆ Xt(p)))− ˆY (p)
t =
M
LΣΩ
Y (p),ˆ ∀p∈M,
and therefore, it holds N ΦXtY (p)ˆ
=exp
t
M
LΣΩ Yˆ
ΦXt (p)
, (4.9)
for everyp∈Mand everyt∈R.
Notice that by Eq. (4.9),
MLΣΩ is a Lyapunov exponent of the linear cocycle{N ΦXt }. So, let us suppose that
MLΣΩ=0. In this case, the one-dimensional sub-bundleEΣ⊂N Xis uniformly hyperbolic.
On the other hand, by Theorem 4.6 we know that K0 and KΣ are the only two minimal sets in P(N X), and moreover, we can find a pointp0∈Msuch thatθ∈P(N X)is the only point inK0∩πP−1(p0).
Observe that sinceK0andKΣ are disjoint and closed, there exists a real constantC >0 such that distP
P ΦXt (θp0), P ΦXt (θ)
> C, ∀t∈R, (4.10)
where distPdenotes the distance function onP(N X)induced by the Riemannian structure·,·N X.
Then, by conservativeness proved in Lemma 4.5, estimate (4.10) and Eq. (4.9) we have that any vectorvˆ∈N Xp0 whose prX-projection is equal toθ∈K0∩πP−1(p0)will satisfy the following estimate:
N Φxt(v)ˆ
N XCexp
−t
M
LΣΩ
ˆvN X, ∀t∈R, (4.11)
for some real constantC>0, which just depends on constantCin (4.10).
From Eq. (4.9) and estimate (4.11) (and supposing that
MLΣΩ=0), we clearly conclude that Oseledets splitting (see [19]) of{N ΦXt }({N ΦXt}can be thought as a linear cocycle over{ΦXt }) is not just measurable, but continuous and uniformly hyperbolic. This implies that{N ΦXt}is an Anosov cocycle, and by Theorem 2.10 we know thatXmust be Anosov, which is clearly impossible since{ΦXt }does not have any periodic orbit.
Therefore, the contradiction arises from our assumption that
MLΣΩ=0. Finally, Eq. (4.9) lets us assure thatYˆ is aN ΦX-invariant section, as desired. 2
Now, we are ready to complete the
Proof of Theorem 4.7. Let K0⊂P(N X),p0∈M,Yˆ ∈Γ (EΣ)andθ∈K0∩πP−1(p0)⊂P(N X)as above. Let ˆ
v∈N Xp0such that prP(v)ˆ =θ. We can rewrite estimate (4.10) as
tinf∈RYˆ ΦXt (p)
, N ΦXt (v)ˆ
>0. (4.12)
Putting together Eq. (4.6), estimate (4.12) and Lemma 4.5 we see that there exists a real constantC>1 so that 1
C <N ΦXt(v)ˆ
N X< C, ∀t∈R. (4.13)
Now, consider another vectorwˆ∈N Xp0\{0}such that prP(w)ˆ ∈KΣ∪K0. SinceKΣ andK0are the only minimal sets for{P ΦXt }, we know that theω-limit of prP(w)ˆ must intersects eitherK0orKΣ. Let us suppose that the positive semi-orbit of prP(w)ˆ accumulates onKΣ. This implies that
lim inf
t→+∞Yˆ
ΦXt (p0)
, N ΦXt (w)ˆ
=0. (4.14)
Once again, taking into account that{N ΦXt }preserves the symplectic formκand the sectionYˆ∈Γ (N X), we see that Eq. (4.14) implies that
lim sup
t→+∞
N Φxt(w)ˆ
N X= ∞. (4.15)
Finally, we clearly see that estimates (4.12), (4.13) and (4.15) violate conservativeness.
Analogously we can get a contradiction supposing that theω-limit of prP(w)ˆ intersectsK0. In this way we conclude thatKΣ is the only minimal set for{P ΦXt}. 2
4.2.3. Dynamics of the normal flow
In Section 4.2.1 we begun the analysis of the dynamics of the normal flow{N Φxt}. After what we have just done in Section 4.2.2, here we shall see that some of those results can be considerably improved. In fact, we will completely characterize the dynamics of{N ΦXt }, showing that it exhibits a parabolic behavior.
In Lemma 4.4 we showed that there was some non-null vector inN Xsuch that its wholeN ΦX-orbit was bounded.
On the other hand, in Lemma 4.9, under the assumption that there were two different minimal sets for{P ΦXt }, we proved there existedYˆ∈Γ (EΣ)which was invariant under the action of{N ΦXt }. Our first result of this paragraph consists in proving that we can get the same invariant section assuming, in this case, thatKΣ is the only minimal set:
