Dominated splitting and zero volume for incompressible three-flows
aa r X i v : . [ m a t h . D S ] A ug DOMINATED SPLITTING AND ZERO VOLUME FORINCOMPRESSIBLE THREE-FLOWS
VITOR ARAUJO AND M ´ARIO BESSAA
BSTRACT . We prove that there exists an open and dense subset of theincompressible 3-flows of class C such that, if a flow in this set has apositive volume regular invariant subset with dominated splitting for thelinear Poincar´e flow, then it must be an Anosov flow. With this result weare able to extend the dichotomies of Bochi-Ma˜n´e (see [26, 13, 9]) andof Newhouse (see [30, 10]) for flows with singularities. That is we obtainfor a residual subset of the C incompressible flows on 3-manifolds that:(i) either all Lyapunov exponents are zero or the flow is Anosov, and (ii)either the flow is Anosov or else the elliptic periodic points are dense inthe manifold. Keywords: generic incompressible flows, Lyapunov exponents, dominatedsplitting, hyperbolicity.
Primary: 37D25; Secondary:37D35,37D50, 37C40. C
ONTENTS
1. Introduction 21.1. Definitions and statement of the results 31.2. Overview of the arguments and organization of the paper 7Acknowledgments 72. Generic dichotomies for incompressible flows 73. Dominated splitting and regularity 123.1. Bounded angles, eigenvalues and Lorenz-like singularities 133.2. Invariant manifolds of a positive volume set with dominatedsplitting for the Linear Poincar´e Flow 144. Uniform hyperbolicity 204.1. Positive volume hyperbolic sets and conservative Anosovflows 20References 21
Date : October 30, 2018.V.A. was partially supported by CNPq, FAPERJ and PRONEX-Dynamical Systems(Brazil) and FCT (Portugal) through CMUP and POCI/MAT/61237/2004. M.B. was par-tially supported by FCT-FSE, SFRH/BPD/20890/2004.
1. I
NTRODUCTION
Incompressible flows are a traditional subject from Fluid Mechanics, seee.g. [20]. These flows are associated to divergence-free vector fields, theypreserve a volume form on the ambient manifold and thus come equippedwith a natural invariant measure. On compact manifolds this provides aninvariant probability giving positive measure (volume) to all nonempty opensubsets. Therefore for vector fields X in this class we have Ω ( X ) = M bythe Poincar´e Recurrence Theorem, where Ω ( X ) denotes the non-wanderingset. In particular such flows can have neither sinks nor sources, and ingeneral do not admit Lyapunov stable sets, either for the flow itself or forthe time reversed flow.Let X r ( M ) be the space of C r vector fields, for any r ≥
1, and X rµ ( M ) thesubset of divergence-free vector fields defining incompressible (or conser-vative) flows. It is natural to study these flows under the measure theoreticpoint of view, besides the geometrical one.The device of Poincar´e sections has been used extensively to reduce sev-eral problems arising naturally in the setting of flows to lower dimensionalquestions about the behavior of a transformation. Recent breakthroughs onthe understanding of generic volume-preserving diffeomorphisms on sur-faces have non-trivial consequences for the dynamics of generic incom-pressible flows on three-dimensional manifolds.The Bochi-Ma˜n´e Theorem [13] asserts that, for a C residual subset ofarea-preserving diffeomorphisms, either the transformation is Anosov (i.e.globally hyperbolic), or the Lyapunov exponents are zero Lebesgue almosteverywhere (i.e. there is no asymptotic growth of the length of vectors inany direction for almost all points). This was announced by Ma˜n´e in [24]but only a sketch of a proof was available in [26]. The complete proofpresented by Jairo Bochi in [13] admits extensions to higher dimensions,obtained by Bochi and Viana in [15], stating in particular that either theLyapunov exponents of a C generic volume-preserving diffeomorphism arezero Lebesgue almost everywhere, or else the system admits a dominatedsplitting for the tangent bundle dynamics. A survey of this theory can befound in [14].Recently (see Theorem 1.1 below) one of the coauthors was able to use,adapt and fully extend the ideas of the original proof by Bochi to the settingof generic conservative flows on three-dimensional compact boundarylessmanifolds without singularities , in [9]. The presence of singularities im-poses some differences between discrete and continuous systems. The ideasfrom the Bochi-Ma˜n´e proof were partially extended to a dense subset of C incompressible flows (see Theorem 1.2 below) admitting singularities but OMINATED SPLITTING AND ZERO VOLUME 3 without a full dichotomy between zero exponents and global hyperbolicityin the same work [9].There are related results from Arbieto-Matheus in [4], where it is provedthat C robustly transitive volume-preserving 3-flows must be Anosov, withthe help of a new perturbation lemma for divergence-free vector fields, andalso from Horita-Tahzibi in [22], where it is proved that robustly transitivesymplectomorphisms must be partially hyperbolic. One of the coauthors to-gether with Rocha proved in [11] that robustly transitive volume-preserving n -flows must have dominated splitting.There are older C dichotomy results for low dimensional transforma-tions. A result of fundamental importance in the theory of generic conserva-tive diffeomorphisms on surfaces was obtained by Newhouse in [30]. New-house’s theorem states that C generic area-preserving diffeomorphisms onsurfaces either are Anosov, or else the elliptical periodic points are dense.A refined version of this results was presented by Arnaud in [5] in the fam-ily of 4-dimensional symplectomorphisms. Even more recently Saghin-Xia [38] generalized Arnauld result for the multidimensional symplecticcase, and in [10] one of the coauthors together with Duarte obtained a simi-lar dichotomy for C -generic incompressible flows without singularities on3-manifolds: either the flow is Anosov, or else the elliptic periodic orbitsare dense in the manifold.Here we complete the results of [9] and [10] fully extending the di-chotomy from generic non-singular vector fields to generic vector fieldsin the family of C all incompressible flows on 3-manifolds.The main step is our arguments is to show that if a C incompressible flowon a 3-manifold admits a positive volume invariant subset (not necessarilyclosed) formed by regular orbits with a very weak form of hyperbolicity,known as dominated decomposition , then there cannot be any singularity onthe closure of this set, under a mild non-resonant conditions on the possibleeigenvalues at the singularities. This leads easily to the conclusion that theclosure of this invariant subset is a positive volume hyperbolic subset.Adapting arguments from Bochi-Viana [14] to the flow setting it is provedthat incompressible C flows with positive volume compact invariant hyper-bolic sets must be globally hyperbolic. Finally using standard argumentsfrom Bochi-Viana [14], [9] and [10] these results imply the C genericdichotomies mentioned above for incompressible flows without any extracondition on the singularities.1.1. Definitions and statement of the results.
In what follows M willalways be a C ¥ compact connected boundaryless three-dimensional Rie-mannian manifold. We denote by µ a volume form on M and by dist the dis-tance induced on M by the Riemannian scalar product, denoted by < · , · > . VITOR ARAUJO AND M ´ARIO BESSA
We begin by recalling Oseledets’ Theorem for measure preserving flowsand the notion of
Linear Poincar´e Flow first introduced by Doering in [18].Consider X ∈ X µ ( M ) and the associated flow X t : M → M . Oseledets’Theorem [31] guarantees that we have, for µ -a.e. point x ∈ M , a measur-able splitting of the tangent bundle at x , T x M = E x ⊕ ... ⊕ E k ( x ) x , called the Oseledets splitting and real numbers l ( x ) > ... > l k ( x ) ( x ) called Lyapunovexponents such that DX tx ( E ix ) = E iX t ( x ) andlim t →± ¥ t log k DX tx · v i k = l i ( x ) for any v i ∈ E ix \ ~ i =
1, ..., k ( x ) . Oseledets’ Theorem allow us to con-clude also that lim t →± ¥ t log | det ( DX tx ) | = k ( x ) (cid:229) i = l i ( x ) . dim ( E ix ) (1.1)which is related to the sub-exponential decrease of the angle between anysubspaces of the Oseledets splitting along µ -a.e. orbits. Since DX tx ( X ( x )) = X ( X t ( x )) the direction of the vector field is one of the Oseledets subspacesand it is associated to a zero Lyapunov exponent. The full µ -measure subsetof points where these exponents and directions are defined will be referredto as the set of Oseledets points of X .In the volume-preserving setting we have | det ( DX tx ) | =
1. Hence on 3-manifolds by (1.1) either l ( x ) = − l ( x ) > l ( x ) > E ux and E sx respectively associated to l ( x ) and l ( x ) which we denote by l u ( x ) and l s ( x ) .We say that s ∈ M is a singularity of X if X ( s ) = ~ S ( X ) the set of all singularities of X . The complement M \ S ( X ) is the setof regular points for the flow of X . For a regular point z of X denote by N z = { v ∈ T z M : < v , X ( z ) > = } the orthogonal complement of the flow direction [ X ] z = [ X ( z )] : = R · X ( z ) in T z M . Denote by O z : T z M → N z the orthogonal projection of T z M onto N z . For every t ∈ R define P tX ( z ) : N z → N X t ( z ) by P tX ( z ) = O X t ( z ) ◦ DX tz .It is easy to see that P = { P tX ( z ) : t ∈ R , X ( z ) , ~ } satisfies the cocycleidentity P s + tX ( z ) = P sX ( X t ( z )) ◦ P sX ( z ) for every t , s ∈ R .The family P is called the Linear Poincar´e Flow of X .If we have an Oseledets point x < S ( X ) and l ( x ) >
0, the Oseledetssplitting on T x M induces a P tX -invariant splitting on N x , say O x ( E ⋄ x ) = N ⋄ x OMINATED SPLITTING AND ZERO VOLUME 5 for ⋄ = u , s . If l ( x ) =
0, then the P tX -invariant splitting is trivial. Using(1.1) it is easy to see that the Lyapunov exponents of P tX ( x ) associated tothe subspaces N ux and N sx are respectively l u ( x ) ≥ l s ( x ) ≤ regular invariant subset Λ for X ∈ X ( M ) , that is Λ ∩ S ( X ) = /0 , an in-variant splitting N ⊕ N of the normal bundle N Λ for the Linear Poincar´eFlow P tX is said to be m-dominated , if there exists an integer m such that for every x ∈ Λ we have the domination relation (cid:13)(cid:13) P mX ( x ) | N (cid:13)(cid:13)(cid:13)(cid:13) P mX ( x ) | N (cid:13)(cid:13) ≤
12 . (1.2)Dominated splittings are automatically continuous on the Grassmanian ofplane subbundles of the tangent bundle, see e.g. [21, 16] for an expositionof the theory. In particular the dimensions of the subbundles are constanton each connected component of Λ .As is traditional we say that a vector field is Anosov if the flow preserves aglobally defined hyperbolic structure, that is, the tangent bundle
T M splitsinto three continuous DX t -invariant subbundles E ⊕ [ X ] ⊕ G where [ X ] isthe flow direction, the sub-bundle E , ~ G , ~ DX t for t >
0. Note that foran Anosov flow X the entire manifold is m -dominated for some m ∈ N .The fact that the dimensions of the subbundles are constant on the entiremanifold implies that S ( X ) = /0 for an Anosov vector field.Denote by X rµ ( M ) ⋆ the subset of X rµ ( M ) of C r incompressible flows but without singularities . Theorem 1.1. [9, Theorem 1]
There exists a residual set R ⊂ X µ ( M ) ⋆ suchthat, for X ∈ R , either X is Anosov or else for Lebesgue almost every p ∈ Mall the Lyapunov exponents of X t are zero. Developing the ideas of the proof of this result one can also obtain thefollowing statement on denseness of dominated splitting, now admittingsingularities.
Theorem 1.2. [9, Theorem 2]
There exists a dense set D ⊂ X µ ( M ) suchthat for X ∈ D , there are invariant subsets D and Z whose union has fullmeasure, such that • for p ∈ Z the flow has only zero Lyapunov exponents; • D is a countable increasing union Λ m n of invariant sets admittingan m n -dominated splitting for the Linear Poincar´e Flow, where m n is a strictly increasing integer sequence. We recall another C -type result for incompressible three-dimensionalflows without fixed points. Preliminary versions for the discrete symplectic VITOR ARAUJO AND M ´ARIO BESSA case were presented in [30, 5, 38] respectively for surfaces, 4-dimensionalmanifolds and 2 n -dimensional manifolds. Theorem 1.3. [10, Theorem 1.2]
Given e > , any open subset U of M anda non Anosov vector field X ∈ X µ ( M ) ⋆ , there exists Y ∈ X µ ( M ) ⋆ such thatY is e -C -close to X and Y has an elliptic closed orbit intersecting U . We are able to extend Theorem 1.1 and Theorem 1.3 to the full family ofincompressible C flows. Here are our main results. Theorem A.
There exists a generic subset R ⊂ X µ ( M ) such that for X ∈ R • either X is Anosov, • or else for Lebesgue almost every p ∈ M all the Lyapunov exponentsof X t are zero. Theorem B.
Let e > , an open subset U of M and a non Anosov vector fieldX ∈ X µ ( M ) be given. Then there exists Y ∈ X µ ( M ) such that Y is C - e -closeto X and Y t has an elliptic closed orbit intersecting U . From Theorem B we can follow ipsis verbis the proof of Theorem 1.3of [10] to deduce the next generic result.
Corollary 1.4.
There exists a C residual set R ⊂ X µ ( M ) such that if X ∈ R , then X is Anosov or else the elliptic closed orbits of X are dense in M. It is well known that a C dynamical system admitting a hyperbolicset with positive measure must be globally hyperbolic: see e.g. Bowen-Ruelle [17] and Bochi-Viana [14]. Recently in [2] this was extended totransitive sets having a weaker form of hyperbolicity called partial hyper-bolicity with the extra assumption of non-uniform expansion along the cen-tral direction. Also in [1] similar results where obtained for positive volume singular-hyperbolic sets for C (not necessarily incompressible) flows.We extend these results for an even weaker type of hyperbolicity, i.e. forsets with a dominated splitting. Both Theorems A and B are deduced fromthe following result. Theorem C.
There exists an open and dense subset G ⊂ X µ ( M ) such thatfor every X ∈ G with a regular invariant set Λ (not necessarily closed) sat-isfying: • the Linear Poincar´e Flow over Λ has a dominated decomposition;and • Λ has positive volume: µ ( Λ ) > ;then X is Anosov and the closure of Λ is the whole of M. OMINATED SPLITTING AND ZERO VOLUME 7
Overview of the arguments and organization of the paper.
Theproofs of Theorems A and B follow standard arguments from Bochi [6], [9]and [10] assuming Theorem C together with the denseness of C incom-pressible flows among C incompressible ones given by Zuppa in [40]. Wepresent these arguments in the following Section 2.We give now an outline of the proof of Theorem C. Fix X ∈ X µ ( M ) and assume that there exists an invariant subset Λ for X (not necessarilycompact) without singularities (i.e. formed by regular orbits of X ) and withpositive volume: µ ( Λ ) >
0. We show that(1) the closure A of Λ cannot contain singularities.This is done in Section 3 combining arguments from the characterization ofrobustly transitive attractors in [28], with properties of positive volume in-variant subsets from [2] and of hyperbolic smooth invariant measures from Pesin’s Theory [33, 8], together with the arguments from [14, Appendix B].(2) If A is a compact invariant set without singularities and with dom-inated decomposition of the Linear Poincar´e Flow, then A is a uni-formly hyperbolic set.This is a well-known result from [9] and the work of Morales-Pacifico-Pujals in [28].(3) a uniformly hyperbolic set A with positive volume for a C incom-pressible flow must be the whole M .For the last item above we adapt the arguments from [14, Appendix B] tothe flow setting. Acknowledgments.
V.A. wishes to thank IMPA for its hospitality, excel-lent research atmosphere and access to its superb library. M.B. wishes tothank CMUP and the Pure Mathematics Department of University of Portofor access to its facilities and library during the preparation of this work.2. G
ENERIC DICHOTOMIES FOR INCOMPRESSIBLE FLOWS
Here we prove Theorems A and B assuming Theorem C.We start with a sequence of simple lemmas. We say that the vector field X is aperiodic if the volume of the set of all closed orbits for the correspondingflow is zero. Lemma 2.1.
There exists a C -dense set D ⊆ X µ ( M ) such that if X ∈ D ,then • X is aperiodic; • X is of class C r for some r ≥ ; and • every invariant m-dominated set Λ has zero or full measure, for anym ∈ N . VITOR ARAUJO AND M ´ARIO BESSA
Proof.
Let
K S be the C r generic subset given by [36, Theorem 1(i)], forsome r ≥
2, so that X ∈ K S is C r and admits countably many closed orbitsonly, all of which are hyperbolic or elliptic. According to the results in [40], X rµ ( M ) is also C -dense on X µ ( M ) , for r ≥
2. Therefore, we can find a set D such that X ∈ D is aperiodic, of class at least C and given any m -dominatedinvariant subset Λ of M for X , by Theorem C we have that either Λ has zerovolume, or X is Anosov, and so Λ = M . (cid:3) We define as in [15] or [9], the integrated upper Lyapunov exponent L ( X ) = lim n → + ¥ Z M n log k P nX ( x ) k dµ ( x ) ,which is an upper semicontinuous function L : X µ ( M ) → R .The proof of the next result follows [9, Proposition 3.2] step by step, onlyreplacing hyperbolic invariant subset with m -dominated invariant subset inthe relevant places of the argument. Proposition 2.2.
Let X ∈ X µ ( M ) be a aperiodic vector field and assumethat every m-dominated invariant subset has zero volume.For every given e , d > there exists a incompressible C vector field Ysuch that Y is e -C -close to X and L ( Y ) < d .Proof of Theorem A. Let D be given by Lemma 2.1. Denote by A the C r -stable subset of Anosov incompressible flows. By upper semicontinuity of L , for every k ∈ N , the set A k = { X ∈ X µ ( M ) : L ( X ) < / k } is open. ThenProposition 2.2 implies that A k dense in the complement A c of A in X µ ( M ) .We define a C residual set by R = \ k ∈ N (cid:0) A ∪ A k (cid:1) .It is straightforward to check that R satisfies the statement of Theorem A. (cid:3) Now we start the proof of Theorem B. But first we recall a basic re-sult which is a consequence of the persistence of dominated splittings, seee.g. [16].
Lemma 2.3.
Given a subset Λ with m-dominated splitting for a vector fieldX , there exists a neighborhood U of Λ and d > such that the set Λ Y ( U ) : = ∩ t ∈ R Y t ( U ) has a ( m + ) -dominated splitting for any vector field Y whichis d -C -close to X . This means that perturbing the original flow X to Y around an invariant m -dominated set, we can in (1.2) switch from 1 / / + e for a verysmall e and for every regular orbit of Y which remains nearby Λ . OMINATED SPLITTING AND ZERO VOLUME 9
The following perturbation lemmas from [10] are the main tools in ourarguments to prove Theorem B.
Lemma 2.4. (Small angle perturbation [10, Proposition 3.8] ) Let X ∈ X µ ( M ) and e > be given. There exists q = q ( e , X ) > such that if a hyperbolicperiodic orbit O for X has angle between its stable and unstable directionssmaller than q , then we can find an e -C -close volume-preserving vectorfield Y such that O is an elliptic periodic orbit for Y t . Another setting where one can create a nearby elliptic periodic orbit isthe following.
Lemma 2.5. (Large angle perturbation [10, Proposition 3.13] ) Let X ∈ X µ ( M ) and e , q > be given. There exists m = m ( e , q ) ∈ N and T ( m ) > such that if O is a hyperbolic periodic orbit for X with • angle between its stable and unstable directions bounded from be-low by q ; • period larger than T ( m ) , and • the Linear Poincar´e Flow along O is not m-dominated,then we can find a e -C -close vector field Y such that O is an elliptic peri-odic orbit for Y t . Conversely the absence of elliptic periodic orbits for all nearby perturba-tions implies uniform bounds on hyperbolic orbits with big enough period.This is an easy consequence of the two previous Lemmas 2.4 and 2.5 whichwe state for future reference.
Lemma 2.6.
Let X ∈ X µ ( M ) and e > be given and set q = q ( e , X ) , m = m ( e , q ) and T = T ( m ) given by Lemmas 2.4 and 2.5.Assume that all divergence-free vector fields Y which are e -C -close toX do not admit elliptic closed orbits. Then for every such Y all closed or-bits with period larger than T are hyperbolic, m-dominated and with anglebetween its stable and unstable directions bounded from below by q .Proof of Theorem B. Let P be the residual set given by Pugh’s GeneralDensity Theorem in [35], that is P is the family of all divergence-free vec-tor fields X such that Ω ( X ) is the closure of the set of periodic orbits, allof them hyperbolic or elliptic, and Ω ( X ) = M by the Poincar´e RecurrenceTheorem.We take any X ∈ X µ ( M ) which is not approximated by an Anosov flow.Then by a small C perturbation we can assume that X belongs to P andthat X is still not approximated by an Anosov flow. We fix some open set U and e > X intersects U there is nothing to prove,just set Y = X . Otherwise we fix e > (A) All closed orbits of X which intersect U are hyperbolic, and some ofthem has a small angle, less than q = q ( e , X ) provided by Lemma 2.4,between the stable and unstable directions.(B) All closed orbits of X which intersect U are hyperbolic, with an-gle between stable and unstable directions bounded from bellowby q , but some of them, with period larger than T , do not admitany m -dominated splitting for the Linear Poincar´e Flow, where m = m ( e , q ) and T = T ( m ) are given by Lemma 2.5, and q = q ( e , X ) was given as before by Lemma 2.4.(C) All closed orbits of X which intersect U and have period larger than T are hyperbolic, with m -dominated splitting, and with the anglebetween the stable and unstable directions bounded from bellow by q , where m = m ( e , q ) and T = T ( m ) are given by Lemma 2.5, and q = q ( e , X ) was given as before by Lemma 2.4.Case (A) implies the desired conclusion for some zero-divergence vectorfield Y e - C -close to X by Lemma 2.4. Analogously for case (B) by thechoice of the bounds m , T and by Lemma 2.5.Finally, we use Theorem A to show that if X is in case (C) and we as-sume that every C -nearby vector field Y does not admit elliptic periodicorbits through U , then we get a contradiction. This proves the statement ofTheorem B.If X is in case (C), then from Lemma 2.6 we know that every periodicorbit intersecting U , for every vector field Y e - C -close to X , with periodlarger than T , is hyperbolic with uniform bounds on m and q .From Theorem A, since X is not approximated by an Anosov flow, thereexists an incompressible vector field Y , which is e / C -close to X , admit-ting a full µ -measure subset Z where all Lyapunov exponents for Y are zero.Moreover we can assume that Y is aperiodic, that is the set of all periodicorbits has volume zero.Let ˆ U ⊂ U be a measurable set with positive measure. Let R ⊂ ˆ U be theset given by Poincar´e Recurrence Theorem (see e.g. [25]) with respect to Y . Then every x ∈ R returns to ˆ U infinitelly many times under the flow Y t and is not a periodic point. Denote by T the set of positive return times to ˆ U under Y t .Given x ∈ Z ∩ R and 0 < d < log 2 / m , there exists t x ∈ R such that e − d t < k P tY ( x ) k < e d t for every t ≥ t x .Let us choose t ∈ T such that t > max { t x , T } . OMINATED SPLITTING AND ZERO VOLUME 11
The Y t -orbit of x can be approximated for a very long time t > C -close flow Z : given r , t > e / C -neighborhood U of Y in X µ ( M ) , a vector field Z ∈ U , a periodic orbit p of Z with period ℓ and a map g : [ t ] → [ ℓ ] close to the identity such that • dist (cid:0) Y t ( x ) , Z g ( t ) ( p ) (cid:1) < r for all 0 ≤ t ≤ t ; • Z = Y over M \ S ≤ t ≤ ℓ (cid:0) B ( p , r ) ∩ B ( Z t ( p ) , r ) (cid:1) .This is Pugh’s C Closing Lemma adapted to the setting of conservativeflows, see [35]. Letting r > e − d ℓ < k P ℓ Z ( p ) k < e d ℓ with ℓ > T . (2.1)Now it is easy to see that Z is e - C -close to X , so that the orbit of p under Z satisfies the conclusion of Lemma 2.6. In particular we have that (cid:13)(cid:13) DP mZ | N sx (cid:13)(cid:13)(cid:13)(cid:13) DP mZ | N ux (cid:13)(cid:13) ≤
12 for all x ∈ O Z ( p ) ,for otherwise we would use Lemma 2.5 and produce an elliptic periodic or-bit for a flow e - C -close to X . Since the subbundles N s , u are one-dimensionalwe write p i : = Z im ( p ) for i = . . . , [ ℓ / m ] with [ t ] : = max { k ∈ Z : k ≤ t } and (cid:13)(cid:13) DP ℓ Z | N sp (cid:13)(cid:13)(cid:13)(cid:13) DP ℓ Z | N up (cid:13)(cid:13) = (cid:13)(cid:13) DP ℓ − m · [ ℓ / m ] Z | N sp (cid:13)(cid:13)(cid:13)(cid:13) DP ℓ − m · [ ℓ / m ] Z | N up (cid:13)(cid:13) · [ ℓ / m ] (cid:213) i = (cid:13)(cid:13) DP mZ | N sp i (cid:13)(cid:13)(cid:13)(cid:13) DP mZ | N up i (cid:13)(cid:13) ≤ C ( p , Z ) · (cid:18) (cid:19) [ ℓ / m ] ,(2.2)where C ( p , Z ) = sup ≤ t ≤ m (cid:0) k DP tZ | N sp k · k DP tZ | N up k − (cid:1) depends continu-ously on Z in the C topology. There exists then a uniform bound on C ( p , Z ) for all vector fields Z which are C -close to X .We note that we can take ℓ > T arbitrarily big by letting r > k DP ℓ Z ( p ) k = k DP ℓ Z | N up k and also1 ℓ log (cid:13)(cid:13) DP ℓ Z | N sp (cid:13)(cid:13) ≤ ℓ log C ( p , Z ) + [ ℓ / m ] ℓ log 12 + ℓ log (cid:13)(cid:13) DP ℓ Z | N up (cid:13)(cid:13) .Moreover since Z is volume preserving we have that the sum of the Lya-punov exponents along O Z ( p ) is zero, that is (we recall that ℓ is the periodof p ) 1 ℓ log k DP ℓ Z | N sp k = − ℓ log k DP ℓ Z | N up k . The constants in (2.2) do not depend on ℓ so taking the period very big wededuce that 1 ℓ log k DP ℓ Z ( p ) k ≥ m log 2 > d .This contradicts (2.1) and completes the proof of Theorem B. (cid:3)
3. D
OMINATED SPLITTING AND REGULARITY
Here we prove that positive volume regular invariant subsets with domi-nated splitting cannot admit singularities in its closure and thus are essen-tially uniformly hyperbolic sets. This result will be used to prove Theo-rem C.We denote by X + ( M ) the set of all C vector fields X whose derivative DX is H¨older continuous with respect to the given Riemannian norm, andwe say that X ∈ X + ( M ) is of class C + . We clearly have X ( M ) ⊃ X + ( M ) ⊃ X r ( M ) , for every r ≥ Proposition 3.1.
Let X ∈ X + µ ( M ) be given. Assume that Λ is a regularX t -invariant subset of M with positive volume and admitting a dominatedsplitting. Then the closure A of the set of Lebesgue density points of Λ doesnot contain singularities. We recall that a compact invariant subset Λ of X ∈ X µ ( M ) is (uniformly) hyperbolic if T Λ M = E ⊕ [ X ] ⊕ G is a continuous DX t -invariant splitting with the sub-bundle E , ~ G , ~ DX t for t > C three-dimensional vector field admitting a dominated splittingfor the Linear Poincar´e Flow is a uniformly hyperbolic set. Then we obtainthe following. Corollary 3.2.
Let X ∈ X + µ ( M ) and Λ be a regular X t -invariant subsetof M with positive volume and admitting a dominated splitting. Then theclosure A of the set of Lebesgue density points of Λ is a hyperbolic set. This implies in particular that there are neither singular-hyperbolic sets(e.g. Lorenz-like sets or singular-horseshoes) nor partially hyperbolic sets(see e.g. [16] or [28] for the definitions) with positive volume for C + incompressible flows on three-dimensional manifolds. A similar conclusionfor singular-hyperbolic sets was obtained by Arbieto-Matheus in [4] butassuming that the invariant compact subset is robustly transitive.The proof of Proposition 3.1 is divided into several steps, which we stateand prove as a sequence of lemmas in the following subsections. OMINATED SPLITTING AND ZERO VOLUME 13
Bounded angles, eigenvalues and Lorenz-like singularities.
Denoteby D ( Λ ) the subset of the Lebesgue density points of Λ , that is, x ∈ D ( Λ ) if x ∈ Λ and lim r → + µ ( Λ ∩ B ( x , r )) µ ( B ( x , r )) = µ ( Λ \ D ( Λ )) =
0. Moreoversince every nonempty open subset of M has positive µ -measure, we see that D ( Λ ) is contained in the closure of Λ .Assume that Λ is a X t -invariant set without singularities such that µ ( Λ ) > A for the closure of D ( Λ ) in what follows. Note that A is con-tained in the closure of Λ . Lemma 3.3.
Suppose that the Linear Poincar´e Flow over Λ has a dom-inated splitting for X . Then there exist a neighborhood V of Λ , a neigh-borhood U of X in X ( M ) (not necessarily contained in the space of con-servative flows) and h > such that for every Y ∈ U , every periodic orbitcontained in U is hyperbolic of saddle type and its eigenvalues l and l satisfy l < − h and l > h . Moreover the angle between the unstable andstable directions of these periodic orbits is greater than h .Proof. The Dominated Splitting for the Linear Poincar´e Flow extends bycontinuity to every regular orbit O which remains close to Λ for a C nearbyflow Y , this is Lemma 2.3. The domination implies that the eigenvalues l ≤ l of O satisfy l + k ≤ l for some k > Λ . Since the flow Y is close to being conservative,we have | l + l | ≤ e , where we can take e < k / Y be in asmall C -neighborhood of X .Thus we have − l − e ≤ l which implies − l − e + k ≤ l + k ≤ l and so 2 l ≥ k − e > l ≤ e − l implies l ≤ e − ( k − e / ) = e / − k < h >
0, independent of Y in a C neighborhood of X ,and independent of the periodic orbit O of Y in a neighborhood of Λ , suchthat l < − h and l > h , as stated.For the angle bound we argue by contradiction as in [28]: assume thereexists a sequence of flows Y n C −−−−→ n → + ¥ X and of periodic orbits O n of Y n con-tained in the neighborhood V of Λ such that the angle a n between the un-stable subspace and the stable direction satisfies a n −−−−→ n → + ¥ C perturbation Z n of Y n , for all big enough n ≥ sending the stable direction close to the unstable direction along the peri-odic orbit, such that the orbit of O n becomes a sink or a source for Z n . Thiscontradicts the first part of the statement of the lemma. (cid:3) We say that a singularity s is Lorenz-like for X if DX ( s ) has three realeigenvalues l ≤ l ≤ l satisfying l < l < < − l < l . Lemma 3.4.
Assume that X ∈ X µ ( M ) is such that all singularities are hy-perbolic with no ressonances (real eigenvalues are all distinct). Then thesingularities S ( X ) ∩ A are all Lorenz-like for X or for − X .
Remark 3.5.
The assumptions of the lemma above hold true for an openand dense subset of all C r vector fields, both volume preserving or not. Proof.
Fix s in S ( X ) ∩ A if this set is nonempty (otherwise there is nothingto prove). By assumption on X we known that s is hyperbolic. As in [28]we show first that s has only real eigenvalues. For otherwise we wouldget a conjugate pair of complex eigenvalues w , w and a real one l and,by reversing time if needed, we can assume that l < < Re ( w ) . Since µ ( A ) > Λ passing through everygiven neighborhood of s , for each regular orbit of a flow is a regular curve,and so does not fill volume in a three-dimensional manifold.Using the Connecting Lemma of Hayashi adapted to conservative flows(see e.g. [39]) we can find a C -close flow Y preserving the same measure µ with a saddle-focus connection associated to the continuation s Y of thesingularity s . By a small perturbation of the vector field we can assume that Y is of class C ¥ and still C -close to X (see e.g. [40]).We can now unfold the saddle-focus connection as in [12] to obtain aperiodic orbit with all Lyapunov exponents equal to zero (an elliptic closedorbit) for a C -close flow and near A . This contradicts Lemma 3.3, sincesuch orbit will be contained in a neighborhood of Λ . This shows that com-plex eigenvalues are not allowed for any singularity in A .Let then l ≤ l ≤ l be the eigenvalues of s . We have l < < l because s is hyperbolic. The preservation of volume implies that l = − ( l + l ) < − l < l . We have now two cases: l < : this implies l < l < < − l < l by the non-resonanceassumption, and s is Lorenz-like for X ; l > : since l = − ( l + l ) > l < − l < < l < l , so s is Lorenz-like for − X .The proof is complete. (cid:3) Invariant manifolds of a positive volume set with dominated split-ting for the Linear Poincar´e Flow.
OMINATED SPLITTING AND ZERO VOLUME 15
Invariant manifolds and (non-uniform) hyperbolicity.
An embeddeddisk g ⊂ M is a (local) strong-unstable manifold , or a strong-unstable disk , ifdist ( X − t ( x ) , X − t ( y )) tends to zero exponentially fast as t → + ¥ , for every x , y ∈ g . In the same way g is called a (local) strong-stable manifold , ora strong-stable disk , if dist ( X t ( x ) , X t ( y )) → n → + ¥ , for every x , y ∈ g . It is well-known that every point in a uniformlyhyperbolic set possesses a local strong-stable manifold W ssloc ( x ) and a localstrong-unstable manifold W uuloc ( x ) which are disks tangent to E x and G x at x respectively with topological dimensions d E = dim ( E ) and d G = dim ( G ) respectively. Considering the action of the flow we get the (global) strong-stable manifold W ss ( x ) = [ t > X − t (cid:16) W ssloc (cid:0) X t ( x ) (cid:1)(cid:17) and the (global) strong-unstable manifoldW uu ( x ) = [ t > X t (cid:16) W uuloc (cid:0) X − t ( x ) (cid:1)(cid:17) for every point x of a uniformly hyperbolic set. Similar notions are de-fined in a straightforward way for diffeomorphisms. These are immersedsubmanidfolds with the same differentiability of the flow or the diffeomor-phism. In the case of a flow we also consider the stable manifold W s ( x ) = ∪ t ∈ R X t (cid:0) W ss ( x ) (cid:1) and unstable manifold W u ( x ) = ∪ t ∈ R X t (cid:0) W uu ( x ) (cid:1) for x ina uniformly hyperbolic set, which are flow invariant.We note that these notions are well defined for a hyperbolic periodic orbit,since this compact set is itself a hyperbolic set.Now we observe that since A has positive volume, the dominated splittingof the Linear Poincar´e Flow implies that the Lebesgue measure µ A normal-ized and restricted to A is a (non-uniformly) hyperbolic invariant probabil-ity measure , see e.g. [7]: every Lyapunov exponent of µ A is non-zero, exceptalong the direction of the flow. Indeed, (recall the arguments in the proof ofLemma 3.3) the Lyapunov exponents l ≤ ≤ l along every Oseledet’sregular orbit satisfy l + l = Λ (a non-empty set because Λ has positivevolume) the exponents also satisfy l + k ≤ l for some k > k does not depend on theorbit chosen inside Λ . Thus there exists h > l = − l > h along every Oseledet’s regular orbit inside Λ . Assuming from now on that X ∈ X + ( M ) we have, according to the non-uniform hyperbolic theory (see [32, 33, 7]), that there are smooth strong-stable and strong-unstable disks tangent to the directions corresponding tonegative and positive Lyapunov exponents, respectively, at µ A almost every point. The sizes of these disks depend measurably on the point as well as therates of exponential contraction and expansion. We can define as before thestrong-stable, strong-unstable, stable and unstable manifolds at µ A almostall points.In addition, since µ is a smooth invariant measure, we can use [8, The-orem 11.3] and conclude that there are at most countably many ergodiccomponents of µ A . Therefore we assume from now on that µ A is ergodicwithout loss of generality. In addittion, hyperbolic smooth ergodic invariant probability measuresfor a C + dynamics are in the setting of Katok’s Closing Lemma, see [23]or [8, Section 15]. In particular we have that the support of µ A is containedin the closure of the closed orbits inside A supp ( µ A ) ⊂ Per ( X ) ∩ A , (3.1)where the periodic points in our setting are all hyperbolic by Lemma 3.3.3.2.2. Almost all invariant manifolds are contained in A.
Now we adaptthe arguments in [14] to our setting to deduce the following. Let µ u and µ s denote the measure induced on (strong-)unstable and (strong-)stable mani-folds by the Lebesgue volume form µ . Lemma 3.6.
For µ A almost every x the corresponding invariant manifoldssatisfy µ s (cid:0) W ss ( x ) \ A (cid:1) = and µ u (cid:0) W uu ( x ) \ A (cid:1) = that is, the invariant manifolds are µ u , s mod 0 contained in A. In addition, since A is closed and every open subset of either W ss ( x ) or W uu ( x ) has positive µ s or µ u measure, respectively, then we see that in fact W ss ( x ) ⊂ A and W uu ( x ) ⊂ A for µ − almost every x . (3.2)To prove Lemma 3.6 we need a bounded distortion property along invari-ant manifolds which is provided by [8, Theorems 11.1 & 11.2]. To state thisproperly we need the notion of hyperbolic block for a hyperbolic invariantprobability measure.3.2.3. Hyperbolic blocks and bounded distortion along invariant manifolds.
The measurable dependence of the invariant manifolds on the base pointmeans that for each k ∈ N we can find a compact hyperbolic block H ( k ) and positive numbers C x satisfying • dist ( X t ( y ) , X t ( x )) ≤ C x e − t t · dist ( y , x ) for all t > y ∈ W ssloc ( x ) ,and analogously for y ∈ W uuloc ( x ) exchanging the sign of t ; • C x ≤ k and t x ≥ k − for every x ∈ H ( k ) ; • H ( k ) ⊂ H ( k + ) for all k ≥ µ A (cid:0) H ( k ) (cid:1) → k → + ¥ ; OMINATED SPLITTING AND ZERO VOLUME 17 • the C strong-stable and strong-unstable disks W ssloc ( x ) and W uuloc ( x ) vary continuously with x ∈ H ( k ) (in particular the sizes of thesedisks and the angle between them are uniformly bounded from zerofor x in H ( k ) ).Now we have the bounded distortion property. Theorem 3.7. [8, Theorems 11.1 & 11.2]
Fix k ∈ N such that µ A ( H ( k )) > . Then the functionh s ( x , y ) : = (cid:213) i ≥ (cid:12)(cid:12) det D f | E s ( f i ( x )) (cid:12)(cid:12)(cid:12)(cid:12) det D f | E s ( f i ( y )) (cid:12)(cid:12) is H¨older-continuous for every x ∈ H ( k ) and y ∈ W ssloc ( x ) , where f : = X is the time- map of the flow X t and E s is the direction corresponding tonegative Lyapunov exponents.An analogous statement is true for a function h u on the unstable disksin H ( k ) exchanging E s with the direction E u corresponding to positiveLyapunov exponents and reversing the sign of i in the product h s above. Note that since H ( k ) is compact, there exists 0 < h k < ¥ such thatmax { h u , h s } ≤ h k on H ( k ) .3.2.4. Recurrent and Lebesgue density points.
We are now ready to startthe proof of Lemma 3.6.Let us take a strong-unstable disk W uu ( x ) satisfying simultaneously • x ∈ H ( k ) , • µ u (cid:0) W uu ( x ) ∩ A (cid:1) > • x is a µ u density point of W uu ( x ) ∩ A .For this it is enough to take k big enough since by the absolute continuityof the foliation of strong-unstable disks a positive volume subset, as H ( k ) ,must intersect almost all strong-stable disks on a subset of µ u positive mea-sure, see e.g. [34].Using the Recurrence Theorem we can also assume without loss of gener-ality that x is recurrent inside H ( k ) , that is, there exists a strictly increasingsequence of integers n < n < . . . such that x k : = f n k ( x ) ∈ H ( k ) for all k ∈ N and x k −−−→ k → ¥ x .Therefore we can consider the disk W k = f − n k (cid:0) W uuloc ( x k ) (cid:1) . Observe that W k ⊂ W uuloc ( x ) is a neighborhood of x and since the sizes of the strong-unstable disks on H ( k ) are uniformly bounded we see that diam (cid:0) W k ) → k → + ¥ . Now W uuloc ( x ) is one-dimensional in our setting and thus the shrinking of W k to x together with the f -invariance of A are enough to ensure µ u (cid:16) f − n k (cid:0) W uuloc ( x k ) \ A (cid:1)(cid:17) µ u (cid:16) f − n k (cid:0) W uuloc ( x k ) (cid:1)(cid:17) = µ u (cid:0) W k \ A (cid:1) µ u ( W k ) −−−→ k → ¥ µ u (cid:16) f − n k (cid:0) W uuloc ( x k ) \ A (cid:1)(cid:17) µ u (cid:16) f − n k (cid:0) W uuloc ( x k ) (cid:1)(cid:17) = R W uuloc ( x k ) \ A | det D f − n k | E u ( z ) | dµ u ( z ) R W uuloc ( x k ) | det D f − n k | E u ( z ) | dµ u ( z ) ≥ h u k · µ u (cid:0) W uuloc ( x k ) \ A (cid:1) µ u ( W uuloc ( x k )) ,which means that µ u (cid:0) W uuloc ( x k ) \ A (cid:1) µ u ( W uuloc ( x k )) ≤ h k · µ u (cid:0) W k \ A (cid:1) µ u ( W k ) for all k ≥
1. Hence we get µ u ( W uuloc ( x ) \ A ) = x k andthe continuous dependence of the strong-unstable disks on the points of thehyperbolic block H ( k ) . The argument for the stable direction is the same.Since the points of a full µ A measure subset have all the properties we used,this concludes the proof of Lemma 3.6 and of the property (3.2).3.2.5. Dense invariant manifolds of a periodic orbit.
Now we use the den-sity of periodic points in A (property (3.1)). Consider again a hyperbolicblock H ( k ) with a big enough k ∈ N such that µ A ( H ( k )) >
0. For anygiven x ∈ H ( k ) and d > O ( p ) in-tersecting B ( x , d ) . Because the sizes and angles of the stable and unstabledisks of points in H ( k ) are uniformly bounded away from zero, we canensure that we have the following transversal intersections W u ( p ) ⋔ W s ( x ) , /0 , W s ( p ) ⋔ W u ( x ) .This together with the Inclination Lemma implies that W u ( p ) = W u ( x ) ⊂ A and W s ( p ) = W s ( x ) ⊂ A . (3.3)Moreover since we can pick any x ∈ H ( k ) we can assume without loss that x has a dense orbit in A (since we took µ A to be ergodic) and then we canstrengthen (3.3) to: there exists a periodic orbit O ( p ) inside A such that W u ( p ) = A and W s ( p ) = A . (3.4) Recall the difference between W uu ( p ) and W u ( p ) etc in the flow setting. OMINATED SPLITTING AND ZERO VOLUME 19
Absence of singularities in A.
We recall that the alpha-limit set of apoint p ∈ M with respect to the flow X is the set a ( p ) of all limit points of X − t ( p ) as t → + ¥ . Likewise the omega-limit set is the set w ( p ) of limitpoints of X t ( p ) when t → + ¥ . Both these sets are flow-invariant.Using property (3.4) we consider, on the one hand, the invariant compactsubset of A given by L = a X ( W ss ( p )) the closure of the accumulation points of backward orbits of points in thestrong-stable manifold of the periodic orbit O ( p ) . By (3.4) we have L = A .On the other hand, considering N = w X ( W uu ( p )) we likewise obtain that N = A .Let us assume that s is a singularity contained in A . By Lemma 3.4 s iseither Lorenz-like for X or Lorenz-like for − X .In the former case, we would get W ss ( s ) ⊂ A because any compact partof the strong-stable manifold of s is accumulated by backward iterates of asmall neighborhood g inside W ss ( x ) . Here we are using that the contractionalong the strong-stable manifold, which becomes an expansion for negativetime, is uniform. In the latter case we would get W uu ( s ) ⊂ A by a similarargument reversing the time direction.We now explain that each one of these possibilities leads to a contradic-tion with the dominated splitting of the Linear Poincar´e Flow on the regularorbits of A , following an argument in [28]. It is enough to deduce a contra-diction for a Lorenz-like singularity for X , since the other case reduces tothis one through a time inversion.If W ss ( s ) ∩ A \ { s } ⊃ { y } for some point y ∈ A and for some singularity s ∈ A , then we have countably distinct regular orbits of Λ accumulatingon y ∈ W ss ( s ) (by the definition of A ) and on a point q ∈ W u ( s ) (by thedynamics of the flow near s ).Applying the Connecting Lemma, we obtain a saddle-connection asso-ciated to the continuation of s for a C -close vector field Y , known as“orbit-flip” connection, that is, there exists a homoclinic orbit Γ associ-ated to s Y such that W cu ( s Y ) intersects W s ( s Y ) transversely along Γ , i.e. Γ = W cu ( s Y ) ⋔ W s ( s Y ) , and also Γ ∩ W ss ( s Y ) , /0 .These connections can be C approximated by “inclination-flip” connec-tions for another C nearby vector field Z , not necessarily conservative , seee.g. [27, 3]. This means that the continuation s Z of the singularity has anassociated homoclinic orbit g such that W cu ( s Z ) intersects W s ( s Z ) along g but not transversely , and g ∩ W ss ( s Z ) = /0 . However the presence of “inclination-flip” connections is an obstructionto the dominated decomposition of the Linear Poincar´e Flow for nearby reg-ular orbits. This contradicts Lemma 2.3 and concludes the proof of Propo-sition 3.1. 4. U
NIFORM HYPERBOLICITY
Here we conclude the proof of Theorem C, showing that proper invari-ant hyperbolic subsets of a C + incompressible flow cannot have positivevolume. Proposition 4.1.
Let A be a compact invariant hyperbolic subset for X ∈ X + µ ( M ) . Then either µ ( A ) = or else X is an Anosov flow and A = M. The proof of Proposition 4.1 is given as a sequence of intermediate re-sults along the rest of this section. Assuming this result we easily have thefollowing.
Proof of Theorem C.
From Corollary 3.2 we have that a regular invariantsubset with positive volume with dominated splitting for the Linear Poincar´eFlow admits a positive volume subset which is hyperbolic. Therefore theflow of X is Anosov from Proposition 4.1. (cid:3) Positive volume hyperbolic sets and conservative Anosov flows.
We start the proof by recalling the notion of partial hyperbolicity.Let Λ be a compact invariant subset for a C flow on a compact bound-aryless manifold M with dimension at least 3. We say that Λ is partiallyhyperbolic if there are a continuous invariant tangent bundle decomposition T Λ M = E s ⊕ E c and constants l , K > x ∈ Λ and for all t ≥ • E s dominates E c : k DX t ( x ) | E sx k · k DX − t | E cX t ( x ) k ≤ Ke − l t • E s is uniformly contracting: k DX t | E sx k ≤ Ke − l t .We note that for a partially hyperbolic set of a flow the flow direction mustbe contained in the central bundle. Now we recall the following result.
Theorem 4.2. [1, Theorem 2.2]
Let f : M → M be a C + diffeomorphismand let Λ ⊂ M be a partially hyperbolic set with positive volume. Then Λ contains a strong-stable disk. Now we can use an argument similar to the one presented in Subsec-tion 3.2.6.
Lemma 4.3.
Let X ∈ X + µ ( M ) and Λ be a compact invariant partially hy-perbolic subset containing a strong-stable disk g . Then L = a X ( g ) = { a ( z ) : z ∈ g } contains all stable disks through its points. OMINATED SPLITTING AND ZERO VOLUME 21
Proof.
The partial hyperbolic assumption on A ensures that every one of itspoints has a strong-stable manifold. Moreover W ss ( z ) ⊂ Λ for every z ∈ a ( g ) , (4.1)since any compact part of the strong-stable manifold of z is accumulated bybackward iterates of any small neighborhood of x ∈ g inside W ss ( x ) . Herewe are using that the contraction along the strong-stable manifold, whichbecomes an expansion for negative time, is uniform. (cid:3) Proof of Proposition 4.1.
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