Dominated splittings for flows with singularities

Dominated splittings for flows with singularities

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Nonlinearity26(2013) 2391–2407 doi:10.1088/0951-7715/26/8/2391

Dominated splittings for flows with singularities

Vitor Araujo1, Alexander Arbieto2and Luciana Salgado3

1_{Instituto de Matem´atica, Universidade Federal da Bahia, Av. Adhemar de Barros, S/N , Ondina,}

40170-110-Salvador-BA-Brazil

2_{Universidade Federal do Rio de Janeiro, Instituto de Matem´atica, PO Box 68530, 21945-970}

Rio de Janeiro, Brazil

3_{Instituto de Matem´atica Pura e Aplicada-Estrada Dona Castorina, 110, Jardim Botˆanico,}

22460-320 Rio de Janeiro, Brazil

E-mail:[email protected],[email protected]@impa.br Received 24 October 2012, in final form 2 July 2013 Published 18 July 2013

Online atstacks.iop.org/Non/26/2391

Recommended by R de la Llave

Abstract

We obtain sufficient conditions for an invariant splitting over a compact invariant subset of aC1 _{flow X}

t to be dominated. In particular, we reduce

the requirements to obtain sectional hyperbolicity and hyperbolicity.

Mathematics Subject Classification: 37D30, 37D25

1. Introduction

The theory of hyperbolic dynamical systems is one of the main paradigms in dynamics. Developed in the 1960s and 1970s after the work of Smale, Sinai, Ruelle, Bowen [9,10,26,27], among many others, this theory deals with compact invariant setsfor diffeomorphisms and flows of closed finite-dimensional manifolds with hyperbolic splitting of the tangent space. That is, ifXis a vector field andXt is the generated flow then we say that an invariant and

compact setis hyperbolic if there exists a continuous splitting of the tangent bundle over

,TM=Es⊕EX⊕Eu, whereEXis the direction of the vector field, the subbundles are

invariant under the derivativeDXtof the flowXt

DXt·E∗x =E

∗

Xt(x), x∈, t ∈R, ∗ =s, X, u;

Esis uniformly contracted byDXtandEuis uniformly expanded: there areK, λ >0 so that

DXt |Es

x Ke

−λt_{, DX}

−t |Eu

x Ke

−λt_{, x∈, t ∈R.}

(1.1)

Very strong properties can be deduced from the existence of such a structure; see for instance [13,25].

Weaker notions of hyperbolicity, such as the notions of dominated splitting, partial hyperbolicity, volume hyperbolicity and singular or sectional hyperbolicity (for singular flows),

have been proposed to try to enlarge the scope of this theory to classes of systems beyond the uniformly hyperbolic ones; see [1] and [5] for singular or sectional hyperbolicity. However, the existence of dominated splittings is the weaker one.

Many researchers have studied the relations of the existence of dominated splittings with other dynamical phenomena, mostly in the discrete time case, such as robust transitivity, homoclinic tangencies and heteroclinic cycles, and also the possible extension of this notion to endomorphisms; see for instance [5,6,15,19–21,33].

However, the notion of dominated splittings deserves attention. Several authors used this notion for linear Poincar´e flow, see [11,16,17], and this is useful in the absence of singularities, as in [12]. We remark that this flow is defined only in the set of regular points. However, in the singular case, it is a non-trivial question to obtain a dominated splitting for the derivative of the flow, and thus allowing singularities. Indeed, it is difficult to obtain it from a dominated splitting for linear Poincar´e flow. See [14], for an attempt to solve this, in the context of robustly transitive sets, using the extended linear Poincar´e flow.

Definition 1.1. Adominated splitting over a compact invariant setofX is a continuous

DXt-invariant splittingTM =E⊕F withEx = {0},Fx = {0}for everyx ∈and such

that there are positive constantsK, λsatisfying

DXt|Ex · DX−t|FXt (x)< Ke

−λt_{, for all x∈, and allt >0. (1.2)}

The purpose of this article is to study the existence of dominated splittings for flows with singularities. On one hand, we study this question in the presence of some sectional hyperbolicity, as this theory was built to understand flows like the Lorenz attractor, which are the prototype of non-hyperbolic dynamics with singularities, but with robust dynamical properties. On the other hand, we present several examples to clarify the role of the condition on the singularities.

This article is organized as follows. In section2, we present the definitions and auxiliary results needed in the proof of theoremA, and some examples showing that the domination condition on the singularities is necessary. In section3, we present applications of our main result and more examples implying that these results are not valid for diffeomorphisms. In section4theoremBis proved assuming theoremA. In section5, we prove theoremAand theorem2.4. TheoremCand corollary2are proved in section6using the previously proved results. Finally, in section7, we present the examples stated in propositions1and2.

2. Statements of the results

Let M be a connected compact finite n-dimensional manifold, n 3, with or without a boundary. We consider a vector fieldX, such thatXis inwardly transverse to the boundary

∂M, if∂M= ∅. The flow generated byXis denoted by{Xt}.

Aninvariant setfor the flow ofX is a subset ofM which satisfiesXt() = for

allt ∈R_{. Themaximal invariant set of the flowisM(X):= ∩t0Xt(M), which is clearly a}

compact invariant set.

Asingularityfor the vector fieldX is a pointσ ∈ M such that X(σ ) = 0. The set formed by singularities is denoted by Sing(X). We say that asingularity is hyperbolicif the eigenvalues of the derivativeDX(σ )of the vector field at the singularityσ have a non-zero real part.

A compact invariant setis said to bepartially hyperbolicif it exhibits a dominated splittingTM=E⊕Fsuch that subbundleEisuniformly contracted, i.e. there existC >0

andλ >0 such thatDXt|Ex Ce

−λ_{fort 0. In this caseF is thecentral subbundle}

We say that aDXt-invariant subbundleF ⊂TMis asectionally expandingsubbundle

if dimFx 2 is constant forx ∈and there are positive constantsC, λsuch that for every x∈and every two-dimensional linear subspaceLx ⊂Fxone has

|det(DXt|Lx)|> Ce

λt_{, for allt >0. (2.1)}

Definition 2.1 ([18, definition 2.7]). Asectional-hyperbolic setis a partially hyperbolic set whose singularities are hyperbolic and whose central subbundle is sectionally expanding.

This is a generalization of hyperbolicity for compact invariant sets with singularities accumulated by regular orbits, since the Lorenz attractor and higher dimensional examples are sectional-hyperbolic [1,7,18,30] and because of the following result the proof of which can be found in [1,5,19].

Lemma 2.2 (Hyperbolic lemma). Every compact invariant subset of a sectional-hyperbolic set without singularities is a hyperbolic set.

The main result of this paper is the following.

Theorem A.Letbe a compact invariant set ofX such that every singularity in this set is hyperbolic. Suppose that there exists a continuousDXt-invariant splitting of the tangent bundle of,TM =E⊕F, whereEis uniformly contracted,F is sectionally expanding and for some constantsC, λ >0we have

DXt|Eσ · DX−t|Fσ< Ce −λt

for all σ ∈∩Sing(X) and t 0. (2.2)

ThenTM=E⊕F is a dominated splitting.

As a corollary, we obtain sufficient conditions to obtain hyperbolicity in the non-singular case using the hyperbolic lemma.

Corollary 1. Letbe a compact invariant set without singularities for the vector fieldX. Suppose that there exists a continuousDXt-invariant decomposition of the tangent bundle of ,TM=E⊕F, whereEcontracts andF is sectionally expanding. Thenis a hyperbolic set.

We note that in definition2.1domination is required. As a consequence of theoremA, the domination assumption is only necessary at the singularities, so that we obtain the following equivalent definition of sectional-hyperbolicity.

Definition 2.3. A compact invariant set ⊂ M is a sectional-hyperbolic set for X if all singularities in are hyperbolic, there exists a continuous DXt-invariant splitting of the

tangent bundle onTM =E⊕F with constantsC, λ >0 such that for everyx ∈ and

everyt >0 we have

(1) DXt|ExCe −λt_;

(2) |detDXt|Lx|> C

−1_eλt_{, for every two-dimensional linear subspaceL} x ⊂Fx;

(3) DXt|Eσ · DX−t|Fσ< Ce

−λt_{for allσ ∈∩Sing(X).}

2.1. Hyperbolicity versus sectional-expansion

We present here some motivation for theoremA. We observe that, for a hyperbolic set, both splittings(Es_⊕EX_)⊕Eu_andEs_⊕(EX_⊕Eu_{)are dominated. Moreover, it is easy to see}

Our next result shows that we cannot have a hyperbolic splitting without the flow direction, unless we restrict ourselves to finitely many singularities, all of them isolated on the non-wandering set.

Theorem 2.4. Letbe a compact invariant set ofX. Suppose that there exists a continuous DXt-invariant decomposition of the tangent bundleTM =E⊕F overand constants C, λ >0,such that for everyx ∈and allt >0

DXt|ExCe

−λt _{and DX}

−t|F_{Xt (x)}Ce−λt. Thenconsists of finitely many hyperbolic singularities.

We cannot replace the assumptions on theorem2.4either by sectional-expansion or by sectional-contraction, as the following examples show.

Example 1. Consider a Lorenz-like singularityσ for aC1 flow{Xt}t∈Ron a3-manifoldM,

that is,σ is a hyperbolic singularity of saddle-type such that the eigenvalues ofDX(σ )are real and satisfyλ2 < λ3<0<−λ3< λ1andλ1+λ2>0.

LetEi be the eigenspace associated to the eigenvalueλi,i = 1,2,3, and setE =E3 andF =E1⊕E2. Then the decomposition is trivially continuous, not dominated (F admits vectors more sharply contracted than those ofE) butEuniformly contracts the lengths of the vectors andF uniformly expands the area, that is,F is sectionally expanded.

Example 2. Consider a hyperbolic saddle singularity,σ, for aC1flow{Xt}t∈Ron a4-manifold

Msuch that the eigenvalues ofDX(σ )are real and satisfy

λ2< λ3<0< λ4 < λ1, λ1+λ3>0 and λ2+λ4 <0.

LetEibe the eigenspace associated to the eigenvalueλi,i=1,2,3,4, and setF =E2⊕E4 andE=E1⊕E3. Then the decomposition is trivially continuous, not dominated as before, Eis uniformly area contracting, sinceDX−t |Eexpands area fort >0; andF is uniformly area expanding. In other words,F is sectionally expanded andEis sectionally contracted.

In both examples above we have sectional-expansion and sectional-contraction along the subbundles of a continuous splitting but the splitting is not dominated. These examples involve a trivial invariant set: an equilibrium point. But there are examples with compact invariant sets having singularities accumulated by regular orbits and also with a dense regular orbit; see section7.

This suggests that theorem 2.4might be generalized if we assume domination at the singularities oftogether with sectional-expansion alongF and uniform contraction along

Eover regular orbits. This is precisely the content of theoremA.

3. Applications and other examples

To present the next result characterizing sectional-hyperbolic sets through sectional Lyapunov exponents, we first have to present more definitions.

We say that a probability measure µ is X-invariant if µ(Xt(U )) = µ(U ), for every

measurable subsetU ⊂Mand everyt ∈R. Given a compactX-invariant setwe say that a subsetY ⊂Mis atotal probability subsetofifµ(Y )=1 for everyX-invariant measure

µsupported in . We note that each singularityσ ofbelongs to Y sinceµ = δσ is a X-invariant probability measure.

LetA:E×R_{→Ebe a Borel measurable map given by a collection of linear bijections}

whereMis the base space (we assume it is a manifold) of the finite-dimensional vector bundle

E, satisfying the cocycle property

A0(x)=I d, At+s(x)=At(Xs(x))◦As(x), x∈M, t, s∈R,

with{Xt}t∈Ra smooth flow overM. We note that for each fixedt >0 the mapAt :E → E, vx ∈Ex →At(x)·vx∈EXt(x)is an automorphism of the vector bundleE.

The natural example of a linear multiplicative cocycle over a smooth flowXton a manifold

is the derivative cocycleAt(x)=DXt(x)on the tangent bundleT M of a finite-dimensional

compact manifoldM.

A concrete instance of this structure we will use is the derivative cocycleAt(x)=DXt | Exrestricted to a continuousDXt-invariant subbundleEof the tangent bundleT M.

According to the multiplicative ergodic theorem of Oseledets [3,4], since in this setting we have that sup_−1t1logAt(x)is a bounded function ofx ∈, there exists a subsetR

ofwith total probability such that for everyx ∈Rthere exists a splitting

Ex =E1(x)⊕ · · · ⊕Es(x)(x) (3.1)

which isDXt-invariant and the following limits, known as theLyapunov exponents atx, exist

λi(x)= lim

t→±∞

1

t logDXt(x)·v,

for everyv∈Ei(x)\{0}, i=1, . . . , s(x). We order these numbers asλ1(x) < . . . < λs(x)(x).

One of these subbundles is given by the flow direction (at non-singular points of the flow) and the corresponding Lyapunov exponent is zero for almost every point.

The functionssandλiare measurable and invariant under the flow, i.e.,s(Xt(x))=s(x)

andλi(Xt(x))=λi(x)for allx ∈Randt ∈R. The splitting (3.1) also depends measurably

on the base pointx ∈ R. IfF is a measurable subbundle of the tangent bundle then by ‘the Lyapunov exponents ofE’ we mean the Lyapunov exponents of the non-zero vectors inF.

Given a vector spaceE, we denote by∧2_{Ethe second exterior power ofE, defined as}

follows. For a basisv1, . . . , vn of E, then ∧2E is generated by{vi ∧vj}i=j. Any linear

transformationA:E→F induces a transformation∧2_A:∧2_{E→ ∧}2_{F. Moreover,v} i∧vj

can be viewed as the 2-plane generated byvi andvj ifi =j; see for instance [3] for more

information.

In [2], a notion of sectional Lyapunov exponents was defined and a characterization of sectional-hyperbolicity was obtained based on this notion.

Definition 3.1 ([2, definition 2.2]). Given a compact invariant subsetof X with aDXt

-invariant splittingTM = E⊕F, thesectional Lyapunov exponentsofx alongF are the

limits

lim

t→+∞ 1

t log ∧ 2_DX

t(x)·v

whenever they exists, wherev∈ ∧2_F x− {0}.

As explained in [2], if{λi(x)} s(x)

i=1are the Lyapunov exponents, then the sectional Lyapunov

exponents at a pointx ∈ R are{λi(x)+λj(x)}1i<js(x)and they represent the asymptotic

sectional-expansion in theF direction.

As an application of theoremA, we have the following result, which is an extension of the main result in [2] assuming continuity of the splitting, instead of a dominated splitting.

If the Lyapunov exponents in theEdirection are negative and the sectional Lyapunov exponents in the F direction are positive on a set of total probability within , then the splitting is dominated andis a sectional-hyperbolic set.

We now apply the main result to the setting of weakly dissipative three-dimensional flows. We recall that an open subsetUis atrapping regionifXt(U )⊂U, t >0 and a compact

invariant subsetisattractingif it is the maximal invariant subset(U )= ∩t0Xt(U )inside

the trapping regionU. We say that a compact invariant subsetfor the flow of the vector field

Xisweakly dissipativeif div(X)(x)0 for allx ∈, that is, the flow nearinfinitesimally does not expand in volume.

We also say that a hyperbolic singularity σ of X is C1_{-linearizable if there exists a}

diffeomorphismh:Vσ →Bfrom an open neighbourhood ofσ ∈Mto an open neighbourhood Bof the origin inR3_{such that}h(Xt(x))=et A·h(x), Xs(x)∈Vσfor|s|tandA=DX(σ ).

For this it is enough that the spectrum ofDX(σ )satisfies a finite number of non-resonance conditions; see e.g. [28]. Hence this is aCr_{-generic condition for all sufficiently larger1.}

Theorem C. Let X be a C1 vector field on a three-dimensional manifold M admitting a trapping regionU whose singularities (if any) are hyperbolic andC1 linearizable. Let us assume that the compact invariant subset=(U )is weakly dissipative and endowed with a one-dimensional continuous fieldF of asymptotically backward contracting directions, that is,x∈→Fxis continuous and for eachx∈,Fxis a one-dimensional subspace ofTxM, and also

lim inf

t→+∞ 1

t logDX−t |Fx<0 for allx∈. (3.2) If at each singularityσ ∈ Uthere exists a complementaryDXt-invariant directionEσ such thatEσ⊕Fσ =TσMis a dominated splitting, thenis a hyperbolic set (in particular,has no singularities).

Since we use the attracting and dominated splitting assumption only to prove the non-existence of singularities in, as a consequence of the proof we obtain the following.

Corollary 2. LetXbe aC1_{vector field on a three-dimensional manifoldMadmitting compact} invariant subset, without singularities, which is weakly dissipative and endowed with a one-dimensional continuous fieldF of asymptotically backward contracting directions.

Thenis a hyperbolic set.

These results are extremely weak versions of the following conjecture of Viana, presented in [31].

Conjecture 1. If an attracting set(U )of smooth map/flow has a non-zero Lyapunov exponent at Lebesgue almost every point of its isolating neighbourhoodU, that is

lim sup

t→+∞ 1

t logDXt>0 for Lebesgue almost everyx ∈U, (3.3)

then(U )has a physical measure: there exists an invariant probability measureµsupported in(U )such that for all continuous functionsϕ:U →R

lim

t→+∞ 1

t

t

0

ϕ(Xt(x))dt =

ϕdµ for Lebesgue almost every x ∈U.

Indeed, ifUis a trapping region and=(U )satisfies the assumptions of theoremC, then

We now provide several examples which clarify the role of the assumption of domination at the singularities in theorem A and the relations between dominated splittings for diffeomorphisms and flows.

Proposition 1.There exist the following examples of vector fields and flows.

(1) A vector field with an invariant and compact set containing singularities accumulated by regular orbits inside the set, with a non-dominated and continuous splitting of the tangent space, but satisfying uniform contraction and sectional-expansion.

(2) A vector field with an invariant and compact set with a continuous and invariant splitting E⊕F such thatEis uniformly contracted,F is area expanding but it is not dominated. (3) A suspension flow whose base map has a dominated splitting but the flow does not admit

any dominated splitting.

Item (3) above is based on a diffeomorphism described in [8] and similar to another suggested by Pujals in [5, example B.12].

An important remark is that our results are not valid for diffeomorphisms, as the following result shows.

Proposition 2.There exists a transitive hyperbolic set for a diffeomorphism with a non-dominated splitting which is sectionally-expanding and sectionally contracting.

4. Proof of theoremB

We follows the lines of [2]. The following proposition, whose proof can be found in [2], is the main auxiliary result in the proof.

We fix a compactXt-invariant subset. We say that a family of functions{ft :→R}t∈R is subadditive if for everyx ∈ M andt, s ∈ Rwe have thatft+s(x) fs(x)+ft(Xs(x)).

The subadittive ergodic theorem (see e.g. [32]) shows that the functionf (x)=lim inf

t→+∞

ft(x)

t

coincides withf (x) = lim

t→+∞

1

tft(x)in a set of total probability in.

Proposition 3.Let {t → ft : → R}t∈Rbe a continuous family of continuous function

which is subadditive and suppose that f (x) < 0 in a set of total probability. Then there exist constants C > 0 and λ < 0 such that for everyx ∈ and every t > 0 we have

exp(ft(x))Cexp(λt /2).

Proof.See [2, proposition 3.4].

Remark 1. In [2] only =M(X)was considered to conclude that{Xt}t∈Ris a sectional-Anosov flow. However all statements are valid for a compact invariant subset ofM.

Proof of theoremB.Define the following families of continuous functions on φt(x)=logDXt|Ex and ψt(x)=log ∧

2_DX

−t|F_{Xt (x)}, (x, t )∈×R.

Both familiesφt, ψt :→Rare easily seen to be subadditive. From proposition3applied

toφt(x)and the hypothesis on the Lyapunov exponents, there are constantsC >0 andγ <0

such that exp(φt(x))= DXt|ExCexp(γ t )for allt >0 showing thatEis a contractive subbundle.

Analogously, the hypothesis on the sectional Lyapunov exponents and proposition 3

applied to the function ψt(x) provides constants D > 0 and η < 0 for which ∧2 DX−t|F_{Xt (x)}Deηt, soF is a sectionally expanding subbundle. Now theoremAensures that

5. Proof of theoremAand theorem2.4

We need an auxiliary lemma for which the following definition is necessary. Given a pair of subspacesEx, Fx ofTxMsuch thatEx∩Fx = {0}, the angle (Ex, Fx)between the two

subspaces is defined by

sin _{(Ex, Fx):=} 1

π(Ex)

, x∈,

whereπ(Ex):Ex⊕Fx →Ex is the projection ontoEx parallel toFx defined in the vector

spaceEx⊕Fx.

We say that an invariant splittingE⊕F =TMof the tangent bundle over a invariant

subsethas angle uniformly bounded away from zeroif the dimensions of the fibersEx,Fx

are constant for allxinand there existsθ0>0 such that sin (Ex, Fx)θ0for eachxin.

The next lemma specifies the subbundle which contains the flow direction.

Lemma 5.1. Letbe a compact invariant set forX. Given an invariant splittingE⊕F of TMwith the angle bounded away from zero over, such thatE is uniformly contracted, then the flow direction is contained in theF subbundle, for allx ∈.

Proof. We denote byπ(Ex): TxM →Ex the projection onEx parallel toFx atTxM, and

likewiseπ(Fx):TxM→Fxis the projection onFxparallel toEx. We note that forx ∈

X(x)=π(Ex)·X(x)+π(Fx)·X(x)

and fort ∈R_{, by linearity ofDXtandDXt-invariance of the splittingE⊕F}

DXt·X(x)=DXt·π(Ex)·X(x)+DXt·π(Fx)·X(x)

=π(EXt(x))·DXt·X(x)+π(FXt(x))·DXt·X(x)

Let z be a limit point of the negative orbit of x. That is, we assume without loss of generality sinceis compact, that there is a strictly increasing sequence tn → +∞such

that lim

n→+∞xn:=n→lim+∞X−tn(x)=z. Thenz∈and, ifπ(Ex)·X(x)is not the zero vector, we get

lim

n→+∞DX−tn·X(x)=_n→lim

+∞X(xn)=X(z), but also lim

n→+∞DX−tn·π(Ex)·X(x)_n→lim

+∞c

−1_eλtn_π(E

x)·X(x) =+∞. (5.1)

This is possible only if the angle betweenExn andFxntends to zero whenn→+∞.

Indeed, using the Riemannian metric onTyM, the angleα(y)=α(Ey, Fy)betweenEy

andFyis related to the norm ofπ(Ey)as follows:π(Ey) =1/sin(α(y)). Therefore

DX−tn·π(Ex)·X(x) = π(Exn)·DX−tn·X(x)

1

sin(α(xn))

· DX−tn·X(x)

= 1

sin(α(xn))

· X(xn)

for alln1. Hence, if the sequenceDX−tn·π(Ex)·X(x)in the left hand side of (5.1) is unbounded, then lim

n→+∞α(X−tn(x))=0.

However, since the splittingE⊕Fhas an angle bounded away from zero over the compact

, we have obtained a contradiction. This contradiction shows thatπ(Ex)·X(x)is always

The proof of theorem2.4follows from lemma5.1.

Proof of theorem 2.4. We assume, arguing by contradiction, that there exists a point

x ∈ \Sing(X), that is, x is a regular point: X(x) = 0. Hence lemma 5.1ensures that

X(x)∈Fx. But the same lemma5.1applied to the reversed flowX−t generated by the field −Xon the same invariant setshows thatX(x)∈ Ex. ThusX(x)∈Ex∩Fx = {0}. This

contradiction ensures that⊂Sing(X). But our assumptions on the splitting show that each

σ ∈is a hyperbolic singularity: the eigenvalues ofDX|Eσ are negative and the eigenvalues ofDX |Fσ are positive. It is well known that hyperbolic singularitiesσ are isolated in the ambient manifoldM; see e.g. [22]. We conclude that the compactis a finite set of hyperbolic

singularities.

Before proving theoremAwe need some extra notions and definitions.

We recall that, for a invertible continuous linear operatorL, the minimal norm (or co-norm) is equal tom(L) := L−1−1_{. Hence condition (1.2) in definition1.1is equivalent}

to

DXt|Ex< Ke

−λt_·m(DX

t|Fx) for all x∈and for allt >0. (5.2)

The multiplicative ergodic theorem of oseledets [3,4] provides the following properties of Lyapunov exponents and Lyapunov subspaces, in addition to the existence of Lyapunov exponents on the total probability subsetR of, as presented in section3. For any pair of disjoint subsetsI, J ⊂ {1, . . . , s(x)}, the angle between the bundlesEI(x)=i∈IEi(x)and EJ(x)=j∈JEj(x)decreases at most subexponentially along the orbit ofx, that is

lim

t→±∞ 1

t log sin

_{(EI(Xt(x)), EJ(Xt(x)))=0, x ∈R;}

which implies, in particular, that for any pair i, j ∈ {1, . . . , s(x)} withi = j and vi ∈ Ei(x)\ {0}, vj ∈Ej(x)\ {0}

lim

t→±∞ 1

t log|det(DXt |span{vi, vj})| =λi(x)+λj(x).

Proof of theorem A. Let x ∈ R ⊂ be a regular point of the flow of X. From lemma5.1we know thatX(x) ∈ Fx. SinceF is aDXt-invariant subbundle, considering

the linear multiplicative cocycleAt(z)=DXt |Fzforz∈, t ∈R, there exists a splitting Fx =s(x)j=1Fj(x)ofFxinto a direct sum of Lyapunov subspaces. One of these subspaces is

EX_{generated byX(x)= 0, which we renameF}

1(x)=ExXin what follows. We also have the

corresponding Lyapunov exponentsλF_j(x), j =1, . . . , s(x).

Fixing i = 2, . . . , s(x) andv ∈ Fi(x)\ {0}, we consider span{X(x), v}. From the

assumption of sectional expansion of area together with the subexponential control on angles at regular points we obtain

0< λlim inf

t→+∞ 1

t log|det(DXt|span{X(x), v})| =λ F 1(x)+λ

F i (x)=λ

F i(x).

HenceλF

i (x)λ >0 for alli=2, . . . , s(x), x∈RandX(x)= 0.

If we consider the cocycleBt(z)=DXt | Ez, x ∈ , t ∈R, then we find the splitting Ex = r(x)_i=1Ej(x)into Lyapunov subspaces and the corresponding Lyapunov exponents λE

i (x), i=1, . . . , r(x). It is easy to see that the assumption of uniform contraction alongE

implies thatλE

Thenφt(x)=log

DXt|Ex

m(DXt|Fx) is a subadditive family of continuous functions satisfying

φ(x)=lim inf

t→+∞

φt(x)

t lim infn→+∞ 1

t logDXt|Ex −lim supn→+∞ 1

t logm(DXt|Fx)

=max{λE_i (x),1ir(x)} −min{λF_i (x),1is(x)}−λ−0= −λ

for allx∈Rsuch thatX(x)= 0.

Forx =σ for some singularityσ ∈ we haveφ(σ )−λas a direct consequence of the assumption of domination at the singularities.

We have shown thatφ(x) <0 for allx ∈ R. Applying proposition3we see that there existsC > 0 such that expφt(x) Ce−λt /2 for allx ∈ andt > 0, which gives us the

condition (5.2). This concludes the proof of theoremA.

The proof of corollary1follows from theoremAand the hyperbolic lemma2.2.

6. Proof of theoremCand corollary2

We start by showing that the one-dimensional asymptotically backward contracting subbundle

Foveris in fact a uniformly expanding subbundle. Indeed, we note thatft(x)=logDX−t | Fx is a subadditive family of continuous functions satisfying f (x) < 0 for all x ∈ .

In particular, f (x) < 0 in a set of total probability. Hence by proposition3 we obtain

ft(x) Ce−t λ/2 for all t > 0 and some constants λ, C > 0. (This trivially implies

DXt | FXt(x) e

t λ/2_{/C, x ∈ , t 0 and soDX}

T | Fx > 2 for allx ∈ and T 2 log(2C)/λ.)

Now we use the assumption thatis an attracting set whose possible singularitiesσ are hyperbolic,C1linearizable and admit a complementaryDXt-invariant directionEσsuch that Eσ⊕Fσ =TσMis a dominated splitting.

Lemma 6.1. The attracting setdoes not contain singularities.

Proof. By theC1-linearization assumption and the stable/unstable manifold theorems, there exists a correspondingXt-invariant (weak- or strong-)unstable one-dimensional manifoldWσu

contained in (sinceis attracting) such that Fσ = TσWσu andX−t(z) −→

t→+∞σ for each

z∈Wu σ.

Claim 1. For allz∈Wu

σ we haveX(z)∈Fz.

We note that this is trivially true for z = σ. Let us assume, by contradiction, that

X(z) /∈Fzfor somez∈Wσu\ {σ}(and thus for all suchzbyDXt-invariance). By assumption

there exists a diffeomorphismh:Vσ→Bfrom an open neighbourhood ofσ ∈Mto an open

neighbourhoodBof the origin inR3such thath(Xt(x))=et A·h(x), Xs(x)∈Vσfor|s|t

andA=DX(σ ). By a linear change of coordinates we can assume without loss of generality thatDh(σ )·Fσ =R×02andDh(σ )·Eσ =0×R2. Letv : Vσ ∩→ R3 denote the

continuous unitary vector field such that span(vx)=Dh(x)·Fx, x∈Vσ∩.

We are assuming thatvz has a non-zero component along theEσ ≈ 0×R2 direction

for z ∈ W_σu\ {σ}. SinceX−t(z) →

t→+∞σ the domination of the splittingEσ ⊕Fσ ensures that _(et A_·v

z, Eσ) −→

t→−∞0. However the invariance ofF ensures thatFX−t(z)_t→−∞−→ Fσ and sincehis aC1conjugation we also have _(et A_·v

z, Fσ) −→

Using the claim together with the previous estimate on expansion alongF, we have that

∞>sup

x∈

X(x)X(XkT(z)) = DXkT ·X(z)2kX(z), k1

which is a contradiction and completes the proof of the lemma.

From now we use only the assumptions of corollary2, that is, thatis a weakly dissipative compact invariant subset without singularities having a continuous field of asymptotically backward contracting directions.

In this setting, the linear Poincar´e flow is well defined over . If we denote by

O_x :TxM→Nxthe orthogonal projection fromTxMtoNx = {v∈TxM:< v, X(x) >=0}

the orthogonal complement of the field direction atx ∈ , then thelinear Poincar´e flowis given byPt

x =OXt(x)◦DXt :TxM→NXt(x), x∈, t ∈R.

We note that since the flow direction is invariant, from the splittingTxM = ExX⊕NX

we can writeDXt = _X(X

−t(x)) X(x) ⋆

0 P−t

and since|detDX−t|1 by the weak dissipative

assumption, we get|det(P−t)|m0=minz∈(X(x)/X(X−t(x))) >0.

Claim 2. The subbundleF˜of the normal bundle given by{ ˜Fx =Ox(Fx)}x∈isPt-invariant and uniformly backward contracting.

Indeed, the continuity of the subbundleFensures that _{(Fx, X(x))is bounded away from} zero, and so there existsκ > 0 such that for everyv ∈ ˜Fx we have|β| κvsuch that βX(x)+v=u∈Fx. This ensures that

P_x−t·v = O_X−t(x)(DX−t·(u−βX(x)))OX−t(x)(DX−t·u−βX(X−t(x)))

DX−t·uCe−t λ/2v+βX(x)Ce−t λ/2(v+κvX(x))

C

1 +κsup

z∈

X(z)

e−t λ/2v = ˜Ce−t λ/2v.

This proves claim2.

This is enough to guarantee the existence of a complementaryP−t-invariant and expanding subbundle through a graph transform technique. To this end, let us consider the one-dimensional subbundleGxofNx orthogonal toF˜x. We can then writeNx = ˜Fx⊕Gxand so,

sinceF˜ isPt-invariant, we getP_x−t =

at(x) bt(x)

0 ct(x)

, whereat(x):F˜x → ˜FX−t(x), ct(x):

Gx → GX−t(x) andbt(x) : Gx → FX−t(x). We know that |at(x)| C˜e

−t λ/2 _{and since}

0< m0|det(P−t)| = |at(x)| · |ct(x)|we obtain|ct(x)|m0et λ/2/C, x˜ ∈, t >0. We

fixT >0 such thatm0eT λ/2/C˜ 2.

Let us now consider the family of continuous one-dimensional complementary subspaces ofF˜xinsideNxgiven by the graph of a linear mapL:Gx → ˜Fx, i.e., we consider the vector

spaceGof allL= {Lx :Gx → ˜Fx :x ∈}with the normL1−L2 =supx∈L1x−L2x.

We have thatGwith this norm is a Banach space.

It is easy to see that the map P−t _{transforms the graph of L}

x into the graph of ˆ

LX−t(x) = (P −t

x ·Lx +πX−t(x)·P

−t_{)◦[(I −π}

X−t(x))·P

−t_]−1_{, where π}

z : Nz → ˜Fz is

the orthogonal projection fromNzintoF˜zatz∈. In this way we get a mapP−T :G→G

which we claim is a contraction with respect to the norm given above. Indeed, for any pair

Li _{= {L}i

x}x∈, i=1,2 we get ˆL1_X−

T(x)− ˆL

2

X−T(x) = P −t

x ·(L1x−L2x)◦[(I−πX−t(x))·P −t_]−1

|aT(x)| · L1x−L 2

x/|ct(x)|

1 2L

1

x−L

s

W

E

F

Figure 1.A double homoclinic saddle connection expanding area.

for eachx ∈ . We thus have aP−T _{fixed pointL}0 _{defining aP}−T_{-invariant subbundle} ˜

E= { ˜Ex= {(u, L0x(u)):u∈Gx}x∈of the normal bundle. SinceP−T commutes with each

Psfor alls∈R_{, we see thatE}˜ _isPs_{-invariant for alls∈R.}

Claim 3. The subbundleE= ˜E⊕EX_isDX

t-invariant.

Indeed, for anyv=u+βX(x)withu∈ ˜Ex, β∈R, we haveDXt·v=DXt·u+βX(Xt(x))

andDXt·u=Pxt·u+γ X(Xt(x))for someγ ∈R, thusDXt·v=Pxt·u+(β+γ )X(Xt(x))

withPt

x·u∈ ˜EXt(x, henceDXt·v∈EXt(x)and the claim is proved.

At this point we have a continuous splitting TxM = Ex ⊕Fx, x ∈ . Using this

invariant splitting we can writeDX−t =

At(x) 0

0 Bt(x)

and the continuity of the splitting

ensures that _{(Ex, Fx)is bounded away from zero. This ensures that 1 |detDX−t| =}

|detAt(x)| · |Bt(x)| ·sin (Ex, Fx)and sinceBt(x)=DX−t|Fxwe have|Bt(x)|Ce−t λ/2,

hence|detAt(x)|grows exponentially fast.

This shows thathas a continuous invariant splittingE⊕F withFuniformly contracting andEarea expanding for the backward flow. In a three-dimensional manifold this shows that

Eis sectionally expanding for−X. From corollary1we have thatis a hyperbolic set for

−X, henceis a hyperbolic set and the proof of theoremCand corollary2is complete.

7. Examples

This section is devoted to presenting some examples.

7.1. Proof of proposition1

Here we present examples stated in proposition1.

Item (1). Consider a vector field on the planeR2_{with a double saddle homoclinic connection}

which expands volume and multiply this vector field by a contraction along the vertical direction; see figure1. For the construction of the planar vector field, see e.g. [23, chapter 4, section 9].

λ2+ λ₂−

W 1

W 4

s 3

−1 1

s s

4 _s 4

s

1

2

W

W 2

3

S1

λ+₁ λ

1 −

Figure 2.A sketch of Bowen’s example flow.

Indeed, since we assume that the singularitysexpands area, i.e.,|detDXt(x)|Lx|

ceλt_{for allt >0 and all 2-planeL}

x ⊂Fx, for some fixed constantsc, λ >0, we can

fix a neighbourhoodUofssuch that

|detDXt(x)|Lx|ce

λt_{, ∀}

0t 1, ∀x ∈U

and consequently

|detDXt(x)|Lx|ce

λt_{, ∀t >0}

whenever the positive orbit ofxis contained inU. Now we observe that there exists

T >0 such that for allx∈W\Uwe haveXt(x)∈Ufor allt∈Rsatisfying|t|> T.

Hence we can choosek=inf{c−1e−λt_|detDX

t(x)|Lx|:x ∈W\U,0t T}>0 and write|detDXt(x)|Lx|k·ce

λt_{for 0tT;and fort > T} |detDXt(x)|Lx| = |detDXt−T(XT(x))|LXT (x)| · |detDXT(x)|Lx|ce

λ(t−T )_· k·ceλT

=kc2eλt_.

So we can find a constantκ >0 so that|detDXt(x)|Lx| κe

λt _{for allx ∈W and}

everyt 0.

To obtain anon-dominatedsplitting, just choose the contraction rate alongEto be weaker than the contraction rate of the singularitys.

Example 3. Considering, in the previous item(1),as the union of the saddleswith only one homoclinic connection, we haveTM =E⊕F a non-dominated continuous splitting over a compact invariant and isolated set containing a singularity of the flow. Moreover,is theα-limit set of all points of the planeR2× {0}in the region bounded by the curveexcept the sink; see [23, chapter 4, section 9].

Item (2). Consider the flow known as the ‘Bowen example’; see e.g. [29] and figure2. We have chosen to reverse time with respect to the flow studied in [29] so that the heteroclinic connectionW = {s1, s2} ∪W1∪W2 ∪W3∪W4 is now a repeller. The past orbit

under this flowφt of every pointz∈ S1×[−1,1] =Mnot inW accumulates on

either side of the heteroclinic connection, as suggested in the figure, if we impose the conditionλ−₁λ−₂ < λ₁+λ+₂ on the eigenvalues of the saddle fixed pointss1 ands2

W

N

S

s

1

s

2

s

s

Figure 3.The Bowen flow embedded in a sphere.

We can now embed this system in the 2-sphere putting two sinks at the ‘north and south poles’ ofS2_{; see figure3. Let us denote the corresponding vector field onS}2

byX.

Let Y be a vector field in S1 _{corresponding to the ‘north-south’ system: the}

corresponding flow has only two fixed pointsN andS,N is a source andSa sink. We can choose the absolute value of the eigenvalues ofY atN, Sto be between the absolute value of the eigenvalues ofs1, s2.

In this way, for the flow{Zt}t∈Rassociated to the vector fieldX×Y onS2×S1, the singularities{si×S}i=1,2of the invariant set:=W× {S}are hyperbolic saddles,

and the splittingT(S2×S1)=E⊕F given by

E(p,S)= {0} ×TSS1 and F(p,S)=TpS2× {0} forp∈W,

is continuous,DZt-invariant,Eis uniformly contracted andF is area expanding (by

an argument similar to the previous example), but it is not dominated.

Example3and items(1)and(2)of proposition1show that theoremAis false without the dominating condition on the splittingE⊕F at the set of singularities within.

Item (3). Now we present a suspension flow whose base map has a dominated splitting but the flow does not admit any dominated splitting.

Letf : T4 _×T4 _{be the diffeomorphism described in [8] which admits a continuous}

dominated splittingEcs_⊕Ecu_onT4_{, but does not admit any hyperbolic (uniformly contracting}

or expanding) subbundle. There are hyperbolic fixed points off satisfying (see figure4):

• dimEu_(p)=2=dimEs_{(p)and there exists no invariant one-dimensional subbundle of} Eu_(p);

• dimEu_(p)=2=dimEs_{(p) and there exists no invariant one-dimensional subbundle of} Es_(p);

• dimEs_{(q)=3 and dimE}u_(q)=3.

s

q

p

Figure 4.Saddles with real and complex eigenvalues.

splitting overT4_{×[0,1] with bundles of least dimension, and is not dominated since at the}

pointpthe flow directionEX_{(p)dominates theE}cs_(p)=Es_{(p)direction, but at the pointq}

this domination is impossible.

This completes the proof of proposition1.

7.2. Proof of proposition2

We present first some auxiliary notions.

Letf :M→Mbe aC1_{-diffeomorphism andŴ⊂Ma compactf-invariant set, that is,} f (Ŵ)=Ŵ. We say thatŴistransitiveif there existsx ∈Ŵwith dense orbit: the closure of

{fn_{(x):n1}equalsŴ. The invariant setŴistopologically mixingif, for each pairU, V}

of non-empty open sets ofŴ, there existsN =N (U, V )∈ Z+_{such that}U∩fn_{(V )= ∅for}

alln > N.

We say thatŴis an attracting set if there exists a neighbourhoodUofŴinMsuch that

∩n0fn(U )=Ŵ; in this case we say thatUis an isolating neighbourhood forŴ. Anattractor

is a transitive attracting set and arepelleris an attractor for the inverse diffeomorphismf−1. It is well known that hyperbolic attractors or repellers can be decomposed into finitely many compact subsets which are permuted and each of these pieces is topologically mixing for a power of the original map; see e.g. [25].

Now, we present the example stated in proposition2. Consider a hyperbolic attractor (a Plykin attractor, see [24]) Ŵ, with an isolating neighbourhood U, defined on the two-dimensional sphereS2 _{for a diffeomorphismf :S}2 _{→ S}2_{, with a splittingT}

ŴS2 =E⊕F

satisfyingDf |Eλs andDf |F λuwhere 0< λs<1< λuandλs·λu<1.

Now we consider the following diffeomorphism

g=f1×f2:M→M, withM=S2×S2,

wheref1 := f2 andf2 := f−1. We note thatŴ1 := ∩n0f1n(U )is a hyperbolic attractor

with respect to f1 whose contraction and expansion rates along its hyperbolic splitting TŴ1S

2 _{= E}

1⊕F1 are λ2s, λ2u, respectively. Likewise Ŵ2 := ∩n0f2n(U ) is a hyperbolic

repeller with respect tof2whose contraction and expansion rates along its hyperbolic splitting TŴ2S

2 _{= E}

2⊕F2 areλu−1, λ−s1, respectively. This implies in particular thatDf1 contracts

area nearŴ1, that is,λ2s ·λ2u<1 and thatDf2expands area nearŴ2, that is,(λs·λu)−1>1.

Moreover, the set

Ŵ:=Ŵ1×Ŵ2

is a hyperbolic set forg, whose hyperbolic splitting is given by(E1×E2)⊕(F1×F2).

the contraction rate alongE1, butEis uniformly contracted andF is uniformly

sectionally-expanded, since the Lyapunov exponents 2 logλu,−logλu,−logλs are such that each pair

has positive sum.

On the other hand, the decompositionTŴM=TŴ1S

2_×T Ŵ2S

2_{=E⊕FwhereE=E} 1⊕F1

andF =E2⊕F2 is continuous, since each subbundle is continuous; is alsoDg-invariant;

but it is not a dominated decomposition. In fact, we have the contraction/expansion rates 0 < λ2_s < 1 < λ2_u associated toE and the corresponding rates λ−_u1, λ−_s1 associated to F, satisfying the relationλ2_s < λ−_u1<1< λ−_s1< λ2_u(since 0< λs·λu<1 and 0< λs <1< λu

imply thatλ2_s < λ−_u1 andλ_s−1 < λ2_u). Therefore the splittingE⊕F cannot be dominated. In addition, as noted above, bothDg−1 |EandDg |F expand area, so thatEis sectionally

contracted andF is sectionally expanded.

Finally, since bothf1 |Ŵ1andf2 |Ŵ2are topologically mixing, theng|Ŵis transitive; see e.g. [32].

This example shows that there are transitive hyperbolic sets with a non-dominated, although continuous, splitting satisfying the sectional-expansion and uniform contraction or sectional-contraction conditions. This completes the proof of proposition2.

Acknowledgments

This is a part of the PhD thesis of LS at Instituto de Matem´atica-Universidade Federal do Rio de Janeiro under a CNPq (Brazil) scholarship and she is now supported by an INCTMat-CAPES post-doctoral scholarship at IMPA.

The authors thank C Morales and L D´ıaz for many observations that helped improve the statements of the results.

Authors were partially supported by CNPq, PRONEX-Dyn.Syst., FAPERJ and FAPESB.

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