LYAPUNOV FUNCTIONS, STRONGLY HOMOGENEOUS SETS AND STAR FLOWS

LUCIANA SALGADO

ABSTRACT. We say that a flow or vector fieldX∈X1₍M)isstarif there exists a neighborhood U_⊂X1₍M)ofX for which every closed orbit of every vector fieldY inU _{is hyperbolic. In this} work, it is presented a characterization of star condition for flows based on Lyapunov functions. It is obtained conditions to strong homogeneity for singular sets for aC1flow by using the notion of infinitesimal Lyapunov functions. As an application we obtain some results related to singular hyperbolic sets for flows.

CONTENTS

1. Introduction and statement of results 1

1.1. Preliminary definitions and Main results 3

2. Fields of quadratic forms 6

2.1. J-separated linear maps 8

2.2. Lyapunov exponents 10

3. Some applications 10

3.1. Proof of Corollaries 3.3 and 3.5 11

4. Proof of Theorems 12

References 14

1. INTRODUCTION AND STATEMENT OF RESULTS

Since Morales, Pac´ıfico and Pujals in [26] defined the so calledsingular hyperbolic systems,

many researchers have worked about this notion in order to understand it as an extension of the hyperbolic theory for invariant sets for flows which are not (uniformly) hyperbolic, but which have some robust properties, certain kind of weaker hyperbolicity and also admit singularities.

Date: June 26, 2017.

2000Mathematics Subject Classification. Primary: 37D30; Secondary: 37D25.

Key words and phrases. Dominated splitting, partial hyperbolicity, sectional hyperbolicity, Lyapunov function. L.S. is partially supported by a Fapesb-JCB0053/2013, PRODOC/UFBA 2014, CNPq, INCTMat-CAPES. This is the last version of the paper presented at the 11th AIMS Conference on Dynamical Systems, Differential Equations and Applications 2016 and L.S. thanks Alexander Arbieto for his comments and suggestions and Federal University of Rio de Janeiro for 2017 postdoc position, whose support and hospitality helped obtain deep improvements of the previous version. She also thanks Instituto de Matematica Pura e Aplicada - IMPA for 2012 postdoc financial support, where the seminal version of this paper has been structured.

In [25], the same authors proved that every robustly transitive singular set for a three dimensional flow is a partially hyperbolic attractor or repeller and the singularities in this set must be Lorenz-like. In [11], Gan, Li and Wen generalized the result in [25] assuming that the set is also strongly

homogeneous. We recall that a compact invariant setΛisrobustly transitivefor a vector fieldX

if there exist a neighborhoodU ofΛand a neighborhood

U

Y ∈

U

non-trivially (not a single orbit) transitive. Astrongly homogeneousset of index 0≤ind(Λ)≤n−1

for a flow Xt is such that it cannot beC1-approximated by flows which have some hyperbolic

periodic orbit of index different of ind(Λ)in a neighborhoodU ofΛ.

In this paper, we prove a relation between the J-algebra of Potapov [29, 30, 38], star flows

and strong homogeneity. Then, we apply this to obtain some results about singular hyperbolic systems.

TheJ-algebra here means a pseudo-euclidean structure given byC1non-degenerate quadratic form J, defined onΛ, which generates positive and negative cones with maximal dimension p andq, respectively, with p+q=dim(M).

The maximal dimension of a cone inTxMis the maximal dimension of the subspaces contained

in there.

This algebraic/geometric approach has been very useful in the study of weak and uniform hyperbolicity, see [16], [17], [38]. In [4], this author jointly with V. Ara´ujo, obtained

charac-terizations of partial and singular/sectional hyperbolicity based on J-algebra. In [5], the same

authors proved an equivalence between dominated splittings for the flow and dominated

split-tings for thek-th exterior powers of the tangent cocycle.

More results relating geometric and algebraic features of singular hyperbolicity can be view in [4], [5], [32], for the classical sectional and singular hyperbolicity definitions, and [34] for singular hyperbolicity in a broad sense involving sectional expansion of intermediate dimensions between two and the full dimension of the central subbundle.

The main theorem rounds about of the so calledstar flows. Star systems has been studied by

many renowned researchers, among them R. Ma˜n´e and S. Liao, whom many years ago used it in

order to prove the famousstability conjecturefrom Palis and Smale. For more details about star

systems, see for instance [19],[22],[28],[23],[12],[13],[7].

For now, it plays a crucial role in Palis’ Conjecture, which leads with global behaviour of

dynamical systems involving hyperbolicity.

So, we hope that this work contributes is this direction.

Definition 1. A flowXt is said to be star if it cannot beC1-approximated by ones exhibiting

nonhyperbolic periodic orbits.

The second result guarantees that a compact connected invariant set for a flow is strongly

homogeneous under the existence of a field of non-degenerate quadratic forms J defined on a

neighborhood of this set such a way that the flow derivativeDXt keeps positive conesC+(x):=

{0} ∪ {v∈TxM;J(x)v>0}invariants, i.e.,DXt(C+(x))⊂C+(Xt(x)), for allt>0,x∈Λand the

As an application, we obtain some results about partial hyperbolicity for robustly transitive strongly homogeneous singular sets of [11] and transitive set of [7].

The text is organized as follow. In first section, it is given the main definitions and stated

the results. In second section, it is presented the main tools by using the notion of J-algebra

of Potapov. In third section, it is given some applications concerning singular hyperbolicity. In fourth section is proved the main theorems.

1.1. Preliminary definitions and Main results.

Before presenting the main statements, we give some definitions.

Let M be a connected compact finite d-dimensional manifold, d ≥3, without boundary,

to-gether with a flowXt :M→M,t ∈Rgenerated by aC1vector fieldX:M→T M.

Aninvariant setΛfor the flow ofX is a subset ofMwhich satisfiesXt(Λ) =Λfor allt∈R.

Atrapping region Ufor a flowXt is an open subset of the manifoldMwhich satisfies: Xt(U)

is contained inU for allt>0; and there existsT >0 such thatXt(U)is contained in the interior

ofU for allt >T. The maximal invariant setΛX(U):=∩t≥0Xt(U)ofU is called anattracting

set. An attracting set forX which is transitive is called anattractorforX. ArepellerforX is an

attractor for−X.

We say that a setΛisLyapunov stableif for every neighborhoodU ofΛthere is another one

V ⊂U such that every point p∈V has its forward orbit contained inU.

A singularity for the vector field X is a point σ∈M such that X(σ) =0 or, equivalently,

Xt(σ) =σfor allt∈R. The set formed by singularities is thesingular set of X denoted Sing(X)

and Per(X) is the set of periodic points of X. We say that a singularity is hyperbolic if the

eigenvalues of the derivative DX(σ) of the vector field at the singularity σ have nonzero real

part. The set of critical elements ofX is the union of the singularities and the periodic orbits of

X, and will be denoted by Crit(X).

We recall that an invariant setΛfor a flowXtis an invariant subset ofMwith a decomposition

T_ΛM=Es⊕EX⊕Euof the tangent bundle overΛwhich is a continuous splitting, whereEX is

the direction of the vector field, the subbundles are invariant under the derivativeDXt of the flow

DXt·Ex∗=EX∗t(x), x∈Λ, t∈R, ∗=s,X,u;

Es is uniformly contracted byDXt andEuis uniformly expanded: there areK,λ>0 so that

kDXt|Es

xk ≤Ke

−λt_{, kDX}

−t|Eu

x k ≤Ke

−λt_{, x∈Λ, t∈}

R. (1.1)

Now, we present the definition of strong homogeneity. Recall that the index of a hyperbolic periodic orbit of a flow is the dimension of the contracting subbundle of its hyperbolic splitting.

Definition 2. We say that a setΛ is strongly homogeneous of index ind for a flow Xt, if there

exist neighborhoodsUofΛandUofX such that all periodic orbits inUwith respect to any flow

inUhave index ind.

We say that a point x∈M is nonwandering for X provided for every neighborhoodU ofx

there ist>0 such thatXt(x)∩U 6= /0, i.e., there exists a pointy∈U withXt(y)∈U. We denote

byΩ(X)the non-wandering set ofX.

An ε-chain from x0 to xl forX is a sequence {x0,x1,· · ·,xl}such that for all 0≤ j≤l, the

We define thechain recurrent setofX byR(X) ={x∈M; there is anε−chain fromxtox}.

We say that two points are chain equivalent provided, givenε>0, there is an ε-chain fromx

to y and from y to x. It is known that this is an equivalence relation, the equivalence classes

are calledchain componentsofR(X)and, for flows, the components are actually the connected

components ofR(X). IfX admits a single chain component on an invariant setΛ, we say thatX

ischain transitiveonΛ. See [31], for instance.

Finally, recall that a point p∈ M is said to be a C1 preperiodic point of X if for any C1

neighborhood

V

V

q∈Per(g). We denote this set byP∗(X), and it is easy to see that it is closed andX-invariant.

We can assume that the periodic points are hyperbolic and we can define aC1i-preperiodic point

pofX, 0≤i≤d, if there are sequencesXnof flows and pnof periodic points ofXnwith indexi

such that

lim

n→∞Xn=X and limn→∞pn=p.

We denote byP_∗i(X)the set ofC1i-preperiodic ofX and, then,P∗(X) =∪di=0P∗i(X).

FromC1Pugh’s closing lemma we haveΩ(X)⊂P∗(X)⊂R(X).

Remark 1.1. As proved by Wen [36],C1preperiodic sets do not explode underC1perturbations,

i.e., for any 0≤i≤d and for any neighborhoodU ofP∗(X), there is aC1neighborhood

V

such thatP∗(Y)⊂U, for anyY ∈

V

Indeed, for example in [35], we can see that the recurrent set do not explode underC0

pertur-bations., i.e., ifU is a neighborhood ofR(X)there is a neighborhood

V

for allY ∈

V

Our main results are the following.

LetΛ⊂M be a compact invariant subset forX.

Theorem A. A vector field X∈X1_{(M)is star if, and only if, satisfy all of the next properties:}

(1) there is a neighborhood U of P∗(X)and a field of quadratic formsJwith index0≤ind≤ dim(M)−1defined on the preperiodic set P∗(X), C1 along the flow direction over each preperiodic orbit, such that X is strictlyJ-separated on every p∈P∗(X);

(2) for everyσ∈Sing(X|U)and∀v∈TσM,J′(v)>0;

(3) the linear Poincar´e flow Pt associated to each preperiodic orbit γ of X|U is strictly

J-monotone.

The next result give us a characterization of strong homogeneity based on quadratic forms.

Theorem B. A compact invariant setΛfor X ∈X1_{(M)is strongly homogeneous with indexind}

if and only if there is a neighborhood U ⊂ M of Λ and a continuous field of non-degenerate quadratic forms J on U with fixed index ind(J) =ind such that the preperiodic set P∗(X) of X|U is strictlyJ-separated and the associated linear Poincar´e flow Pt is strictlyJ-monotone on

P∗(X|U).

Note that in the last result monotonicity is only required on preperiodic orbits. If we require

it over any nonsingular compact invariant subsets inΛ, it is possible evaluate the index of

Corollary C. Let Λ be a maximal invariant set of a neighborhood U for X ∈X1_{(M). Then,} Λ is strongly homogeneous with index indandind(σ)≥indfor all σ∈Sing(X|_Λ)if there is a field of non-degenerate C1quadratic formsJon U with indexind(J) =indsuch that X is strictly J-separated, the associated linear Poincar´e flow Pt is strictly J-monotone on every compact invariant nonsingular subset K of U and for everyσ∈Sing(X|U)and∀v∈TσM,J′(v)>0.

We may ask if the converse is valid. But, just by supposing strongly homogeneity, we couldn’t obtain the field of quadratic forms, because we need some kind of decomposition on the tangent bundle for create the cones. However, if we have certain orbit recurrence in the set we get the following result.

Theorem D. Let Λ be a compact invariant set whose singularities are hyperbolic (if any) and accumulated by regular orbits for a C1vector field X , which is strongly homogeneous with index indandind(σ)>indfor allσ∈Sing(X|_Λ). Then, there exists a field of non-degenerate quadratic formsJ onΛ with indexind(J) =ind(Λ)for which X isJ-separated and the associated linear Poincar´e flow Pt is strictlyJ-monotone on every compact invariant nonsingular subsetΓofΛ.

Remark 1.2. The definitions concerning the quadratic forms are given in the next section. Definition 3. A dominated splittingover a compact invariant set Λ of X is a continuousDXt

-invariant splittingT_ΛM=E⊕F withEx6={0},Fx6={0}for everyx∈Λand such that there are

positive constantsK,λsatisfying

kDXt|Exk · kDX−t|F_Xt(x)k<Ke

−λt_{, for allx∈Λ, and allt>0. (1.2)}

A compact invariant setΛis said to bepartially hyperbolicif it exhibits a dominated splitting

T_ΛM =E⊕F such that subbundle E is uniformly contracted. In this case F is the central

subbundleofΛ.

A compact invariant setΛis said to besingular-hyperbolicif it is partially hyperbolic and the

action of the tangent cocycle expands volume along the central subbundle, i.e.,

|det(DXt|Fx)|>Ceλ

t_{,∀t>0, ∀x∈Λ. (1.3)}

The following definition was given as a particular case of singular hyperbolicity.

Definition 4. Asectional hyperbolic set is a singular hyperbolic one such that for every

two-dimensional linear subspaceLx⊂Fxone has

|det(DXt|Lx)|>Ceλ

t_{,∀t>0. (1.4)}

If we require, in addition, that the field direction must be inside the non-positive cone, we obtain an equivalence between the existence of such a quadratic forms and singular hyperbolicity. In [34], this author give another definition of singular hyperbolicity encompassing two previ-ous as follow and improving a similar one contained in [1].

Definition 5. A compact invariant setΛ⊂Mis p-sectional hyperbolic or singular hyperbolic of order pforX if all singularities inΛare hyperbolic, there exists a partially hyperbolic splitting

of the tangent bundle onT_ΛM=E⊕F and constantsC,λ>0 such that for everyx∈Λand every

(1) kDXt|Exk ≤Ce−λ

t_;

(2) | ∧p_DX

t|Lx|>C

−1_eλt_{, for every p-dimensional linear subspaceL}

x⊂Fx.

In our applications here, we only deal with two dimensional singular hyperbolic case, but we

conjecture that analogous results hold for singular hyperbolic sets of any order p, with 2≤p≤

dimF.

From now on, we consider M a connected compact finite dimensional riemannian manifold

and all singularities ofX (if they exist) are hyperbolic.

2. FIELDS OF QUADRATIC FORMS

In this section, we introduce the quadratic forms and its properties.

Let J: EU → R be a continuous family of quadratic forms Jx : Ex → R which are

non-degenerate and have index 0<q<dim(E) =n, whereU⊂Mis an open set such thatXt(U)⊂U

for a vector fieldX. We also assume that(J_x)x∈U is continuously differentiable along the flow.

The continuity assumption onJjust means that for every continuous sectionZofEU the map

U →Rgiven byx7→J(Z(x))is continuous. TheC1assumption onJalong the flow means that

the mapx7→J_Xt(x)(Z(Xt(x)))is continuously differentiable for allx∈U and eachC1 sectionZ

ofEU.

The assumption thatM is a compact manifold enables us to globally define an inner product

inE with respect to which we can find the an orthonormal basis associated toJ_x for each x, as

follows. Fixing an orthonormal basis onExwe can define the linear operator

Jx:Ex→Ex such that Jx(v) =<Jxv,v> for all v∈TxM,

where <, >=<, >x is the inner product at Ex. Since we can always replaceJx by (Jx+Jx∗)/2

without changing the last identity, where J_x∗ is the adjoint of Jx with respect to <, >, we can

assume that Jx is self-adjoint without loss of generality. Hence, we represent J(v) by a

non-degenerate symmetric bilinear form<J_xv,v>x. Now we use Lagrange’s method to diagonalize

this bilinear form, obtaining a base{u1, . . . ,un}ofExsuch that

J_x(

∑

i

αiui) = q

∑

i=1

−λiα2i + n

∑

j=q+1

λjα2j, (α1, . . .,αn)∈Rn.

Replacing each element of this base according tovi=|λi|1/2uiwe deduce that

J_x(

∑

i

αivi) = q

∑

i=1

−α2i + n

∑

j=q+1

α2

j, (α1, . . . ,αn)∈Rn.

Finally, we can redefine <, > so that the base {v1, . . .,vn} is orthonormal. This can be done

smoothly in a neighborhood ofxin Msince we are assuming that the quadratic forms are

non-degenerate; the reader can check the method of Lagrange in a standard Linear Algebra textbook and observe that the steps can be performed with small perturbations, for instance in [21].

In this adapted inner product we have thatJxhas entries from{−1,0,1}only,Jx∗=Jxand also

Having fixed the orthonormal frame as above, thestandard negative subspaceatxis the one

spanned byv1, . . .,vqand thestandard positive subspaceatxis the one spannedvq+1, . . . ,vn.

2.0.1. Positive and negative cones. LetC_±={C±(x)}x∈U be the family of positive and negative

cones

C±(x):={0} ∪ {v∈Ex:±Jx(v)>0} x∈U

and also letC₀={C0(x)}x∈U be the corresponding family of zero vectorsC0(x) =J−1x ({0})for

allx∈U. In the adapted coordinates obtained above we have

C0(x) ={v=

∑

i

αivi∈Ex: n

∑

j=q+1

α2

j = q

∑

i=1

α2

i}

is the set ofextreme pointsofC±(x).

The following definitions are fundamental to state our main result.

Definition 6. Given a continuous field of non-degenerate quadratic formsJwith constant index

on the trapping regionU for the flowXt, we say that the flow is

• J-separatedifDXt(x)(C+(x))⊂C+(Xt(x)), for allt>0 andx∈U;

• strictlyJ-separatedifDXt(x)(C+(x)∪C0(x))⊂C+(Xt(x)), for allt>0 andx∈U;

• J-monotoneifJ_Xt(x)(DXt(x)v)≥Jx(v), for eachv∈TxM\ {0}andt>0;

• strictlyJ-monotoneif∂t JXt(x)(DXt(x)v)

|t=0>0, for allv∈TxM\ {0},t>0 andx∈U;

• J-isometryifJ_Xt(x)(DXt(x)v) =Jx(v), for eachv∈TxMandx∈U.

Thus,J-separation corresponds to simple cone invariance and strictJ-separation corresponds

to strict cone invariance under the action ofDt(x).

Remark 2.1. If a flow is strictly J-separated, then for v∈TxM such that Jx(v)≤0 we have

J_X−t(x)(DX−t(v))<0 for all t >0 and x such that X−s(x)∈U for every s∈[−t,0]. Indeed,

otherwiseJ_X

−t(x)(DX−t(v))≥0 would implyJx(v) =Jx DXt(DX−t(v))

>0, contradicting the

assumption thatvwas a non-positive vector.

This means that a flowXt is strictlyJ-separated if, and only if, its time reversalX−t is strictly

(−J)-separated.

A vector fieldX isJ-non-negativeonU ifJ(X(x))≥0 for allx∈U, andJ-non-positiveonU

ifJ(X(x))≤0 for allx∈U. When the quadratic form used in the context is clear, we will simply

say thatX is non-negative or non-positive.

We apply this notion to the linear Poincar´e flow defined on regular orbits ofXt as follows.

Suppose that the vector field X is non-negative onU. Then, the span E_xX of X(x)6=0 is a

J-non-degenerate subspace.

According to item (1) of Proposition2.3, we have thatTxM=ExX⊕Nx, whereNxis the

pseudo-orthogonal complement ofE_xX with respect to the bilinear formJ, andNxis also non-degenerate.

Moreover, by the definition, the index ofJrestricted toNx is the same as the index ofJ. Thus,

Define the Linear Poincar´e FlowPtofXt along the orbit ofx, by projectingDXt orthogonally

(with respect toJ) overNXt(x) for eacht∈R:

Ptv:=ΠXt(x)DXtv, v∈TxM,t∈R,X(x)6=0,

whereΠXt(x):TXt(x)M→NXt(x) is the projection onNXt(x) parallel toX(Xt(x)). We remark that

the definition ofΠxdepends onX(x)andJX only. The linear Poincar´e flowPt is a linear

multi-plicative cocycle overXton the setU with the exclusion of the singularities ofX.

In this setting we can say that the linear Poincar´e flow is (strictly)J-separated and (strictly)

J-monotonous using the non-degenerate bilinear formJrestricted toNx for a regularx∈U. More

precisely: Pt isJ-monotonous if∂tJ(Ptv)|t=0≥0, for eachx∈U,v∈TxM\ {0}andt>0, and

strictlyJ-monotonous if∂tJ(Ptv)|t=0>0, for allv∈TxM\ {0},t>0 andx∈U.

Proposition 2.2. Let L:V →V be aJ-separated linear operator. Then

(1) L can be uniquely represented by L=RU , where U is aJ-isometry and R isJ-symmetric (orJ-pseudo-adjoint; see Proposition2.3) with positive spectrum.

(2) the operator R can be diagonalized by a J-isometry. Moreover the eigenvalues of R satisfy

0<rq₋≤ · · · ≤r1₋=r−≤r+=r+₁ ≤ · · · ≤r+p.

(3) the operator L is (strictly)J-monotonous if, and only if, r− ≤(<)1and r+≥(>)1.

2.1. J-separated linear maps.

2.1.1. J-symmetrical matrixes and J-selfadjoint operators. The symmetrical bilinear form

de-fined by (v,w) =hJxv,wi, v,w∈Ex for x∈M endows Ex with a pseudo-Euclidean structure.

SinceJ_x is non-degenerate, then the form(·,·)is likewise non-degenerate and many properties

of inner products are shared with symmetrical non-degenerate bilinear forms. We state some of them below.

Proposition 2.3. Let(·,·):V×V →Rbe a real symmetric non-degenerate bilinear form on the real finite dimensional vector space V .

(1) E is a subspace of V for which(·,·)is non-degenerate if, and only if, V =E⊕E⊥. We recall that E⊥ :={v∈V :(v,w) =0 for all w ∈E}, the pseudo-orthogonal space of E, is defined using the bilinear form.

(2) Every base{v1, . . .,vn}of V can be orthogonalized by the usual Gram-Schmidt process of

Euclidean spaces, that is, there are linear combinations of the basis vectors{w1, . . . ,wn}

such that they form a basis of V and (wi,wj) =0 for i6= j. Then this last base can be

pseudo-normalized: letting ui=|(wi,wi)|−1/2wiwe get(ui,uj) =±δi j,i,j=1, . . . ,n.

(3) There exists a maximal dimension p for a subspace P+ ofJ-positive vectors and a

maxi-mal dimension q for a subspace P− ofJ-negative vectors; we have p+q=dimV and q is known as theindexofJ.

(4) For every linear map L:V →_Rthere exists a unique v∈V such that L(w) = (v,w)for each w∈V .

(6) Every pseudo-self-adjoint L:V →V , that is, such that L=L+, satisfies

(a) eigenspaces corresponding to distinct eigenvalues are pseudo-orthogonal; (b) if a subspace E is L-invariant, then E⊥is also L-invariant.

The proofs are rather standard and can be found in [21].

The following simple result will be very useful in what follows.

Lemma 2.4. Let V be a real finite dimensional vector space endowed with a non-positive definite and non-degenerate quadratic formJ:V →R.

If a symmetric bilinear form F:V×V →Ris non-negative on C0then

r+= inf

v∈C+

F(v,v)

hJv,vi ≥usup∈C−

F(u,u)

hJu,ui =r−

and for every r in[r−,r+]we have F(v,v)≥rhJv,vifor each vector v.

In addition, if F(·,·)is positive on C0\ {0}, then r−<r+and F(v,v)>rhJv,vifor all vectors

v and r∈(r−,r+).

Remark 2.5. Lemma 2.4 shows that if F(v,w) =h_Jv˜_{,wi for some self-adjoint operator ˜J and}

F(v,v)≥0 for allvsuch thathJv,vi=0, then we can finda∈Rsuch that ˜J≥aJ. This means

precisely thath_Jv˜_{,vi ≥ahJv,vifor allv.}

If, in addition, we have F(v,v) >0 for all v such that hJv,vi=0, then we obtain a strict

inequality ˜J>aJ for some a∈R since the infimum in the statement of Lemma2.4 is strictly

bigger than the supremum.

The (longer) proofs of the following results can be found in [38] or in [30]; see also [39].

For aJ-separated operatorL:V →V and ad-dimensional subspace F+⊂C+, the subspaces

F+ andL(F+)⊂C+ have an inner product given byJ. Thus both subspaces are endowed with

volume elements. Letαd(L;F+) be the rate of expansion of volume of L|F+ and σd(L)be the

infimum ofαd(L;F+)over alld-dimensional subspacesF+ofC+.

Proposition 2.6. We haveσd(L) =r+1· · ·r+d, where ri+ are given by Proposition2.2(2).

Moreover, if L1,L2areJ-separated, thenσd(L1L2)≥σd(L1)σd(L2).

The following corollary is very useful.

Corollary 2.7. ForJ-separated operators L1,L2:V →V we have

r₊1(L1L2)≥r+1(L1)r1+(L2) and r−1(L1L2)≤r1−(L1)r1−(L2).

Moreover, if the operators are strictlyJ-separated, then the inequalities are strict.

Remark 2.8. Another important property about the singular values of aJ-separated operatorL is that

r₊1 =r+≥1(>1) and r−1 =r−≤1(<1)

if, and only if,Lis (strictly)J-monotone.

2.2. Lyapunov exponents.

It is well known that under conditions of measurability, by Oseledec’s Ergodic Theorem [27],

there exist a full probability setX such that for everyx∈Y there is an invariant decomposition

TxM=hXi ⊕E1(x)⊕ · · · ⊕El(x)(x)

and numbersχ1<· · ·<χl correponding to the limits

χj= lim t→+∞

1

t logkDXt(x)·vk, for everyv∈Ei(x)\ {0},i=1,· · ·,l(x).

In this setting, Wojtkowski [38] proved that the logarithm of the pseudo-Euclidean singular

values 0≤r_q−≤ · · · ≤r₁−≤r+₁ ≤ · · · ≤r+_p ofDXt areµ-integrable, and obtained estimates of the

Lyapunov exponents related to the singular eigenvalues of strictlyJ-separated maps.

Theorem 2.9. [38, Corollary 3.7]For1≤k1≤p and1≤k2≤q

χ−

1 +· · ·+χ −

k1 ≤

k1

∑

i=1 Z

logr_i−dµ andχ+₁ +· · ·+χ+_k2 ≥

k2

∑

i=1 Z

logr_i+dµ.

This result will be very useful in proof of TheoremD.

3. SOME APPLICATIONS

3.0.1. Some results about partial and sectional hyperbolicity fromJ-separation.

The author, together with V. Ara´ujo, proved in [4] the following useful theorem which relates

partial hyperbolicity andJ-separated sets for a flow.

Theorem 3.1. [4, Theorem A] A maximal invariant subset Λ of a trapping region U whose singularities are hyperbolic is a partially hyperbolic set for a flow Xt if, and only if, there is a

C1fieldJof non-degenerate quadratic forms with constant index, equal to the dimension of the stable subspace ofΛ, such that Xt is a non-negative strictlyJ-separated flow on U .

This result will be useful in our applications of TheoremB.

In the sequence, we can give another proof of next result from [11].

Theorem 3.2. [11, Theorem A]Let X ∈X1_{(M), and}Λ _{be a robustly transitive singular set of}

X that is strongly homogeneous of index ind. If every singularityσof X is hyperbolic of index

ind(σ)>ind, thenΛhas a partially hyperbolic splitting of contracting dimension Ind. Likewise,

if every singularityσof X is hyperbolic of indexind(σ)≤ind, thenΛhas a partially hyperbolic splitting of expanding dimension n−1−Ind.

Proof. We are going to deal with the case ind(σ)>ind, the other case is analogous.

SinceΛis strongly homogeneous and ind(σ)>ind, by [11, Lemma 4.1] there is a dominated

splittingT_σM=E_σ⊕F_σ such that dim(E_σ) =ind. Hence, TheoremDimplies that there exists a

J-separated and the associated linear Poincar´e flowPt is strictly J-monotone on every compact

invariant subsetγofΛ∗. Therefore, Theorem3.1completes the proof.

Some immediate results follow from the main theorems.

The following consequences of these results follows from the robustness of sectional-hyperbolicity and the theory of sectional-hyperbolic transitive sets for homogeneous flows from [24] and [7].

Corollary 3.3. Let X ∈X1_{(M),dim(M)≥4with a nontrivial transitive compact invariant set}Λ

whose singularities, if any, are hyperbolic. Then the following conditions are equivalent:

(1) There exists a familyJof smooth non-degenerate indefinite quadratic forms with constant index ind(J) on Λ such that X is a non-negative strictly J-separated vector field, for which the linear Poincar´e flow is strictly J-monotonous on every compact invariant set inΛX(U)∗=ΛX(U)\Sing(X)

(2) The setΛis a sectional-hyperbolic subset for X with constant indexind(O) =ind(J)for all periodic orbitsOofΛandind(σ) =Ind(J) +1for all singularitiesσ∈Λ∩Sing(X).

For the next statement, we recall that a hyperbolic singularityσis said to be of codimension

one if its index satisfies either ind(σ) =1 or ind(σ) =n−1, wheren=dim(M).

Remark 3.4. Every attracting set is Lyapunov stable.

Corollary 3.5. LetΛ⊂Mn,n≥4, be a nontrivial transitive set, which is Lyapunov stable for X , with singularities all of them hyperbolic of codimension one. Then, the following properties are equivalent:

(1) Λis sectional-hyperbolic with1≤dim(Es) =ind(J)≤n−2;

(2) There exists a field of non-degenerate quadratic forms with constant index1≤ind(J)≤

n−2such that X is non-negative strictlyJ-separated onΛand every compact invariant subsetΓ⊂Λis strictlyJ-monotone for linear Poincar¨ı¿1₂ flow associated to X .

3.1. Proof of Corollaries3.3and3.5.

proof of Corollary3.3. Indeed, suppose that(1)is true. Then,X is strongly homogeneous onΛ.

By [7, Corollary 8], this is a sectional hyperbolic set forX. To prove the converse statement, we

need just use [4, Theorem D].

The next proof needs the following lemma.

LetΛbe a compact invariant set for a flowX of aC1vector fieldX onM.

Lemma 3.6. [1, Lemma 5.1]Given a continuous splitting T_ΛM=E⊕F such that E is uniformly contracted, then X(x)∈Fx for all x∈Λ.

Proof of Corollary3.5. Suppose that Λ is sectional-hyperbolic with decompositionE⊕F. So,

it is clearly strongly homogeneous. Once the subbundles are non-trivial and E is uniformly

contracting, we must have 1≤dim(E):=ind(J)≤n−2, because by Lemma3.6,hXi ⊂F.

By Theorem 3.1, there exists a field J of differentiable quadratic forms with constant index

Reciprocally, the existence of such a fieldJimplies, by TheoremB, thatΛis strongly

homo-geneous of index ind(J). Thus, once the singularities are hyperbolic of codimension one, it is

enough to use Lemma [7, Corollary 9].

4. PROOF OFTHEOREMS

Now, we prove our mains results.

To prove the first theorem we use the following result from [4].

Proposition 4.1. [4, item 3,Theorem 2.23]LetΓbe a compact invariant set for X with a domi-nated splitting T_ΓM=E⊕F . LetJ be a C1 field of indefinite quadratic forms such that DXt is

strictlyJ-separated. Then, E⊕F is uniformly hyperbolic if, and only if, there is an equivalent fieldJ of quadratic forms on a neighborhood ofΓsuch that J′(v)>0, for all v∈T_ΓM and all x∈Γ.

Proof of TheoremA. If X is a star flow, then each singular point σ is hyperbolic and its well

known that its hyperbolic decomposition E_σs ⊕E_σu is a dominated one. So, by using adapted

metrics (see [15]) we construct the desired quadratic form J_σ such that X is strictly separated

(see [4]) and, by Proposition4.1J′(v)>0 for allv∈T_ΓM.

Analogously, for every periodic orbit γ of X, consider the hyperbolic splitting T_γM=Es⊕

EX⊕Eu. Again, consideringEs⊕(EX⊕Eu)as a dominated splitting we obtain a quadratic form

J for whichX is strictly separated onγ. By construction of the adapted metrics, we have that

JisC1along the flow (see [15] for details about the construction of such a adapted metric). In

Addition, the linear Poincar´e flow associated toX,Pt is hyperbolic and thenJ-monotone onγ.

Ifγ is a sink (respectively, a source) the splittingEs⊕EX (respectively,Eu⊕EX) is a

domi-nated one and we proceed constructing the cones the same way, however the core of the nonega-tive cone is the field direction.

Reciprocally, take a small neighborhoodU ofPer∗(X)such that there is aC1neighborhood

V

ofX for whichPer∗(Y)⊂U and suppose that such a field of quadratic forms is defined onU. By

Proposition4.1, every singularityσ∈U is hyperbolic. The case of periodic orbits is analogous.

ShrinkingU, if necessary, we may suppose that, for each preperiodic orbit and each singularity

inU of eachY ∈

V

Indeed, since the quadratic form on each periodic orbit is C1 along the flow, for anyY ∈

V

shrinking

V

for the linear Poincar´e flow.

If, for someY ∈

V

the linear Poincar´e flowP_Yt associated toY, since it comes from a preperiodic one ofX which is

J-monotone for the linear Poincar´e flowP_Xt, by hyphotesis. See [4, Section 2.5.4].

Hence, every periodic orbit for anyY ∈

V

Now, it is proved the second main result.

Proof of TheoremB. Since the linear Poincar´e flow is strictly J-monotone on each preperiodic

index, which we denote ind(J). Moreover, taking a small enough neighborhoodU ofPer∗(X|_Λ)

there exists some neighborhood

V

Y)⊂U, ∀Y ∈

V

orbitγY is created by a smallC1perturbation ofX, it comes from a preperiodic orbit ofX. Thus,

γY is a hyperbolic closed orbit ofY, and must have index equal to ind(J), once that it is fixed.

Hence, ind(J) does not change by small differentiable perturbations ofX on a neighborhood

ofΛ, so the index of hyperbolic periodic orbits also does not change. Therefore, Λis strongly

homogeneous forX.

Reciprocally, ifΛis strongly homogeneous of index ind, then cannot be there a non-hyperbolic

periodic orbit. Otherwise, we can create two periodic orbits with different indices, by Frank’s

Lemma. Moreover,Xis a star flow in a neighborhood ofΛ. Hence, by TheoremAwe can define

the desired field of quadratic formsJ, with fixed index ind(J) =ind, defined on a neighborhood

UofΛ, whereU is the neighborhood for whichPer∗(Y|ΛY)⊂U for anyY close enough toX.

Proof of CorollaryC. Note that Corollary C follow from Theorem B, just observing now that

if any singularity σ is accumulated by regular orbits, it cannot present ind(σ)<indJ, once

X ∈X1_(M),Jis a continuous field of quadratic forms andX isJ- monotonic over any compact

invariant nonsingular setΓ.

Now, we prove our last main result.

First of all, we recall some definitions which are necessary here.

Let Z be a compact metric space and denote

M

Borelσ-algebra ofZ. If T :Z →Z is a measurable map, we say that a probability measureµis

an invariant measure ofT, ifµ(T−1(A)) =µ(A), for every measurable setA⊂Z. We say thatµ

is an invariant measure of X if it is an invariant measure ofXt for everyt ∈R. We will denote

by

M

µ∈

M

set of points for which the measure is non-zero. An invariant measure is said to beatomicif its

support is either a closed orbit or a singularity.

A probability measureµis anergodic measureif for every invariant setA we haveµ(A) =1

orµ(A) =0. Finally, a certain property is said to be valid inµ-almost every pointif it is valid in

the whole Z except, possibly, in a set of null measure.

We recall the definition ofδ-closable points of [22]. We say that a pointx∈M\Sing(X)is

δ-closable if, for anyC1 neighborhood

U

U

pointz∈MandT >0 such that:

(1) ZT(z) =z,

(2) Z=X onM\B_δ(X[0,T](x))and

(3) dist(Zt(z),Xt(x))<δ,∀0≤t≤T.

We denote byΣ(X)the set of points ofMwhich areδ-closable for anyδsufficiently small.

Proof of TheoremD. If Λ is a strongly homogeneous set for X with singularities all of them

hyperbolic, thenX is a star flow inΛ.

Ifx∈Λis a regularδ-closable point, then it is a pre-periodic point of index ind(Λ).

According the proof of [11, Lemma 5.3], we have a dominated splitting Ex⊕Fx of index

ind(Λ)inTxM, for allx.

By Theorem2.9, we have

χ−

1 +· · ·+χ −

k1≤

k1

∑

i=1 Z

logr−_i dνandχ₁++· · ·+χ+_k2 ≥

k2

∑

i=1 Z

logr+_i dν,

for anyk1≤q,k2≤p.

Also according the proof of [11, Lemma 5.3], the ergodic probability measures are not atomic.

Now, Birkhoff’s ergodic theorem and Corollary2.7imply that the Lyapunov exponents onE

are negative and the sectional Lyapunov exponents are positive, in a total probability subset of

Λ.

Moreover, for singularitiesσ∈Sing(Λ)we have two possibilities:

First case:σis accumulated by recurrent orbits (including periodic orbits), then since ind(σ)≥

ind(Λ), by [11, Lemma 4.1] there is a dominated splittingT_σM =E_σ⊕F_σ, where dim(E) =

ind(Λ).

Second case: Either there exists a dominated splitting on T_σM =E_σ⊕F_σ with dim(E) =

ind(Λ), which guarantees the definition of J such that X is stricly J-separated. Or, otherwise,

sinceσis an isolated hyperbolic singularity with ind(σ)≥ind(Λ), we have an invariant splitting

for which we only guarantee thatJsuch thatX is (not strictly)J-separated.

So, we have an invariant splittingT_ΛM=E_Λ⊕F_Λ which has uniformly angle bounded away

from zero andT_σM=E_σ⊕F_σis dominated for everyσ∈Sing(X).

Now, [1, Theorem C] implies that the corresponding decompositionT_ΛM=E⊕F is

domi-nated of index ind(Λ).

By using the adapted metric for dominated splitting [15], we obtain a field ofC1non-degenerated

quadratic formsJsuch thatX strictlyJ-separated overΛ.

Now, to prove theJ-monotonicity, take a compact invariant setΓinΛ∗. SinceX is a star flow

andΓis nonsingular, by [13, Theorem A], this set must be a hyperbolic one. So, by well known

results, the linear Poincar´e flow associated toX is strictlyJ-monotone on any compact invariant

setΓ∈Λ∗.

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(L.S.) UNIVERSIDADEFEDERAL DABAHIA, INSTITUTO DEMATEMI¨¿1₂TICA- AVENIDAADHEMAR DEBAR