LUCIANA SALGADO
ABSTRACT. We say that a flow or vector fieldX∈X1(M)isstaron a compact invariant setΛif there exist neighborhoodsU⊂X1(M)ofX andU⊂MofΛfor which every closed orbit inU
of every vector fieldY inUis 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 aC1 flow 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 5
2.1. J-separated linear maps 7
2.2. Lyapunov exponents 9
3. Some applications 10
3.1. Proof of Corollaries 3.3 and 3.5 11
4. Proof of Theorems 11
References 13
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. In [25], the same authors proved that every robustly transitive singular set for a three dimensional
Date: April 6, 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, 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 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.
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
∈X1(M)ofX such that, for every Y ∈U
, the maximal invariant setΛY =∩t∈RXt(U)is contained in the interior ofU and is non-trivially (not a single orbit) transitive. Astrongly homogeneousset of index 1≤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, 35], 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.
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], [35]. 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], [31], for the classical sectional and singular hyperbolicity definitions, and [33] 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 called called star 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 famous stability conjecture from 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 bestaron a setΛif there are neighborhoodsU ⊂MofΛand
U
⊂X1(M)ofX such that every periodic orbit and every singularity of anyY ∈U
.We are going to prove sufficient and necessary conditions for a flow to be locally star, in the sense of the above definition.
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 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 n-dimensional manifold, n≥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
TΛM=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∗=EXt∗(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.
Our main results are the following.
LetΛ⊂M be a compact invariant subset forX.
(1) there is C1 field of quadratic forms J with index0≤ind≤dim(M)defined on U such that X is strictlyJ-separated onCrit(X|Λ);
(2) for everyσ∈Sing(X|U)and∀v∈TσM,J′(v)>0; and
(3) the linear Poincar´e flow Pt associated to each periodic orbit γ of X|Λ is strictly J -monotone.
Theorem B. A compact invariant setΛfor X ∈X1(M)is strongly homogeneous with indexind if and only if there are a neighborhood U ofΛand a field of non-degenerate C1quadratic forms
JonΛ with indexind(J) =indsuch that X is strictlyJ-separated and whose associated linear Poincar´e flow Pt is strictlyJ-monotone on every closed orbitΓof X|U.
Note that in the last result monotonicity is only required on closed orbits. If we require it on nonsingular compact invariant subsets inΛ, it is possible evaluate the index of singularities, once it is accumulated by regular orbits.
Corollary C. A compact invariant setΛfor X ∈X1(M)is strongly homogeneous with indexind andind(σ)≥indfor allσ∈Sing(X|Λ)if there is a field of non-degenerate C1quadratic forms
JonΛ with indexind(J) =indsuch that X is strictlyJ-separated and whose associated linear Poincar´e flow Pt is strictlyJ-monotone on every compact invariant nonsingular subsetΓofΛ.
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 1.1. 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 splittingTΛM=E⊕F withEx=6 {0},Fx6={0}for everyx∈Λand such that there are positive constantsK,λsatisfying
kDXt|Exk · kDX−t|FXt(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 TΛM =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)|>Ceλt,∀t>0, ∀x∈Λ. (1.3)
Definition 4. Asectional hyperbolic set is a singular hyperbolic one such that for every two-dimensional linear subspaceLx⊂Fxone has
|det(DXt|Lx)|>Ceλt,∀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 [33], 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 continuous invariant splitting of the tangent bundle onTΛM=E⊕F and constantsC,λ>0 such that for everyx∈Λand every t>0 we have
(1) kDXt|Exk ≤Ce−λt; (2) | ∧pDX
t|Lx|>C−1eλt, for every p-dimensional linear subspaceLx⊂Fx; (3) kDXt|Eσk · kDX−t|Fσk<Ce−λt for allσ∈Λ∩Sing(X).
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(Jx)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→JXt(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 toJx 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,
this bilinear form, obtaining a base{u1, . . . ,un}ofExsuch that
Jx(
∑
i
αiui) = q
∑
i=1−λiα2i + n
∑
j=q+1λjα2j, (α1, . . .,αn)∈Rn.
Replacing each element of this base according tovi=|λi|1/2uiwe deduce that
Jx(
∑
i
αivi) = q
∑
i=1 −α2 i + n∑
j=q+1α2
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 thatJx2=Jx.
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 correspoing family of zero vectorsC0(x) =J−1x ({0})for all
x∈U. In the adapted coordinates obtained above we have
C0(x) ={v=
∑
i
αivi∈Ex: n
∑
j=q+1α2 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-monotoneifJXt(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-isometryifJXt(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
JX
−t(x)(DX−t(v))<0 for all t >0 and x such that X−s(x)∈U for every s∈[−t,0]. Indeed, otherwiseJX−t(x)(DX−t(v))≥0 would implyJx(v) =Jx DXt(DX−t(v))
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. We assume that the vector field X is non-negative onU. Then, the spanExX ofX(x)6=0 is a
J-non-degenerate subspace.
According to item (1) of Proposition 2.3, this means that TxM =ExX⊕Nx, where Nx is the pseudo-orthogonal complement of ExX with respect to the bilinear formJ, and Nx is also non-degenerate. Moreover, by the definition, the index ofJrestricted toNx is the same as the index ofJ. Thus, we can define onNx the cones of positive and negative vectors, respectively,Nx+ and
Nx−, just like before.
Now we define the Linear Poincar´e Flow Pt of Xt along the orbit of x, by projecting DXt 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. SinceJx 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.
(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 .
(5) For each L:V →V linear there exists a unique linear operator L+:V →V (the pseudo-adjoint) such that(L(v),w) = (v,L+(w))for every v,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) =hJv˜,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 thathJv˜,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 [35] or in [30]; see also [36]. 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).
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)≤r−1(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.
This property will be used a lot of times in our proofs.
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 [35] proved that the logarithm of the pseudo-Euclidean singular values 0≤rq−≤ · · · ≤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. [35, Corollary 3.7]For1≤k1≤p and1≤k2≤q
χ−
1 +· · ·+χ−k1 ≤
k1
∑
i=1Z
logri−dµ andχ+1 +· · ·+χ+k2 ≥ k2
∑
i=1Z
logri+dµ.
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), andΛ 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 splittingTσM=Eσ⊕Fσ such that dim(Eσ) =ind. Hence, TheoremDimplies that there exists a field of non-degenerate quadratic formsJonΛwith index ind(J) =ind(Λ)for whichX is strictly
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 setΛ 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).
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´e 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 TΛM=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 equal to the dimension ofE with the required properties.
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.
First of all, we recall some definitions which are necessary here.
A point p∈Λis said to be pre-periodic if there are sequencesXnof flows and pn of periodic points ofXnsuch that
lim
n→∞Xn=X and limn→∞pn=p.
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.
We recall that a probability measure µis anergodic measure if for every invariant set A we haveµ(A) =1 orµ(A) =0. Finally, a certain property is said to be valid inµ-almost every point if it is valid in the whole Z except, possibly, in a set of null measure.
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 TΓM=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∈TΓM 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 Eσs ⊕Eσu is a dominated one. So, by using adapted metrics we construct the desired quadratic formJσsuch thatX is strictly separated (see [4]) and, by Proposition4.1J′(v)>0 for allv∈TΓM.
Analogously, for every periodic orbit γ of X, consider the hyperbolic splitting TγM=Es⊕ EX⊕Eu. Again, consideringEs⊕(EX⊕Eu)as a dominated splitting we obtain a quadratic form
Jfor whichX is strictly separated onγ. In Addition, the linear Poincar´e flow associated toX,Pt is hyperbolic and thenJ-monotone onγ.
Reciprocally, 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 straightforward.
Proof of TheoremB. Since the linear Poincar´e flow is strictlyJ-monotone on each compact in-variant subsetΓ⊂Λ∗ implies thatΓ is a hyperbolic subset ofΛ, with a constant index, which we denote ind(J). Moreover, by Corollary 2.7, asX is strictlyJ-separated onΓ, any hyperbolic closed orbit ofX must have index equal to ind(J), once that it is fixed.
AsJ is a differentiable field of quadratic forms, there exist neighborhoodsV ofΛ and
V
⊂X1(M)ofXsuch that every vector fieldY∈
V
is strictlyJ-separated onV, see [4, Theorem 2.17]. In addition, the linear Poincar´e flow associated toY is strictlyJ-monotone over every compact invariant set contained in the maximal invariant set ofV forY (shrinkingV, if necessary).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.
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. We recall the definition ofδ-closable points of [22]. We say that a pointx∈M\Sing(X)isδ-closable if, for anyC1neighborhood
U
⊂X1(M)ofX, there(1) ZT(z) =z,
(2) Z=X onM\Bδ(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.
By Ergodic Closing Lemma, theδ-closable set ofX has total probability.
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=1Z
logr−i dνandχ1++· · ·+χ+k2 ≥ k2
∑
i=1 Zlogr+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 splittingTσM =Eσ⊕Fσ, where dim(E) =
ind(Λ).
Second case: Either there exists a dominated splitting on TσM =Eσ⊕Fσ 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 splittingTΛM=EΛ⊕FΛ which has uniformly angle bounded away from zero andTσM=Eσ⊕Fσis dominated for everyσ∈Sing(X).
Now, [1, Theorem C] implies that the corresponding decompositionTΛM=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 DEMATEMATICA´ - AVENIDAADHEMAR DEBAR
