VARIOUS ORDERS
LUCIANA SALGADO
ABSTRACT. It is given notions of singular hyperbolicity and of sectional Lyapunov exponents of orders beyond the classical ones, namely, other dimensions besides the dimension 2 and the full dimension of the central subbundle of the singular hyperbolic set. It is obtained a characteriza-tion of singular hyperbolcity in this broad sense, by using the nocharacteriza-tion of infinitesimal Lyapunov functions. Furthermore, it is reduced the requirements to obtain singular hyperbolicity. 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 2
1.2. Singular hyperbolicity of other orders 3
1.3. p-sectional Lyapunov exponents 5
2. Fields of quadratic forms 6
2.1. J-separated linear maps 8
2.2. Lyapunov exponents 10
3. Proof of Theorems 10
References 13
1. INTRODUCTION AND STATEMENT OF RESULTS
Let M be a compact C∞ riemannian n-dimensional manifold, n≥3. Let X1(M) the set of C1 vector fields onM, endowed with theC1 topology. And denote by Xt :M→M theC1 flow generated byX.
In a remarkable work Morales, Pac´ıfico and Pujals [16] defined the so calledsingular hyper-bolic systems, in order to describe the behaviour of Lorenz attractor. 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: November 28, 2016.
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, PRODOC-UFBA2014.
In [17], the same authors proved that every robustly transitive singular set for a three dimen-sional flow is a partially hyperbolic attractor or repeller and the singularities in this set must be Lorenz-like.
In this paper, we prove a relation between theJ-algebra of Potapov [19, 20, 24] a new definition of singular hyperbolicity, envolving intermediate dimensions of the central subbundle.
TheJ-algebra here means a pseudo-euclidean structure given byC1non-degenerate quadratic formJ, 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.
We are going to prove sufficient and necessary conditions for a flow to be singular hyperbolic of some order, in a sense to be clarified below.
It is also given a characterization of singular and sectional hyperbolicity for a flow over a compact invariant set, improving a result in [1].
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.
LetMbe a connected compact finiten-dimensional manifold,n≥3, with or without boundary. We consider a vector fieldX, such thatX is inwardly transverse to the boundary∂M, if∂M6= /0. The flow generated byX is denoted by{Xt}.
Aninvariant setΛfor the flow ofX is a subset ofMwhich satisfiesXt(Λ) =Λfor allt∈R. The
maximal invariant setof the flow is M(X):=∩t≥0Xt(M), which is clearly a compact invariant set.
Atrapping region U for 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.
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
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)
Recall that the index of a hyperbolic periodic orbit of a flow is the dimension of the contracting subbundle of its hyperbolic splitting.
Our main results is the following.
LetΛ⊂Mbe a compact invariant subset forX.
Definition 1. A dominated splitting over a compact invariant set Λ of X is a continuous DXt -invariant splittingTΛM=E⊕F withEx6={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)|>Ce
λt,∀t >0, ∀x∈Λ. (1.3)
The following definition was given as a particular case of singular hyperbolicity.
Definition 2. A sectional hyperbolic set is a singular hyperbolic one such that for every two-dimensional linear subspaceLx⊂Fx one has
|det(DXt|Lx)|>Ce λt
,∀t>0. (1.4)
1.2. Singular hyperbolicity of other orders.
Given E a vector space, we denote by ∧pE the exterior power of order p of E, defined as follows. If v1, . . . ,vn is a basis of E then ∧pE is generated by {vi1∧ · · · ∧vip}1≤i≤n,ij6=ik,j6=k.
Any linear transformation A:E →F induces a transformation∧pA:∧pE→ ∧pF. Moreover,
vi1∧ · · · ∧vip can be viewed as the p-plane generated by{vi1,· · ·,vip}ifij6=ik,j6=k. A reference
for more information about exterior powers is, for instance, [7]. We may define a new kind of singular hyperbolicity.
Definition 3. A compact invariant setΛisp-singular hyperbolic (orp-sectionally hyperbolic) for aC1flowX if there exists a partially hyperbolic splittingTΛM=E⊕F such thatEis uniformly
contracting and the central subbundleF is p-sectionally expanding, with 2≤p≤dim(F). IfLxis a p-plane, we can see it asev∈ ∧p(Fx)\ {0}of norm one. Hence, to obtain the singular expansion we just need to show that for someλ>0 and everyt>0 holds the following inequality
k ∧pDXt(x).vke >Ceλt.
Our first main result concerns in a characterization of singular hyperbolicity of any order via infinitesimal Lyapunov functions, following [1], [2], [23], [24], [20].
invariant measure ofX if it is an invariant measure ofXt for everyt ∈R. We will denote by
M
X the set of all invariant measures ofX. A subsetY ⊂Zhastotal probabilityif for everyµ∈M
Xwe haveµ(Y) =1 (see [14]).
Theorem A. A compact invariant setΛwhose singularities are hyperbolic (withind≥ind(J)) for X ∈X1(M)is a p-singular hyperbolic set if, and only if, there exist a neighborhood U ofΛ and a field of non-degenerate quadratic formsJon U with index1≤ind(J)≤n−2such that X is non-negative strictlyJ-separated and the spectrum of the diagonalized operator DXt satisfies
the properties:
(1) r1−<1; and
(2) Π1pr+i >1, where 2≤p≤dim(M)−ind(J),
in a total probability subset ofΛ. Moreover, if ri+·r+j >1, for all1≤i,j,≤ p,i6= j, in a total probability set, thenΛis a sectional-hyperbolic set.
In [1], the authors proved the next result about sectional hyperbolicity.
As a direct application of Theorem A and Theorem2.9 in Section 2, we reobtain the next one, without the assumption on the singularities.
Corollary B. [1, Theorem D]Suppose that all singularities of the attracting set Λof U are all of them sectional-hyperbolic with indexind(σ)≥ind(Λ). Then, Λis a sectional-hyperbolic set for Xt if, and only if, there is a field of quadratic formsJwith index equal to ind(Λ)such that
Xt is a non-negative strictlyJ-separated flow on U and for each compact invariant subsetΓ in
Λ∗=Λ\Sing(X)the linear Poincar´e flow is strictlyJ0-monotonous for some field of quadratic formsJ0equivalent toJ.
Thus, Theorem A is improvement to this result, once it does not requires a priori sectional hyperbolicity on the singularities.
In [3], this author together with V. Araujo and A. Arbieto, proved that the requirements in the definition of sectional hyperbolicity can be weakened, demanding the domination property only over the singularities, because in this setting the splitting is in fact dominated. More precisely, we proved the next result.
Theorem 1.1. [3, Theorem A]LetΛbe a compact invariant set of X such that every singularity in this set is hyperbolic. Suppose that there exists a continuous invariant splitting of the tangent bundle ofΛ, TΛM=E⊕F, where E is uniformly contracted, F is sectionally expanding and for some constants C,λ>0we have
kDXt|Eσk · kDX−t|Fσk<Ce −λt
for all σ∈Λ∩Sing(X)and t≥0. (1.5)
Then TΛM=E⊕F is a dominated splitting.
Here, it is proved a similar result, but now it is done on p-sectional hypothesis.
Λ, TΛM=E⊕F, where E is uniformly contracted, F is p-sectionally expanding and for some constants C,λ>0we have
kDXt|Eσk · kDX−t|Fσk<Ce
−λt for all σ∈Λ∩Sing(X)and t≥0. (1.6)
Then TΛM=E⊕F is a dominated splitting.
This way, the requirements in the definition of singular hyperbolicity (of any order, including the classical one) demands domination property only over the singularities. Indeed, we have
Definition 4. A compact invariant set Λ⊂M is p-singular hyperbolicforX 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 everyt>0 we have
(1) kDXt|Exk ≤Ce
−λt;
(2) | ∧pDX
t|Lx|>C
−1eλt, for every p-dimensional linear subspaceL x⊂Fx; (3) kDXt|Eσk · kDX−t|Fσk<Ce
−λt for allσ∈Λ∩Sing(X).
The study of conditions to a given splitting of the tangent bundle to have the domination property is an important research line in the area of Dynamical Systems, see [4], [8], [5].
Some progress in this context has been obtained, for instance in [2, Theorem A], with V. Araujo, we given a characterization for dominated splittings based onk-th exterior powers, where
k=dimF.
We note that if E⊕F is a DXt-invariant splitting of TΓM, with {e1, . . . ,eℓ} a family of ba-sis for E and {f1, . . . ,fh} a family of basis for F, then Fe = ∧kF generated by {fi1∧ · · · ∧
fik}1≤i1<···<ik≤his naturally∧
kDX
t-invariant by construction. In addition, ˜E generated by{ei1∧ · · · ∧eik}1≤i1<···<ik≤ℓtogether with all the exterior products ofibasis elements ofE with jbasis
elements ofF, wherei+j=kandi,j≥1, is also∧kDX
t-invariant and, moreover,Ee⊕Fegives a splitting of thekth exterior power∧kT
ΓM of the subbundleTΓM.
Theorem 1.2. [2, Theorem A]Let TΓM=EΓ⊕FΓbe a DXt-invariant splitting over the compact
Xt-invariant subsetΓ such thatdimF =k≥2. LetFe=∧kF be the ∧kDXt-invariant subspace
generated by the vectors of F and E be the˜ ∧kDX
t-invariant subspace such that Ee⊕F is ae
splitting of the kth exterior power∧kT
ΓM of the subbundle TΓM.
Then E⊕F is a dominated splitting if, and only if,Ee⊕F is a dominated splitting fore ∧kDX t. We note that the equivalence is only valid ifk=dimF.
In [2, Example 3], we have an example where evenE⊕F being dominated we do not obtain
e
E⊕Fedominated, fork<dimF.
It is not clear, which are the conditions to guarantees an equivalence in Theorem C.
1.3. p-sectional Lyapunov exponents.
The next definition reminds a previous one from Arbieto [6] which deals with, in his terminol-ogy, the sectional Lyapunov exponents.
Definition 5. Thesingular Lyapunov exponents of order p(orp-sectional Lyapunov exponents) ofxalongF are the limits
lim t→+∞
1
t logk ∧
p
DXt(x).vke whenever they exists, whereve∈ ∧pF
x− {0}.
Following the corresponding result from [3, Theorem B] and [6, Theorem 2.3], just by few modifications in computations, changingk ∧2DX
t(x).vke byk ∧pDXt(x).vke . We obtain, via The-orem 2.9, the analogous for singular hyperbolic sets of the main result of this paper.
Theorem D. Let Λ a compact invariant set for a flow Xt such that every singularity σıΛ is
hyperbolic. Suppose that there is an invariant splitting TΛM =E⊕F such that TσM=E⊕F is dominated, for allσ∈Λ. The set Λ is p-singular hyperbolic for the flow if, and only if, the Lyapunov exponents in the E direction are negative and the singular Lyapunov exponents in the F direction are positive on a set of total probability.
Remark 1.3. The properties ofp-singular hyperbolicity can be expressed in the following equiv-alent forms; see [5] for the classical one. There existsT >0 such that
• kDXT| Exk<
1
2 for allx∈Λ(uniform contraction); and
• | ∧p(DXT|∧pF
x)|>2 for allx∈Λ.
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→JX
t(x)(Z(Xt(x)))is continuously differentiable for allx∈U and eachC
1 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,
assume that Jx is self-adjoint without loss of generality. Hence, we represent J(v) by a non-degenerate symmetric bilinear form<Jxv,v>x. Now we use Lagrange’s method to diagonalize this bilinear form, obtaining a base{u1, . . . ,un}ofExsuch that
Jx(
∑
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
Jx(
∑
i
αivi) = q
∑
i=1
−α2i + n
∑
j=q+1
α2j, (α1, . . . ,αn)∈Rn.
Finally, we can redefine <, > so that the base {v1, . . . ,vn} is orthonormal. This can be done smoothly in a neighborhood ofxinM since 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 [13].
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−x1({0})for all
x∈U. In the adapted coordinates obtained above we have
C0(x) ={v=
∑
iαivi∈Ex: n
∑
j=q+1
α2j =
q
∑
i=1
α2i}
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-separated ifDXt(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.
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))
>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. We assume that the vector fieldX is non-negative onU. Then, the span ExX of X(x)6=0 is aJ-non-degenerate subspace. According to item (1) of Proposition 2.3, this means thatTxM=
ExX⊕Nx, whereNxis the pseudo-orthogonal complement ofExX with respect to the bilinear form J, andNx is also non-degenerate. Moreover, by the definition, the index ofJrestricted toNx is the same as the index ofJ. Thus, we can define onNxthe cones of positive and negative vectors, respectively,Nx+andNx−, 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 ofΠx depends onX(x)andJX only. The linear Poincar´e flowPt is a linear multi-plicative cocycle overXt on 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 toNxfor 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 Proposition 2.3) 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.
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 .
(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 [13].
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 Lemma 2.4 is strictly bigger than the supremum.
The (longer) proofs of the following results can be found in [24] or in [20]; see also [25]. For aJ-separated operatorL:V →V and ad-dimensional subspaceF+⊂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
Proposition 2.6. We haveσd(L) =r1+· · ·r+d, where ri+ are given by Proposition 2.2(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)≤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, 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 corresponding to the limits
χj= lim t→+∞
1
t logkDXt(x)·vk,
for everyv∈Ei(x)\ {0},i=1,· · ·,l(x).
In this setting, Wojtkowski [24] 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. [24, Corollary 3.7]
For1≤k1≤p and1≤k2≤q
χ−1 +· · ·+χ−k 1 ≤
k1
∑
i=1
Z
logri−dµ andχ+1 +· · ·+χ+k 2 ≥
k2
∑
i=1
Z
logri+dµ.
Look that, ifXt is aJ-separated flow on Λ, for each diffeomorphismDXt if we fixt>0, the last theorem holds forri±,t, whereri±,t are the singularJ-values ofDXt.
3. PROOF OFTHEOREMS
In this section, we prove our mains results.
First, we prove Theorem A, by using Theorem D which is proved below. We also need to use the following lemma.
Lemma 3.1. [3]Given a continuous splitting TΛM=E⊕F such that E is uniformly contracted, then X(x)∈Fxfor all x∈Λ.
Proof of Theorem A. Suppose Λ p-singular hyperbolic set of index ind. Then, 1≤ind≤n−2 and there is a dominated splitting TΛM=E⊕F, where E is uniformly contracting and F is
uniformlyp-sectionally expanding. Moreover,hXi ⊂F, by Lemma 3.1. By using adapted metric [9], we construct the quadratic forms J such that X is non-negative strictly J-separated. By Proposition 2.2 and Corollary 2.7, there is aJ-diagonalization ofDXtby aJ-isometry, that we are also denoting byDXt, such that its spectrum has the required properties. In fact, for each singular valuer−i corresponding to the contracting subspace, we must haveri−<1. Analogously, asF is a p-sectionally expanding subbundle, the sum of each p r+i1,· · ·,ri+p corresponding singular value must be greater than one. Even including the corresponding field direction.
Reciprocally, suppose that in a total probability subset ofΛwe haver−1 <1 andΠpj=1r+ij >1, where 2≤p≤dim(M)−ind(J).
LetTΛM=E⊕Fthe corresponding splitting and the decomposition in direct sum of Lyapunov
subspaces
Ex=⊕rj=0Ej(x),Fx=⊕s(x)j=0−1Fj(x). By Theorem 2.9,
χ−1 +· · ·+χ−r ≤
r
∑
i=1
Z
logr−i dµandχ+i
0+· · ·+χ
+ ip ≥
p
∑
j=1
Z
logri+ jdµ.
So, we obtain that the Lyapunov exponents over E are all of them negative and the p-sectional Lyapunov exponents onF are all of them positive, in a total probability subset. By Theorem D,
Λis a p-singular hyperbolic set forX.
Proof of Theorem C. Letx∈Λbe a regular point ofX. by Lemma 3.1, asE is uniformly contracting,X(x)∈F.
SinceFis an invariant subbundle, considerφt(x) =DXt|Fx,t∈R, and there is a decomposition
in direct sum of Lyapunov subspaces
Fx=⊕s(x)j=0−1Fj(x).
One of them, say EX =F0(x), generate by X(x)6=0. Denote by χFj(x),j =1,· · ·s(x)−1 the corresponding Lyapunov exponents.
Fixingi1,· · ·,ip∈ {1,· · ·,s(x)−1}and considering pvectorsv1∈Fi1\ {0},· · ·vp−1∈Fip−1\ {0}, putL=span{X(x),v1,· · ·,vp−1}as the generated p-plane.
From assumption,
0<χ≤lim inf t→+∞
1
t log| ∧
pDX
t|L|=χF0+χFi1+· · ·+χ
F ip−1,
and we obtain
p
∑
j=1
andxsuch thatX(x)6=0.
Similarly, if we considerψt(x) =DXt|Ex,x∈Λ,t ∈R,we can find a splitting Ex=⊕rj=0Ej(x)
which is a direct sum in Lyapunov subspaces with the corresponding Lyapunov exponentsχE j,j= 1,· · ·,r.
By assumption,χE
j(x)≤ −χ<0, for all regular pointxand 1≤ j≤r. Consider now the subadditive family of continuous functions
ft=log
|DXt|Ex| m(DXt|Fx)
such that
f(x) =lim inf t→+∞
ft
t ≤lim inft→+∞
1
t log|DXt|Ex| −lim supt→+∞
1
t logm(DXt|Fx) =
max{χEj(x),1≤ j≤r} −min{ p
∑
j=1
χFij ≥χ>0,∀ij∈ {1,· · ·,s(x)−1},1≤ j≤p} ≤
−χ−0=−χ<0,
for all regular pointx.
For some singularity, σ∈Λ, we must have f(σ)≤ −χ, as a consequence of domination hy-pothesis.
Now applying the following proposition from [6]:
Proposition 3.2. Let{t7→ ft :Λ→R}t∈R be a continuous family of continuous function which is subadditive and suppose that f(x)<0in a set of total probability. Then there exist constants C>0andλ<0such that for every x∈Λand every t>0we haveexp(ft(x))≤Cexp(λ2t),
E⊕F is a dominated splitting.
Finally, the proof of Theorem D.
We fix a compactXt-invariant subset Λ. We say that a family of functions{ft :Λ→R}t∈R
is sub-additive if for every x∈M and t,s∈R we have that ft+s(x)≤ fs(x) + ft(Xs(x)). The Subadittive Ergodic Theorem (see e.g. [21]) shows that the function f(x) =lim inf
t→+∞
ft(x)
t coincides
with ef(x) = lim t→+∞
1
t ft(x)in a set of total probability inΛ.
Proof of Theorem D. Define the following families of continuous functions onΛ φt(x) =logkDXt|Exk and ψt(x) =logk ∧
pDX
−t|∧pF
Xt(x)k, (x,t)∈Λ×R.
So, both familiesφt,ψt:Λ→Rare subadditive, by the logarithm properties.
Proposition 3.2, applied to φt(x) and the hypothesis on the Lyapunov exponents, imply that there are constantsC>0 andη<0 such that exp(φt(x)) =kDXt|Exk ≤Cexp(ηt)for allt >0
Similarly, the assumption on the sectional Lyapunov exponents and Proposition 3.2 applied to the functionψt(x)give us constantsD>0 andη<0 for whichk ∧pDX−t|∧pF
Xt(x)k ≤De
ηt, soF is a sectionally expanding subbundle.
Now Theorem C ensures that the splittingE⊕F is a dominated splitting and we are done.
REFERENCES
[1] V. Araujo and L. Salgado. Infinitesimal Lyapunov functions for singular flows. Mathematische Zeitschrift, 275(3-4):863–897, 2013.
[2] V. Ara´ujo, L. S. Salgado. Dominated splittings for exterior powers and singular hyperbolicity.J. Differential Equations, 259, 3874–3893. 2015.
[3] V. Araujo, A. Arbieto, and L. Salgado. Dominated splittings for flows with singularities. Nonlinearity, 26, 2391–2407. 2013.
[4] A. Arbieto and L. Salgado. On critical orbits and sectional hyperbolicity of the nonwandering set for flows.
Journal of Differential Equations, 250:2927–2939, 2011.
[5] V. Ara´ujo and M. J. Pacifico.Three-dimensional flows, volume 53 ofErgebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer, Heidelberg, 2010. With a foreword by Marcelo Viana.
[6] A. Arbieto. Sectional lyapunov exponents.Proc. of the Amercian Mathematical Society, 138:3171–3178, 2010. [7] L., Arnold. Random Dynamical Systems. Springer-Verlag. 2003.
[8] C. Bonatti, L. J. D´ıaz, and M. Viana.Dynamics beyond uniform hyperbolicity, volume 102 ofEncyclopaedia of Mathematical Sciences. Springer-Verlag, Berlin, 2005. A global geometric and probabilistic perspective, Mathematical Physics, III.
[9] N. Gourmelon. Adapted metrics for dominated splittings.Ergodic Theory Dynam. Systems, 27(6):1839–1849, 2007.
[10] M. Hirsch, C. Pugh, and M. Shub.Invariant manifolds, volume 583 ofLect. Notes in Math.Springer Verlag, New York, 1977.
[11] A. Katok. Infinitesimal Lyapunov functions, invariant cone families and stochastic properties of smooth dy-namical systems. Ergodic Theory Dynam. Systems, 14(4):757–785, 1994. With the collaboration of Keith Burns.
[12] J. Lewowicz. Lyapunov functions and topological stability.J. Differential Equations, 38(2):192–209, 1980. [13] A. I. Mal′cev.Foundations of linear algebra. Translated from the Russian by Thomas Craig Brown; edited by
J. B. Roberts. W. H. Freeman & Co., San Francisco, Calif.-London, 1963. [14] R. Ma˜n´e. An ergodic closing lemma.Annals of Math., 116:503–540, 1982.
[15] R. Metzger and C. Morales. Sectional-hyperbolic systems.Ergodic Theory and Dynamical System, 28:1587– 1597, 2008.
[16] C. A. Morales, M. J. Pacifico, and E. R. Pujals. Singular hyperbolic systems. Proc. Amer. Math. Soc., 127(11):3393–3401, 1999.
[17] C. A. Morales, M. J. Pacifico, and E. R. Pujals. Robust transitive singular sets for 3-flows are partially hyper-bolic attractors or repellers.Ann. of Math. (2), 160(2):375–432, 2004.
[18] V. I. Oseledec. A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems.
Trans. Moscow Math. Soc., 19:197–231, 1968.
[19] V. P. Potapov. The multiplicative structure of J-contractive matrix functions. Amer. Math. Soc. Transl. (2), 15:131–243, 1960. Translation of Trudy Moskovskogo Matematiˇceskogo Obˇsˇcestva 4 (1955), 125–236. [20] V. P. Potapov. Linear-fractional transformations of matrices. InStudies in the theory of operators and their
[22] M. Wojtkowski. Invariant families of cones and Lyapunov exponents. Ergodic Theory Dynam. Systems, 5(1):145–161, 1985.
[23] M. P. Wojtkowski. Magnetic flows and Gaussian thermostats on manifolds of negative curvature.Fund. Math., 163(2):177–191, 2000.
[24] M. P. Wojtkowski. Monotonicity,J-algebra of Potapov and Lyapunov exponents. InSmooth ergodic theory and its applications (Seattle, WA, 1999), volume 69 ofProc. Sympos. Pure Math., pages 499–521. Amer. Math. Soc., Providence, RI, 2001.
[25] M. P. Wojtkowski. A simple proof of polar decomposition in pseudo-Euclidean geometry. Fund. Math., 206:299–306, 2009.
(L.S.) UNIVERSIDADEFEDERAL DABAHIA, INSTITUTO DEMATEMATICA´ - AVENIDAADHEMAR DEBAR
