Lyapunov exponents and entropy for divergence-free Lipschitz vector fields

Let Xν0,1(M)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {X}^{0,1}_\nu (M)$$\end{document} be the subset of divergence-free Lipschitz vector fields defined on a closed Riemannian manifold M endowed with the Lipschitz topology ‖·‖0,1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert \,{\cdot }\,\Vert _{0,1}$$\end{document} where ν\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document} is the volume measure. Let Xν,ℓ0,1(M)⊂Xν0,1(M)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathfrak {X}^{0,1}_{\nu ,\ell }(M)\subset \mathfrak {X}^{0,1}_\nu (M)$$\end{document} be the subset of vector fields satisfying the ℓ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell $$\end{document}-property, a property that implies C1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^1$$\end{document}-regularity ν\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document}-almost everywhere. We prove that there exists a residual subset with respect to ‖·‖0,1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert \,{\cdot }\,\Vert _{0,1}$$\end{document} such that Pesin’s entropy formula holds, i.e. for any the metric entropy equals the integral, with respect to ν\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu $$\end{document}, of the sum of the positive Lyapunov exponents.


Motivations and some back history
The main goal of this paper is to initiate and motivate the study of dynamics of divergence-free Lipschitz vector fields on manifolds of dimension greater than three. Lower dimensions will not be considered because the issues we are adressing in The author was partially supported by FCT-'Fundação para a Ciência e a Tecnologia', through Centro de Matemática e Aplicações (CMA-UBI), Universidade da Beira Interior, project UID/00212/2020, and partially supported by the Project 'New Trends in Lyapunov Exponents' (PTDC/MAT-PUR/29126/2017). The author thanks CMUP for providing necessary conditions in which this work was developed.
B Mário Bessa mario.costa@uab.pt 1 Universidade Aberta -Departamento de Ciências e Tecnologia, Rua da Escola Politécnica 147, 1269-001 Lisboa, Portugal this manuscript are somewhat irrelevant and trivial when analyzed in the context of divergence-free vector fields in low-dimensional manifolds. As it is well known, Lipschitz continuity is a strong form of uniform continuity. However, in the case of vector fields, the Lipschitz condition turns out to be of crucial importance as it implies the uniqueness of integral curves by the well-known Picard-Lindelöf theorem which may fail to hold if one assumes continuity only (cf. [16,Figure 2,p. 19]). On the other hand, Lipschitz continuous vector fields strictly contain C 1 -vector fields hence they form a class of models with a wider range of applications. One major difficulty in proving results of either a genericity or stability character for divergence-free Lipschitz vector fields stems from the fact that the latter are not Lipschitz approximable by C 1divergence-free vector fields (see Example 2.1). Therefore, any attempt to generalize results from the C 1 to the Lipschitz class via an approximation by C 1 -vector fields is not viable. 1 Nonetheless, it is worth pointing out that the Lipschitz topology on the C 1 -class is intrinsically the same as the C 1 -Whitney topology (cf. [2,Section 5]).

Metric entropy
The concept of metric entropy was introduced by Kolmogorov and Sinai in mid twentieth century and it is a central concept in ergodic theory with connections to several areas of physics (e.g., Thermodynamics and Statistical Mechanics) and Information Theory among others. Given a state space equipped with a sigma-algebra of sets and a measurable dynamical system which possesses an invariant measure on that same sigma-algebra, the entropy measures the complexity of this system in the sense that it captures the asymptotic expansion rate characteristic of the flow in the whole set observed by the given measure. Not surprisingly, many definitions of a chaotic system impose positive metric entropy. Given a measure ν and a ν-measurable flow X t , we denote by h ν (X 1 ) the measure-theoretic entropy of the time-1 map X 1 with respect to (w.r.t.) the measure ν (for more details see [10]). It is worth pointing out that several authors defined different concepts of flow entropy which are well behaved when we consider a re-parametrization of the flow [14,37]. A complete survey on entropy can be found in [18].

Lyapunov exponents
Being key objects in smooth ergodic theory, Lyapunov exponents measure the asymptotic growth rate of the tangent map of a dynamical system along orbits when restricted to certain fixed directions. Positive or negative Lyapunov exponents ensure, respectively, exponential divergence or convergence of nearby orbits whereas zero Lyapunov exponents imply lack of exponential behavior. As the behavior of orbits moving apart or closer together can be described topologically one expects to find similar notions for maps satisfying continuity only (see e.g. [7,25,31]). However, measurability alone, as it is required in the case of metric entropy, seems to be quite insufficient for developing an interesting theory of Lyapunov exponents. Furthermore, as a complete theory for C 1 -systems is available since the late sixties [30], it is common to consider this class of differentiability when dealing with Lyapunov exponents.

Pesin's formula
In the late seventies Ruelle [35] obtained an upper bound to the metric entropy of a C 1 -measurable map preserving a probability measure. This upper bound is precisely the integral, w.r.t. the given measure, of the sum of all positive Lyapunov exponents. Indeed, Pesin [33] had already proved that the metric entropy equals the integral of the sum of all positive Lyapunov exponents as long as the map is C 1+α and the invariant measure is equivalent to the Lebesgue measure. Notice that, Morse-Smale systems (see e.g. [32,Section 4]) are examples which attest that the Ruelle inequality may be strict. In order to obtain an equality, which is currently designated by Pesin's entropy formula, Pesin developed an outstanding theory that is paramount in dynamical systems and forms the basis of the so-called nonuniform hyperbolicity. Later in [23], Mañé was able to obtain Pesin's entropy formula without making use of Pesin's invariant manifold theory. Undoubtedly, this is one of the most elegant formulas in the whole theory of dynamical systems and the type of result which is useful having at our disposal.

A generic Pesin's formula
Although Pesin's entropy formula requires that the map is C 1+α and also that C 1generically (i.e., for a set that contains the intersection of a countable collection of dense open sets) C 1 -maps are not. 2 of class C 1+α , Tahzibi obtained in [38] a simple yet interesting result: C 1 -generic area-preserving diffeomorphisms satisfy Pesin's entropy formula. Later, this result was generalized to any dimension by Sun and Tian [36], to volume-preserving flows and low-dimension Hamiltonians by the author and Varandas [11] and to volume-preserving lipeomorphisms displaying some C 1 -regularity of measure theoretic type, by the author, Silva and Vilarinho [8]. Sun and Tian (see [36]) follow Mañé's clever arguments in [20,21] on the proof of Pesin's entropy formula using merely C 1 -regularity of diffeomorphisms, but they appended an additional hypothesis of a dominated splitting for almost every point. The dominated splitting was obtained from an outstanding result having its roots also in the ingenious ideas of Mañé: the Mañé-Bochi-Viana dichotomy [11]. Our main result was motivated by [8] and arguably the main reason for studying maps first was that the proofs of the results are somewhat easier than for vector fields. The step forward is then the study of the continuous-time case. Therefore, the prime goal of this article is to present a proof of Pesin's entropy formula for divergence-free vector fields. Thus, tacitly we are considering the entropy related to the volume measure.
This formula may be summarized by saying that the metric entropy related to the volume measure is equal to the integral over the whole manifold of the sum of the positive Lyapunov exponents which are observed by the volume measure.
The proof follows the same method as in [8] but all the constructions are rearranged to work in a differential equations environment. The proof will be divided into three main steps: • Firstly, we use Ruelle's inequality (Main Proposition) in order to obtain that the metric entropy is less or equal to the integral, over the whole manifold, of the sum of the positive Lyapunov exponents. This formula follows directly from the one obtained in the discrete case in [ splitting zone on which we use Theorem C, (iii) some regularity on the tangent map, which is fundamental in order to obtain Theorem C, (iv) the zero Lyapunov exponents zone where the formula is controlled trivially, (v) a dichotomy between dominated splitting and zero exponents managed via Theorem B, and (vi) the upper semicontinuity 3 of the integral of the Lyapunov exponents which is crucial to the proof of the previous dichotomy. Unfortunately, it seems to be not feasible to generalize Theorem A using the above strategy. In fact, it is very likely that we have reached the threshold of validity of this approach to prove the generic Pesin's formula as far as weakening the topology is concerned because all points (i)-(vi) are compromised. If we try to widen our set of dynamical systems while keeping a 'good' topology then the Baire property may vanish and a 'generic' result would be uninteresting. Therefore, we are left with possible generalizations considering other classes of conservative dynamical systems, e.g., symplectic homeomorphisms, Hamiltonians, contact flows, geodesic flows, etc. In this direction [10] proves that C 2 -generic Hamiltonians, i.e. C 1 -generic Hamiltonian flows, satisfy the Pesin entropy formula. An interesting problem is to find out if we could generalize this result to the class of Lipschitz Hamiltonian flows imbued with the same spirit of the present paper. In divergence-free Lipschitz vector fields we make use of a conceptual idiosyncratic property: they can be characterized both from a differentiable and from a measurable point of view. Indeed, the flow X t preserves the volume form if det D X t = 1 or if ν(X t (B)) = ν(B) for any Borelian set B. When dealing with Hamiltonians we have a conceptual shortcoming because the symplectic invariance is strongly based on a differentiable assumption: the pullback of X t applied to a symplectic form ω is again the symplectic . We could try to use the Rademacher theorem and formally define symplectic invariance ν-a.e. but we have a gap in the literature concerning this 'symplectic ν-a.e. structures'. Yet, considering C 0 -closures of Hamiltonian flows is a subject of growing interest that is related to Gromov-Eliasberg symplectic rigidity [12,28,29,39,40]. It would be interesting to better understand how these two concepts are related.

Statement of the results and proof of Theorem A
The main result in the present paper, stated in Theorem A, is to establish that Pesin's entropy formula holds generically in X 0,1 ν, (M). The set of vector fields X 0,1 ν, (M) will be defined in Sects. 2.1 and 2.4 but briefly they represent Lipschitz vector fields with associated flows preserving the volume ν and with a nice regularity of the derivative ν-a.e. Let X ∈ X 1 ν, (M) and let X t be the flow associated to X . We denote by h ν (X t ) the measure-theoretic entropy of the time-t map X t w.r.t. the measure ν (for more details see [10]). Since Abramov [1] formula says that h ν (X t ) = |t| h ν (X 1 ) for any t ∈ R (for a proof see [13,Theorem 3,p. 255]), it is irrelevant to choose other time-t flow to evaluate the metric entropy. From now on, and for practice convenience, we consider h ν (X 1 ) and denote it simply by h ν (X ). The topology · 0,1 regarding the definition of the residual set in Theorem A will be defined in Sect. 2.2. The numbers λ + i (X , x) are the d positive Lyapunov exponents associated to the time-1 flow and will be defined properly in Sect. 2.5.
In order to obtain equality (1.1) we first consider Ruelle's inequality for Lipschitz vector fields which is a fairly straightforward generalization of [8,Proposition 1]. In [8], it was proved for Lipschitz homeomorphisms but the same conclusion can be drawn for time-1 maps of Lipschitz flows.

Main Proposition
The proof of the inverse inequality of (1.2) has two parts, as in the previous approaches [8,36]. The first is the Lipschitz flows version of a result by the author and Rocha [4,6] and it seeks to validate the generic Mañé-Bochi-Viana dichotomy: dominated splitting versus zero Lyapunov exponents, proved in [11], for our class of dynamical systems. The proof of this dichotomy is given in Sect. 3 and relies, between other arguments, on the upper semicontinuity of the integrated Lyapunov exponents map defined in (2.9) w.r.t. the Lipschitz topology and that a continuity point of the integrated Lyapunov exponents has either a dominated Oseledets splitting for the linear Poincaré map or else a trivial Oseledets spectrum. Let us be formal and state the result that will be instrumental to prove Theorem A but which has its own interest. The set M X ⊂ M is the set of Rademacher points for the lipeomorphism X 1 and the linear Poincaré map (see Sects. 2.1 and 2.3 for details).
Theorem B There exists a · 0,1 -residual subset R ⊂ X 0,1 ν (M) such that for all X ∈ R there exists a ν-full measure subset Z ∪ D ⊂ M X such that every Lyapunov exponent vanishes for all x ∈ Z, and for all x ∈ D the Oseledets splitting has an m x -dominated splitting for the linear Poincaré map along the orbit of x, for some m x ∈ N.
To deal with the dominated component D in the previous result we will prove in Sect. 4 a continuous-time version of a result proved in [8,Theorem 3] which was based on [36, Theorem 2.2] and whose construction goes back to the work [20,21].
Observe that in Theorem C if F is associated to the non-negative (or positive) Lyapunov exponents and E is associated to the negative (or non-positive) Lyapunov exponents, then (1.1) holds for X ∈ X 0,1 ν, (M) with dominated splitting.

Proof of Theorem A
Now we are going to prove Theorem A assuming Theorem B, Main Proposition and Theorem C. By Theorem B we know that there exists a residual set R ⊂ X 0,1 ν (M) such that, for each X ∈ R and ν-almost every x ∈ M, the Oseledets splitting of the linear Poincaré map is either dominated along the orbit of x or else the Lyapunov spectrum of X at x is trivial. Consider X ∈ R. By Main Proposition we have for ν-a.e. x ∈ M that We are left with the task of proving that Let Z ⊆ M stand for the set of points such that the Lyapunov spectrum of X at x is trivial and let D ⊆ M stand for the set of points such that the Oseledets splitting of the linear Poincaré map is dominated. It is convenient to choose ν(Z), ν(D) > 0 because other cases follow from this one. Define for any Borelian A ⊆ M the following measures: Clearly, ν Z (Z) = 1, ν D (D) = 1 and ν = ν(Z)ν Z + ν(D)ν D . Therefore, using the affine property of the metric entropy we get We only have to show that (1.3) holds for h ν Z (X ) and h ν D (X ) separately. Since the metric entropy is always non-negative and the Lyapunov exponents of X are all zero in Z, we get Finally, Theorem C gives that and Theorem A follows.

Definition
Let M be a connected, closed and C ∞ -Riemannian manifold of dimension d 3. Since along this paper we deal with divergence-free vector fields, we assume that M is also a volume-manifold with a volume form V : T M d → R where T M stands for the tangent bundle. Furthermore, we equip M with an atlas . We call Lebesgue measure or volume to the measure related to V and denote it by ν. This relation is established by stand for the distance associated to the Riemannian metric and let C(M, R) be the set of continuous maps on M endowed with the usual metric defined by for all x, y ∈ M. The infimum of such constants is called the Lipschitz constant of F, that we denote by lip(F): x n be written in the coordinate charts A in the following sense: In conclusion we say that X is the expression of X in the coordinate charts A.
If, for every n = 1, . . . , d, each function X n ∈ C(M, R) and it is Lipschitz continuous, then X is said to be a Lipschitz vector field. We will denote by X 0,1 (M) the set of Lipschitz vector fields in M and by X r (M) the set of C r ones. The integral family of curves, is a well-defined function on a ν-full measure subset of M if we assume X to be a Lipschitz vector field. We say that a Lipschitz vector field X is divergence-free if ∇ · X = 0 for ν-a.e. x ∈ M. We denote this set by X 0,1 ν (M). The relation between the volume-preserving property of the flow and the divergence-freeness of the vector field is embodied in the Abel-Jacobi-Liouville formula proved for the Lipschitz class in [5,Proposition 1]. In [5,Proposition 1], it was proved that if X ∈ X 0,1 ν (M) and τ ∈ R, then, for any Borelian B we have ν(B) = ν(X τ (B)). When a vector field X is of class C r (r 1) we say that X is divergence-free if ∇ · X = 0 for all x ∈ M. Let Sing(X ) . . = {x ∈ M : X (x) = 0} denote the set of singularities of X and let R(X ) . . = M \ Sing(X ) denote the set of regular points.

Lipschitz topology
We will denote by We notice that the space of Lipschitz maps from R d to R d has a linear structure, and that defines a complete norm. Let us now introduce the Lipschitz topology on X 0,1 ν (M).

Since any smooth metric is Lipschitz equivalent to the Euclidean metric, F is a Lipschitz map if and only if for any coordinate charts
Unless M displays a structure of linear space, the expression X −Y is meaningless, and its use is only to remind the relation between the corresponding charts representatives of X and Y . Let · 0,1 be the topology generated by the subbasic neighborhoods (2.1).

Consider a finite family {N
Let N be a neighborhood of X obtained as the intersection of the subbasic sets N i . There is a metric d 0,1 compatible with its topology that for each X , Y ∈ N assigns The closure N of N is then a complete metric space and thus a Baire space. Since each X ∈ X 0,1 (M) has a neighborhood which is a Baire space, (X 0,1 (M), · 0,1 ) is itself a Baire space and so is (X 0,1 ν (M), · 0,1 ). In the following simple example we show that vector fields in X 0,1 ν (M) may not be · 0,1 -approximable by vector fields in X ∈ X 1 ν (M). Thus, the attempt of using · 0,1 -approximation by smooth vector fields to study Lipschitz vector fields will fail.
and use [26] to transport it to M = S 2 defining a vector field in X 0,1 ν (M). Assume, by contradiction, that there exists a We observe that, Hence one of these numbers α (0) or −α (0) is 0 which contradicts X −Y 0,1 < 1 above.

The linear Poincaré map
Fix X ∈ X 0,1 (M), x ∈ R(X ) and let N x ⊂ T x M denote the normal fiber at X (x), that is, the subfiber spanned by the orthogonal complement of X (x). As we already said, from [17, Theorem 3.41 and Lemma 3.42] we know that Lipschitz vector fields integrate Lipschitz flows. Moreover, we know that we gain some differentiability along the direction of the vector field. Indeed, X t (x) is a C 1,1 -function of t and so N X t (x) vary in a C 1,1 -way. We denote by N ⊂ T M the normal bundle which is defined on R(X ). Now, let N x and N X t (x) be two (d − 1)-dimensional manifolds contained in M whose tangent spaces at x and X t (x), respectively, are N x and N X t (x) . Let also V x be a small neighborhood of x in N x . The set V x can be taken small enough so that the usual Poincaré map P t is well defined. We recall that by Rademacher's theorem the derivative of a Lipschitz map X τ (τ > 0) is defined for a full ν-measure subset M X τ ⊆ M. In particular, ν(M \ M X τ ) = 0, and since X τ is ν-invariant we conclude that ν(X nτ (M \ M X τ )) = 0 for all n ∈ Z. Therefore the set of orbits with points where X τ is not differentiable is the set the set of orbits with points where X τ is not differentiable has zero measure. In view of this we may assume M X τ ⊂ M invariant by X τ , and the existence of D X τ x implies the existence of D X t X nτ (x) for all n ∈ Z. The linear Poincaré flow of X ∈ X 0,1 (M) is the differential of the Poincaré map P t X and so it exists only for ν-a.e. x ∈ R(X ) once we fix t = τ > 0. To define it properly for each τ > 0 we consider the tangent map D X τ : is called the linear Poincaré map at x associated to the vector field X . Now we will define dominated splitting for the linear Poincaré map associated with X over D if the following inequality holds for any x ∈ D:

The -property and some useful continuous dependence results
Let X ∈ X 0,1 ν (M) and let x 0 ∈ M be a Rademacher point. We consider appropriate coordinate charts A and in a small neighborhood B(x 0 , r ), r > 0, the linearization given by when no ambiguity can arise. We are interested in Lipschitz vector fields such that the Lipschitz constant of the remainder decreases as we consider smaller radius for the corresponding domain. We say that X ∈ X 0,1 We denote by X 0,1 ν, (M) ⊂ X 0,1 ν (M) the set of vector fields such that for ν-almost every x 0 ∈ M the -property holds. Notice that, in X 0,1 ν, (M), the -property is algebraically closed w.r.t. the sum.
We say that X ∈ X 0,1 is continuous when restricted to M X . Observe that being almost C 1 is weaker than saying that D X x is continuous for ν-a.e. x on M, because D X x could not be even defined in some points x of M in the almost C 1 -case.
Proof. Let (X n ) be a sequence of vector spaces in X 0,1 ν, (M) converging w.r.t. · 0,1 to X ∈ X 0,1 ν (M). For each n there is a ν-full measure subset M n ⊂ M of points z where (D X n ) z is defined. Similarly, there is a ν-full measure subset M X ⊂ M of points z where D X z is defined. Let z be any point in the ν-full measure subset M = n M n ∩ M X . We have and, for each n, we also have where ρ = ρ(X , z, r ) and ρ n = ρ(X n , z, r ). Since X n ∈ X 0,1 ν, (M) we have for all n that lim r →0 lip B(z,r ) ρ n = 0. We are going to prove that X ∈ X 0,1 ν, (M), that is, for any given ε > 0 there is R > 0 such that lip B(z,R) ρ < ε and consequently the same holds for all 0 < r < R. Let ε > 0 be given. We can find N 0 > 0 such that The next result shows that the flow inherits the -property from the associated vector field. Lemma 2.5 If X ∈ X 0,1 ν, (M) and τ > 0, then X τ satisfies the -property ν-a.e.

Proof.
Fixing τ > 0 we have by the Rademacher theorem that for ν-a.e. x 0 ∈ M the tangent map D X τ x 0 exists. Near x 0 we write: In the same way we have also that X ( For a.e. t ∈ [0, τ ] we consider L D X X t (x 0 ) . Define two parametric maps and observe that from basic properties of flows we get Therefore, Thus By Gronwall's inequality (see e.g. [32]) we obtain Finally, The next result is a consequence of Lemma 2.5 by using standard 'add and subtract' arguments.
where ρ τ (x) = ρ τ (P τ X , x 0 , r ). We will prove that First notice that Recalling that τ (x) −→ r →0 τ it suffices to prove that We notice that The lemma is proved once we observe that Id − X τ (x 0 ) −→ r →0 0 and D X τ x 0 is bounded.

Lemma 2.7
If X ∈ X 0,1 (M) and τ > 0, then the map X → P τ X is continuous when the domain is equipped with · 0,1 and the codomain is equipped with the uniform norm of operators.
Proof. It is sufficient to prove that X → X τ is continuous when both the domain and the codomain are equipped with · 0,1 . Let L X = lip(X ), Y ∈ X 0,1 (M) and L Y = lip(Y ). We have the two integral canonical identities X t ( From Gronwall's inequality we obtain Therefore, and so We perform the calculation by analysing the local problem in a small ball B ⊂ M

Oseledets' theorem in three scenarios
Let X ∈ X 0,1 ν (M), x ∈ R(X ) and let RX (x) . . = {u ∈ T x M : u = ξ X (x), ξ ∈ R} be the line field direction at x. Recalling that D X t x · X (x) = X (X t (x)) we conclude that the vector field direction is D X t -invariant. The existence of other D X t -invariant fibers is guaranteed, at least for Lebesgue almost every point, by a theorem due to Oseledets (see [30]). However, since we consider vector fields in X 1,0 ν (M), we need to pay attention on the construction of the continuous-time dynamical cocycle.
Take X ∈ X 0,1 ν, (M) and τ > 0. Since from Gronwall's inequality we have D X τ x e τ D X x e τ lip(X ) < ∞ for all x ∈ M X τ , we obtain the integrability condition which is crucial to obtain the Oseledets theorem. Let SL(d, R) denote the d-dimensional special linear group with entries in R. The map is a cocycle in the sense that it satisfies the identities: Here we use the chain rule for Lipschitz maps [19].
Under these assumptions we are in a position to apply the Oseledets theorem to the dynamical cocycle D X τ (see e.g. [3]).

Theorem 2.8 (Oseledets theorem for the flow)
Let X ∈ X 0,1 ν, (M) and τ > 0, then there is a ν-full measure subset O X τ ⊂ M such that, for all x ∈ O X τ : (1) there exists a D X τ -invariant splitting of the fiber T x M = E 1 x ⊕ · · · ⊕ E k(x) x along the orbit of x called Oseledets' splitting. Moreover, one of the E i 0 x is 1-dim and Lyapunov exponents, such that for any v ∈ E i x \ {0} and i = 1, . . . , k(x). We also have a multiplicity one for the Lyapunov exponentλ i 0 (X , x) = 0.

Remark 2.9
A 'Lyapunov exponent' is constant along an orbit even when, for the transition from y to a point x = X τ (y), the map D X τ y does not exist. In fact, we can use the broader definition of Lyapunov exponent (cf. [7]) and the fact that the mentioned transition is bounded by some constant depending on lip(X ) and τ . Moreover, the Lyapunov exponents fulfill a property similar to what Abramov's formula says about the metric entropy. We have λ(X t ) = |t|λ(X 1 ) so it is irrelevant to choose other time-t flow to evaluate the Lyapunov exponents.
Let λ 1 (X , x) · · · λ d (X , x) denote the d Lyapunov exponents counted with multiplicity and set λ + and in the divergence-free context |det D X n x | = 1, then being all equal means being all equal to 0. Due to the fact that for any of these subspaces E i x ⊂ T x M, the angle between this space and RX (x) along the orbit has sub-exponential growth, that is we conclude that the Lyapunov exponentλ i (x) for D X τ with associated subspace E i x is also a Lyapunov exponent for P τ X associated to subspace N i In resume we have the following reformulation of Theorem 2.8.

Remark 2.11
As the Lyapunov exponent in the vector field direction is 0, i.e. λ i 0 = 0, and along this direction no metric entropy is added, then in (1.1) of Theorem A we have that h ν (X ) is equal to the entropy of the linear Poincaré time-1 map. Moreover, the right-hand side of (1.1) can be written as M Now we present some multilinear algebra for the linear Poincaré map. The k th exterior power of the normal space N , denoted by k (N ), is a d−1 k -dimensional vector space. Let {e j } j∈J (# J = d − 1) be an orthonormal basis of N , then the family of exterior powers e j 1 ∧ e j 2 ∧ · · · ∧e j k for j 1 < · · · < j k with j α ∈ J forms an orthonormal basis of k (N ).
This formalism of multilinear algebra reveals to be the adequate to prove Theorem B. This is because we can recover the spectrum and the splitting information of the dynamics of k (P t X (x)) from the one obtained by applying Oseledets' theorem to P t X (x). This is precisely the meaning of the next theorem [ This nondecreasing sequence starts with

Take an Oseledets basis {e
Then the Oseledets space is the subspace of k (N p ) which is generated by the k-vectors e i 1 ∧ · · · ∧e i k such that 1 i 1 < · · · < i k d − 1 and k j=1 λ i j (x) = λ j (x). Now we will consider the integrated upper Lyapunov exponent of the exterior power of the linear Poincaré map. Fixing τ > 0 we consider the following function: (2.9) In the same way we define the function L k (X , ), where ⊆ M is an X t -invariant set, defined by Let k (X , x) denote the sum of the first k Lyapunov exponents of X , that is k (X , x) = λ 1 (X , x) + · · · + λ k (X , x). It is an easy consequence of Theorem 2.12 (recall (2.7)) that for k = 1, . . . , d − 2 we have k (X , x) = λ 1 ( k (X ), x) and so L k (P τ X , ) = L 1 ( k (X ), ), for any X t -invariant set . By using [11,Proposition 2.2] we get immediately that concluding, from Lemma 2.7, that for all k ∈ {1, . . . , d − 2}, the function (2.9) is an upper semicontinuous function when the domain is endowed with the · 0,1 topology.

Dealing with singularities
In the next section we will see that small tailor made perturbations cause a decrease on the Lyapunov exponents. Yet, to perform these perturbations we need 'time' along the orbit and consequently the singularities, as equilibrium points, can make our task difficult. We will now see how we can rule out singularities in a sense that they typically represent a zero measure set. Given X ∈ X 1 ν (M), we observe that the hypothesis ν(Sing(X )) = 0 holds if all the singularities of X are hyperbolic, thus from classical results in [34], it is satisfied for a C 1 -open 4 and C 1 -dense subset of X 1 ν (M). We say that x ∈ M is a hyperbolic singularity for X ∈ X 1 ν (M) if the spectrum of D X x has neither pure complex nor zero as an eigenvalue. It is well known that a hyperbolic singularity is stable/persistent and moreover is isolated. Thus, hyperbolicity is a fundamental tool for checking whether a singularity is indeed an isolated singularity in the C r (r 1) case. Basically, a similar statement holds for the Lipschitz class. However, as hyperbolicity is linked to differentiability, we must characterize a singularity being isolated through a broader analysis. Indeed, hyperbolicity is basically transversality and this property behaves in a similar way in the Lipschitz topology. We observe that thickening from the Lipschitz to the C 0 -topology still captures the persistence of transversality but the 'unicity' is lost. Therefore, being isolated can no longer be guaranteed, as the reader can easily see. The next trivial example is illuminating.

Example 2.13
The Lipschitz function f defined by f (x) = 2x if x 0 and f (x) = 3x if x 0 is not derivable at the fixed point 0. Nevertheless, the fixed point is isolated under Lipschitz perturbations.
Let us consider the following observations: (1) The stability of elliptic singularities in divergence-free vector fields only happens when dim(M) = 2. In fact pure complex eigenvalues (the elliptic case) persists under perturbations within the conservative setting. As we are considering dim(M) 3 we only have to deal with stable hyperbolic singularities (see [34] for more details). (2) Hyperbolic singularities are · 0,1 -stable in a sense that have a unique analytic continuation under · 0,1 perturbations. Notice that we are not saying that the continuationx of x for the perturbed X ∈ X 1 ν (M) is a hyperbolic singularity, because D Xx may not even exist. Still we get X (x) = 0 and there exists a · 0,1open set containingx as the only singularity ofX .
(3) Let be given X ∈ X 1 ν (M), x ∈ M such that X (x) = 0 and U an arbitrarily small open neighborhood of x. Then there exists an arbitrarily · 0,1 -close vector field X ∈ X 1 ν (M) such that X (x) = 0,x is the only singularity in U , D Xx exists andx is hyperbolic for X . The conclusion is trivial when D X x exists and x is hyperbolic.
Let X ∈ X 0,1 ν (M). If all the singularities of X are hyperbolic, then they form a finite set, therefore with ν-measure zero. Let M X be the Rademacher points and Sing(X ) be the set of singularities of X . We are only interested in the case when ν(Sing(X )) > 0. In this case we cannot have hyperbolicity in the whole set M X ∩ Sing(X ). Yet, a small · 0,1 -perturbation can be made so that all singularities are of hyperbolic type and for this perturbed vector field X ∈ X 1 ν (M) we have ν(Sing( X )) = 0.

Toolbox for realizable sequences
Our proof depends on the fact that the absence of a dominated splitting allows the Lyapunov exponents to decrease after a small perturbation. Once we have no dominated splitting then we can rotate the unstable Oseledets fiber until it reaches the stable Oseledets fiber mixing the rates of expansion and contraction. This idea goes back at least as far as the 1970s and the Soviet school (see e.g. [27]). In our precise context we refer to Mañé's seminal ideas [22,24] culminated in the outstanding work by Bochi and Viana [11] for the discrete case. For the continuous-time case, see [4,6].
The main tool to realize, implement and make explicit such perturbations is a flowbox theorem for the divergence-free Lipschitz vector fields which has been proved recently. Let us recall this result but first we begin by defining some useful objects. We say that two vector fields X 1 : U 1 → T U 1 and X 2 : U 2 → T U 2 are locally topologically conjugate near p 1 ∈ U 1 and p 2 ∈ U 2 if there exist two open neighborhoods O i p i (i = 1, 2) and a homeomorphism φ : O 1 → O 2 with φ( p 1 ) = p 2 such that for any x ∈ O 1 and a small interval I containing 0 the integral curve σ x : I → O 1 defined by σ x (0) = x and d dt σ x (t) = X 1 (σ x (t)) for all t ∈ I (i.e. defined by X t 1 (x) for t ∈ I ) is a solution associated to X 1 if and only if the integral curve φ • σ x : I → O 2 is a solution associated to X 2 . Theorem 3.1 (Flowbox theorem for Lipschitz divergence-free vector fields [5]) Let be given X ∈ X 0,1 ν (M), a non-singular point p 1 ∈ M and the trivial vector field T ( x 1 ,x 2 , . . . ,x d ) = (1, 0, . . . , 0) on canonical coordinates (x 1 ,x 2 , . . . ,x d ) of R d . Then: (i) X and T are locally topologically conjugate near p 1 and p 2 =0. The homeomorphism φ which gives the conjugacy is a lipeomorphism. (ii) X and T c = cT are locally topologically (volume-preserving) conjugate near p 1 and p 2 =0 for some c = c(X , p 1 ) > 0. The homeomorphism which gives the conjugacy is a volume-preserving lipeomorphism.
For the reader's convenience, we repeat the relevant material from [6] omitting most of the proofs, thus making the text self-contained. As in [6, Definition 4.1] the realizable sequences will be a key object to obtain Theorem B. By modified volumepreserving we mean the volume on the (d − 1)-dimensional transversal section whose measure is denoted by ν. This is a simple point regarding that ν is not invariant but by considering a density given by the direction of the flow we obtain invariance (see [6,Section 2.4]). The set F X (x)(U ) represents a flowbox obtained by the X t -saturation of the set U contained in a transverse section through x. Definition 3.2 Given X ∈ X 0,1 ν (M), ε > 0, κ ∈ (0, 1), ∈ N, and a non-periodic point x, we say that the modified volume-preserving sequence of linear maps L j : j = 0, . . . , − 1 is an (ε, κ)-realizable sequence of length at x if the following occurs: For all γ > 0, there is r > 0 such that for any (d − 1)-manifold N x (tangent to N x ) and any non-empty open set U ⊆ B(x, r ) ⊆ N x , there exist a measurable set K ⊆ U and Y ∈ X 0,1 ν (M) satisfying the conditions: We denote the (d − 1)-dimensional ball with radius r in the subspace described byx 1 The following result is the basic tool in order to build realizable sequences.
Notice that R 1 = R ξ . DenoteṘ t = d dt R t . Through a simple calculation we obtain thatṘ 1 (0, r )). We define We are left to see that Y − T 0,1 < ε and for that we We will show that the perturbation defined in (3.1) is such that where · 1 stands for the C 1 -Whitney topology. Clearly, the C 0 -norm is controlled essentially by the size of ξ so let us estimate the C 1 -norm. Using the product rule on the derivative of the perturbation we get Observe that the gradient ∇ ġ(x 1 )G x 2 2 +x 2 3 is equal to Let us see that all the terms in (3.2) are controlled. In the second item below we use the polar coordinates representationx 2 = r cos β andx 3 = r sin β.
We have ξg(x 1 )G x 2 2 +x 2 3 2ξ η which is controlled by the size of ξ . We also have which is also controlled by the size of ξ . Hence, all the items (a)-(d) are fulfilled and the lemma is proved.
The next result follows directly from combining Theorem 3.1 and Lemma 3.3. Firstly, Theorem 3.1 provides us with good trivial coordinates and secondly Lemma 3.3 allows us to perform the perturbations which essentially will be done into several small balls that fill the open set U proclaimed in Definition 3.2 (see [6,Lemma 5.3]). Lemma 3.4 Given X ∈ X 0,1 ν (M), ε > 0 and κ ∈ (0, 1), there exists ξ 0 > 0 such that for any ξ ∈ (0, ξ 0 ), any x ∈ M (non-periodic or with period > 1) and any 2-dimensional vector space V x ⊂ N x one has that the time-1 map The next result is the key step that allows us to mix the Oseledets directions in the absence of a dominated splitting. This is the rotation (using Lemma 3.4) from the unstable Oseledets fiber to the stable Oseledets fiber referred to in the first paragraph of this section.

Concluding the proof of Theorem B
Once we have established Proposition 3.5 it remains: ♦ First, use Proposition 3.5 to decrease the Lyapunov exponent in a single orbit by showing that for every ε, δ > 0, k ∈ {1, . . . , d − 2} and ν-almost every point x in a set without dominated splitting with dim E = k we can find a 'time' t > T (x) such that there exists a modified volume-preserving sequence of linear maps L j : N X j (x) → N X j+1 (x) for j = 0, . . . , t − 1 which is an (ε, κ)-realizable sequence of length t at x satisfying This local procedure uses the lack of domination (3.3) and also the different Lyapunov exponents to cause a decrease of the largest Lyapunov exponent of the k th exterior power of the linear Poincaré map by a small · 0,1 -perturbation. Second, we generalize the previous construction from a point to a large measure set using a Kakutani's tower argument. It is possible to prove that, given any ε, δ > 0, where J k (X ) is the jump of the function, namely, the integral of λ k (X , x) − λ k+1 (X , x) over a set without domination between the Oseledets directions associated to λ k (X , x) and λ k+1 (X , x). Of course, if this set without dominated splitting has zero ν-measure we apply this argument again, with k replaced by k + 1 until we reach d − 2.
Finally, to prove Theorem B we follow ipsis litteris the arguments in [11, p. 1467]. More precisely, for k ∈ {1, . . . , d − 2} let E k ⊂ X 0,1 ν (M) be the subset corresponding to the points of continuity of the map L k defined in (2.9). Take E = d−2 k=1 E k . It is well known that the sets E k are residual subsets and so is E. If X ∈ E k then, by the definition of this set and by above, we obtain that J k (X ) = 0. Therefore λ k (X , x) = λ k+1 (X , x) for a.e. x in a set without a dominated splitting. For X ∈ E the subset Z ⊂ M is defined by the Oseledets regular points without any dominated splitting between any fiber, and the set D is defined by the Oseledets regular points with some kind of dominated splitting. Hence, if x ∈ Z then all the Lyapunov exponents of x are equal to zero and this ends the proof of Theorem B.

Preliminary results
We have divided the proof of Theorem C into three lemmas focusing on a suitable choice of the decomposition given by the Oseledets theorem. The idea of the proof of the first lemma goes back at least as far as [20,Lemma 3] where this was proved under a uniform hyperbolic context. It was subsequently proved in [36,Lemma 3.3] under the weaker assumption of a dominated splitting and in [8, Lemma 6.1] it was again proved but in the even weaker context of lipeomorphisms. We will state it and observe that its proof follows closely the arguments in [8,23,36]. In order to present this lemma let us first recall some definitions. The formulation of item (ii) of Lemma 4.1 is equivalent to the one in [8, Lemma 6.1] but substantially different from the original one in [23,36] that was based on C 1 -regularity which is not taken for granted for Lipschitz vector fields. The motivation of using the -property defined in Sect. 2.4 lies in the fact that the next result remains operational. Recalling again Abramov's formula and Remark 2.9 we perform our study for the time-1 Poincaré map and so our proof is very close to the proof of the discrete-time case. Indeed, we are considering metric entropy and Lyapunov exponents, on which the usual considerations about having an R-action instead of a Z-action does not need a specific analysis. Loosely speaking the directions of the vector field do not contribute to the change of both entropy and Lyapunov exponents.
Given a normed space E and a splitting E = E ⊕ F, we define γ (E, F) as the maximum of the norms of the projections π i : E → E i (i = 1, 2). A subset G ⊂ E is said to be an (E, F)-graph if there exists an open set U ⊂ F and a Lipschitz map (i) DF 0 is an isomorphism and γ (DF 0 · E, DF 0 · F) α; (ii) denoting L = DF 0 | E and T = DF 0 | F and for some small r > 0 and (x, y) ∈ B(0, r ), we have with remainders p(x, y) and q(x, y) having Lipschitz constants less than δ; When applying previous lemma and according to item (iii) above we need to have a set with a dominated splitting. This set will be precisely the set D of Theorem B. As the dominated splitting is w.r.t. the linear Poincaré map we will take F and DF 0 of Lemma 4.1 to be, respectivelly, the Poincaré map P t X and the linear Poincaré map P t X (x) associated to a certain X ∈ X 0,1 ν (M). We notice that we are not able to proceed with the arguments considering the tangent map D X t because there is no reason to have a dominated splitting of T M for D X t . Recall that by (2.8) the only control we have over the angle is that the convergence to zero (if exists) is not very fast. The existence of a convergence to zero of this angle forbids a dominated splitting.
Let f : M → M be a homeomorphism, r > 0, x ∈ M and n ∈ N. Let be the Bowen ball. Define also f , r , x)) . In [20,21], it was proved that Lemma 4.2 Let X ∈ X 0,1 ν (M) and D ⊂ M be an X t -invariant set. If there is a 1dominated splitting on D for the linear Poincaré map, say N D = E ⊕ F, then for any c > 0, there exists r > 0 such that for every x ∈ D and any (E x , F x )-graph G ⊂ E ⊕ F with dispersion c contained in a Bowen ball B n (X 1 , x, r ), n 1, P n X (G) is a (P n X (x) · E x , P n X (x) · F x )-graph with dispersion c.
Take a vector field X ∈ X 0,1 ν (M), ε > 0 and a ν-full measure set ⊆ M X 1 with an m x -dominated splitting E ⊕ F for P 1 X of a certain fixed index. Consider N ε ∈ N sufficiently large in order to have ν( ε ) > 1 − ε, where ε = {x ∈ : m x N ε }. Observe that f . . = X N , for N = N ε !, is such that E ⊕ F displays a 1-dominated splitting for • D f say for P N X (see [36, p. 1430]). Recall that a dominated splitting can be extended to the closure of ε wherever it is defined 5 and moreover it varies continuously with the point.
Proof Fix a, c > 0 such that if x ∈ ε and y ∈ B(x, a) is a ν-generic Rademacher point, then for any linear subspace G ⊆ N x which is an (E x , F x )-graph with dispersion c, we have from Remark 2.2 that log |det D f y | G − log |det D f x | F | < ε. 5 We have to exclude singularities and non-Rademacher points for instance.
and so |det D f y | G | > |det D f x | F | e −ε . (4.4) Providing Lemma 4.2 with D = ε (mod ν), the 1-dominated splitting N D = E ⊕ F and the c > 0 above, there exists r > 0 such that for every x ∈ D and any (E x , F x )-graph G with dispersion c contained in a Bowen ball B n ( f , x, r ), n 1, f n (G) is a (P n X (x) · E x , P n X (x) · F x )-graph with dispersion c. Given x ∈ ε and y ∈ E x , we denote by ν E the volume measure in E x and by ν F the volume measure in y + F x . There exists B > 0 such that the disintegration holds for all n 1: (F n (y)) dν E (y), (4.5) where F n (y) = (y + F x )∩ B n ( f , r , x). Here notice that B incorporates also the length of the 1-dimensional component transversal to the (d − 1)-dimensional subspace associated to E ⊕ F. From (4.1) and (4.5) we get Hence, Lemma 4.3 is proved once we obtain that for ν-a.e. x ∈ ε the following inequality holds: Considering that F n (y) = ∅ and using Lemma 4.2 we obtain that f n (F n (y)) is a (P n X (x) · E x , P n X (x) · F x )-graph with dispersion c. Given r ∈ (0, a), we take Notice that for f n (x) ∈ ε we have f n (F n (y)) ⊆ f n (B n ( f , r , x)) ⊆ B( f n (x), r ). Thus, D > Vol( f n (F n (y))) = F n (y) det D f n z | T z F n (y) dν E (z). For any j = 0, 1, . . . , n we have f j (F n (y)) ⊆ f j (B n ( f , r , x)) ⊆ B( f j (x), r ) ⊆ B( f j (x), a), and so given any z ∈ F n (y), we have d( f j (z), f j (x)) < a for all j = 0, 1, . . . , n. By (4.7) we obtain det D f f j (x) | F( f j (x)) e −ε dν E (z) which is equivalent to Taking limits and recalling that f = X N we obtain lim sup and (4.6) is proved and so is the lemma.

Proof of Theorem C
We would like to show that for a given X ∈ X 0,1 ν (M) if for ν-a.e. x ∈ M there is an m x -dominated splitting E ⊕ F for the linear Poincaré map along the orbit of x (where m : M → N is an X t -invariant measurable function), then Notice that it suffices to prove the theorem for a certain fixed dimension of F. Remark 2.11 sheds some light on the fact that we focus on a dominated splitting for the linear Poincaré map.
Let f , N , ε and r > 0 be given like in Lemma 4.3. Let C = lip(X ), and from Gronwall's inequality we get D X t x e t D X x e t lip(X ) = e tC for ν-a.e. x. We have h ν ( f ) (4.2) sup Hence, recalling the Abramov formula, and since ε is arbitrarily small, Theorem C is proved.