---
abstract: 'In this work, it is presented a characterization of star condition for a $C^1$ vector field based on Lyapunov functions. It is obtained conditions to strong homogeneity for singular sets by using the notion of infinitesimal Lyapunov functions. As an application we obtain some results related to singular hyperbolic sets for flows.'
author:
- Luciana Salgado
title: 'Star Flows: a characterization via Lyapunov functions'
---

[^1]

Introduction and statement of results {#sec:int-stat}
=====================================

Star systems has been studied by many renowned researchers, among them R. Mañé 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 [@Liao1980],[@Man82],[@Palis87],[@Man88],[@shi-gan-wen2014],[@GanWen2006],[@ArbMo2013].

\[def:star-flow\] A $C^1$ vector field $X$ (or its flow $X_t$) is said to be *star* if it cannot be $C^1$-approximated by ones exhibiting nonhyperbolic periodic orbits.

The definition for diffeomorphisms is analogous.

In the case of diffeomorphisms, $C^1$ structural stability is equivalent to star condition (and equivalent to Axiom A plus no cycle), see e.g. [@Man88].

However, in the case of flows, the situation is much more complex. In fact, in absence of singularities, Gan and Wen [@GanWen2006] proved that nonsingular star flows satisfy Axiom A and no cycle condition. But, in the singular setting, this is no longer true. One of the most emblematic example that a star flow does not satisfy Axiom A is the geometric Lorenz Attractor [@Guck76; @Lo63].

From this, remained the question about singular star vector fields.

\[ex:geomLorenz\] The geometric Lorenz attractor is an emblematic example of star flow. see Figure \[fig:Lorenz\].

![The geometric Lorenz flow.[]{data-label="fig:Lorenz"}](L3D){width="6cm"}

In an attempt to study the behaviour of robust singular attractors like Lorenz ones, Morales, Pacífico and Pujals in [@MPP99] defined the so called *singular 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.

The authors in [@MPP99] proved that, $C^1$-generically in dimension three, chain recurrence classes are singular hyperbolic. Without the generic assumption this result does not hold, as we can see in [@BaMo14].

In [@shi-gan-wen2014], Gan, Shi and Wen, proved that if a chain recurrent class of a star flow $X$ has homogeneous index for singularities (the same dimension of stable manifold), then this is a singular hyperbolic set of $X$, in any dimension. Moreover, they proved that, $C^1$-generically in dimension four, the chain recurrent set of a star flow is singular hyperbolic.

In [@MPP04], it has been proved that every robustly transitive singular set for a three dimensional flow is a partially hyperbolic attractor or repeller and the singularities in this set must be Lorenz-like. In [@GanLiWen2005], Gan, Li and Wen extended the result in [@MPP04] to higher dimension assuming that the set is also strongly homogeneous. We recall that a vector field $X$ is said to be *strongly homogeneous of index $0 \leq {\operatorname{ind}}(\Lambda) \leq n-1$* over a set $\Lambda$ whether it cannot be $C^1$-approximated by one which has some hyperbolic periodic orbit of index different of ${\operatorname{ind}}(\Lambda)$ in a neighborhood $U$ of $\Lambda$. Here, the index ${\operatorname{ind}}(\cdot)$ means the dimension of the contracting subbundle from the hyperbolic splitting of a hyperbolic periodic orbit .

We recall that a compact invariant set $\Lambda$ is *robustly transitive* for a vector field $X$ if there exist a neighborhood $U$ of $\Lambda$ and a neighborhood ${{\mathcal U}}\in {\mathfrak{X}^{1}(M)}$ of $X$ such that, for every $Y \in {{\mathcal U}}$, the maximal invariant set $\Lambda_Y = \cap_{t \in {{\mathbb R}}} X_t(U)$ is contained in the interior of $U$ and is non-trivially (not a single orbit) transitive.

Many researchers believed that, at least generically, singular star flows should be singular hyperbolic ones. There are many conjectures about this and various results involving the most varied assumptions, see for instance, [@ArbMo2013; @GanLiWen2005; @shi-gan-wen2014].

Another point of view on the influence of the singularities emerged in the recent work of Bonatti and da Luz [@BodL17], where the authors have defined *multisingular hyperbolicity* which admit singularities of different indexes in the same recurrent class, and have shown that this kind of hyperbolicity implies star condition. It is easy to see that the classical singular hyperbolic flows (central dimension $2$ or the full dimension of central subbundle) are a particular case of the multisingular ones. Moreover, they proved that the converse holds on the $C^1$ generic assumption.

In this paper, we prove a characterization of star condition for flows via ${\EuScript{J}}$-algebra of Potapov (infinitesimal Lyapunov functions) [@Pota60; @Pota79; @Wojtk01]. We also show a relation between ${\EuScript{J}}$-algebra and strong homogeneity. Then, we apply this to obtain some results about singular hyperbolic systems.

Recall that a pseudo-euclidean space is a real finite-dimensional vector space endowed with a non-degenerate quadratic form (in case of an euclidean space, a positive definite one).

The ${\EuScript{J}}$-algebra here means a pseudo-euclidean structure given by a $C^1$ non-degenerate quadratic form ${\EuScript{J}}$, defined on $\Lambda$, which generates positive and negative cones with maximal dimension $p$ and $q$, respectively, with $p + q = \dim (M)$.

The maximal dimension of a cone in $T_xM$ is 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 [@BurnKatok94], [@lewow80], [@Wojtk01]. In [@ArSal2012], this author jointly with V. Araújo, obtained characterizations of partial and singular/sectional hyperbolicity based on ${\EuScript{J}}$-algebra. In [@ArSal2015], the same authors proved an equivalence between dominated splittings for the flow and dominated splittings for the $k$-th exterior powers of the tangent cocycle.

More results relating geometric and algebraic features of singular hyperbolicity can be viewed in [@ArSal2012], [@ArSal2015], [@SalgCoelho2017], for the classical sectional and singular hyperbolicity definitions. Also see [@Salg2016] for singular hyperbolicity in a broad sense involving sectional expansion of intermediate dimensions between two and the full dimension of the central subbundle.

Our main result (Theorem A) rounds about of the so called *star flows* and its relation with infinitesimal Lyapunov functions.

We recall that a point $p \in M$ is said to be a $C^1$ preperiodic point of $X$ if for any $C^1$ neighborhood $\mathcal{V}$ of $X$ and any neighborhood $U \subset M$ of $p$, there is $g\in \mathcal{V}$ and $q \in U$ such that $q \in {\operatorname{Per}}(g)$. We denote this set by $P_*(X)$, and it is easy to see that it is closed and $X$-invariant.

We can assume that the periodic points are hyperbolic and we can define a $C^1 i$-preperiodic point $p$ of $X$, $0 \leq i \leq d$, if there are sequences $X_n$ of flows and $p_n$ of periodic points of $X_n$ with index $i$ such that $$\begin{aligned}
\lim\limits_{n \to \infty} X_n = X \ \textrm{and} \ \lim\limits_{n \to \infty} p_n = p.\end{aligned}$$ We denote by $P_*^i(X)$ the set of $C^1 i$-preperiodic of $X$ and, then, $P_*(X) = \cup_{i=0}^{d} P_*^i(X)$.

Precisely, it is proved the following.

\[cor:star-flow\] A vector field $X \in {\mathfrak{X}^{1}(M)}$ is star if, and only if, satisfies all of the next properties:

1.  there is a neighborhood $U$ of $P_*(X)$ and a field of quadratic forms ${\EuScript{J}}$ with index $0 \leq {\operatorname{ind}}\leq \dim(M) - 1$ defined on the preperiodic set $P_*(X)$, $C^1$ along the flow direction over each preperiodic orbit, such that $X$ is strictly ${\EuScript{J}}$-separated on every $p \in P_*(X)$;

2.  for every $\sigma \in {\mathrm{Sing}}(X\vert_U)$ and $\forall v \in T_\sigma M, {\EuScript{J}}'(v) > 0$;

3.  the linear Poincaré flow $P^t$ associated to each preperiodic orbit $\gamma$ of $X\vert_U$ is strictly ${\EuScript{J}}$-monotone.

The definitions concerning the quadratic forms are given in the next section.

Let us explain why we are not requiring strict monotonicity in a whole nonsingular neighborhood of $X$. In fact, under certain assumptions, if we have strict monotonicity over any compact invariant nonsingular subset then it is a hyperbolic subset (see [@ArSal2012 Theorem D]). The idea here is to extend the characterization for any star flow, since it is already known the existence of star flows which are not neither uniformly hyperbolic nor singular/sectional hyperbolic (see [@BodL17]).

Now, we present the definition of strong homogeneity.

\[def:str-hom-set\] We say that a set $\Lambda$ is strongly homogeneous of index ${\operatorname{ind}}$ for a flow $X_t$, if there exist neighborhoods $U$ of $\Lambda$ and ${\EuScript{U}}$ of $X$ such that all periodic orbits in $U$ with respect to any flow in ${\EuScript{U}}$ have index ${\operatorname{ind}}$.

The second result guarantees that a compact connected invariant set for a flow is strongly homogeneous under the existence of a field of non-degenerate quadratic forms ${\EuScript{J}}$ defined on a neighborhood of this set such a way that the flow derivative $DX_t$ keeps positive cones $C_+(x) : = \{0\} \cup \{ v \in T_xM ; {\EuScript{J}}(x) v > 0\}$ invariants, i.e., $DX_t(C_+(x)) \subset C_+(X_t(x))$, for all $t>0$, $x \in \Lambda$ and the projected quadratic forms on the normal bundle, with respect to the flow direction, are monotonic functions.

\[mthm:strong-homog-equiv\] A compact invariant set $\Lambda$ for $X \in {\mathfrak{X}^{1}(M)}$ is strongly homogeneous with index ${\operatorname{ind}}$ if and only if there is a neighborhood $U\subset M$ of $\Lambda$ and a continuous field of non-degenerate quadratic forms ${\EuScript{J}}$ on $U$ with fixed index ${\operatorname{ind}}({\EuScript{J}}) = {\operatorname{ind}}$ such that the preperiodic set $P_*(X)$ of $X\vert_{U}$ is strictly ${\EuScript{J}}$-separated and the associated linear Poincaré flow $P^t$ is strictly ${\EuScript{J}}$-monotone on $P_*(X\vert_{U})$.

If $\Lambda$ is the maximal invariant set of a trapping region $U$ and 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, as in [@ArSal2012 Theorem D].

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

If we require it over any nonsingular compact invariant subsets in $\Lambda$, it is possible to evaluate the index of a singularity, once it is accumulated by regular orbits.

\[mcor:strong-homog\] Let $\Lambda$ be a maximal invariant set of a neighborhood $U$ for $X \in {\mathfrak{X}^{1}(M)}$. Then, $\Lambda$ is strongly homogeneous with index ${\operatorname{ind}}$ and ${\operatorname{ind}}(\sigma) \geq {\operatorname{ind}}$ for all $\sigma \in {\mathrm{Sing}}(X\vert_{\Lambda})$ if there is a field of non-degenerate $C^1$ quadratic forms ${\EuScript{J}}$ on $U$ with index ${\operatorname{ind}}({\EuScript{J}}) = {\operatorname{ind}}$ such that $X$ is strictly ${\EuScript{J}}$-separated, the associated linear Poincaré flow $P^t$ is strictly ${\EuScript{J}}$-monotone on every compact invariant nonsingular subset $K$ of $U$ and for every $\sigma \in {\mathrm{Sing}}(X\vert_U)$ and $\forall v \in T_\sigma M, {\EuScript{J}}'(v) > 0$.

We may ask if the converse of Corollary \[mcor:strong-homog\] is valid. But, just by supposing strongly homogeneity, we could not obtain the field of quadratic forms, because we need some kind of decomposition on the tangent bundle to create the cones.

\[mthm:strong-homog-to-quad\] Let $\Lambda$ be a compact invariant set whose singularities are hyperbolic (if any) and accumulated by regular orbits for a $C^1$ vector field $X$, which is strongly homogeneous with index ${\operatorname{ind}}$ and ${\operatorname{ind}}(\sigma) > {\operatorname{ind}}$ for all $\sigma \in {\mathrm{Sing}}(X\vert_{\Lambda})$. Then, there exists a field of non-degenerate quadratic forms ${\EuScript{J}}$ on $\Lambda$ with index ${\operatorname{ind}}({\EuScript{J}}) = {\operatorname{ind}}(\Lambda)$ for which $X$ is ${\EuScript{J}}$-separated and the associated linear Poincaré flow $P^t$ is strictly ${\EuScript{J}}$-monotone on every compact invariant nonsingular subset $\Gamma$ of $\Lambda$.

As an application, we obtain some results about partial hyperbolicity for robustly transitive strongly homogeneous singular sets of [@GanLiWen2005] and transitive set of [@ArbMo2013].

The text is organized as follow. In the first section, it is given the main definitions and stated the main results. In second section, it is presented the main tools by using the notion of ${\EuScript{J}}$-algebra of Potapov. In third section, it is given some applications concerning singular hyperbolicity. In fourth section is proved the main theorems.

Some definitions and auxiliary results {#sec:prelim-definit}
======================================

Now, we give some definitions.

Let $M$ be a connected compact finite $d$-dimensional manifold, $d \geq 3$, without boundary, together with a flow $X_t : M \to M, t \in \mathbb{R}$ generated by a $C^1$ vector field $X: M \to TM$.

An *invariant set* $\Lambda$ for the flow of $X$ is a subset of $M$ which satisfies $X_t(\Lambda)=\Lambda$ for all $t\in{{\mathbb R}}$.

A *trapping region* $U$ for a flow $X_t$ is an open subset of the manifold $M$ which satisfies: $X_t(U)$ is contained in $U$ for all $t>0$; and there exists $T>0$ such that $\overline{X_t(U)} $ is contained in the interior of $U$ for all $t>T$. The maximal invariant set $\Lambda_X(U):= \cap_{t \geq 0} X_t(U)$ of $U$ is called an *attracting set*. An attracting set for $X$ which is transitive is called an *attractor* for $X$. A *repeller* for $X$ is an attractor for $-X$.

We say that a set $\Lambda$ is *Lyapunov stable* if for every neighborhood $U$ of $\Lambda$ there is another one $V \subset U$ such that every point $p \in V$ has its forward orbit contained in $U$.

A *singularity* for the vector field $X$ is a point $\sigma\in M$ such that $X(\sigma)=0$ or, equivalently, $X_t(\sigma)=\sigma$ for all $t \in {{\mathbb R}}$. The set formed by singularities is the *singular set of $X$* denoted ${\mathrm{Sing}}(X)$ and ${\operatorname{Per}}(X)$ is the set of periodic points of $X$. We say that a *singularity is hyperbolic* if the eigenvalues of the derivative $DX(\sigma)$ of the vector field at the singularity $\sigma$ have nonzero real part. The set of critical elements of $X$ is the union of the singularities and the periodic orbits of $X$, and will be denoted by ${\operatorname{Crit}}(X)$.

We recall that an hyperbolic set $\Lambda$ for a flow $X_t$ is an invariant subset of $M$ with a decomposition $T_\Lambda M= E^s\oplus E^X \oplus E^u$ of the tangent bundle over $\Lambda$ which is a continuous splitting, where $E^X$ is the direction of the vector field, the subbundles are invariant under the derivative $DX_t$ of the flow $$\begin{aligned}
  DX_t\cdot E^*_x=E^*_{X_t(x)},\quad
  x\in\Lambda, \quad t\in{{\mathbb R}},\quad *=s,X,u;\end{aligned}$$ $E^s$ is uniformly contracted by $DX_t$ and $E^u$ is uniformly expanded: there are $K,\lambda>0$ so that $$\begin{aligned}
\label{eq:def-hyperbolic}
\|DX_t\mid_{E^s_x}\|\le K e^{-\lambda t},
  \quad
  \|DX_{-t} \mid_{E^u_x}\|\le K e^{-\lambda t},
  \quad x\in\Lambda, \quad t\in{{\mathbb R}}.\end{aligned}$$

We say that a point $x \in  M$ is [*nonwandering*]{} for $X$ provided for every neighborhood $U$ of $x$ there is $t > 0$ such that $X_t(x) \cap U \neq \emptyset$, i.e., there exists a point $y \in U$ with $X_t(y) \in U$. We denote by $\Omega(X)$ the non-wandering set of $X$.

An ${\varepsilon}$-chain from $x_0$ to $x_l$ for $X$ is a sequence $\{x_0, x_1, \cdots, x_l\}$ such that for all $0 \leq j \leq l$, the distance $d(x_{j-1}, x_j) < {\varepsilon}$.

We define the [*chain recurrent set*]{} of $X$ by $R(X) = \{x \in M; \textrm{there \ is \ an} \ {\varepsilon}-\textrm{chain \ from} \ x \ \textrm{to} \ x\}$. We say that two points are [*chain equivalent*]{} provided, given ${\varepsilon}> 0$, there is an ${\varepsilon}$-chain from $x$ to $y$ and from $y$ to $x$. It is known that this is an equivalence relation, the equivalence classes are called [*chain components*]{} of $R(X)$ and, for flows, the components are actually the connected components of $R(X)$. If $X$ admits a single chain component on an invariant set $\Lambda$, we say that $X$ is [*chain transitive*]{} on $\Lambda$. See [@Rob99], for instance.

From $C^1$ Pugh’s closing lemma we have $\Omega(X) \subset P_*(X) \subset R(X)$.

\[rmk:wen-preperiod\] As proved by Wen [@wen00], $C^1$ preperiodic sets do not explode under $C^1$ perturbations, i.e., for any $0 \leq i \leq d$ and for any neighborhood $U$ of $P_*(X)$, there is a $C^1$ neighborhood $\mathcal{V}$ of $X$ such that $P_*(Y) \subset U$, for any $Y \in \mathcal{V}$.

Indeed, for example in [@shub87], we can see that the recurrent set do not explode under $C^0$ perturbations, i.e., if $U$ is a neighborhood of $R(X)$ there is a neighborhood $\mathcal{V}$ of $X$ such that $R(Y) \subset U$ for all $Y \in \mathcal{V}$.

Next, we explain the kind of hyperbolicities we are dealing.

\[def1\] A *dominated splitting* over a compact invariant set $\Lambda$ of $X$ is a continuous $DX_t$-invariant splitting $T_{\Lambda}M =
  E \oplus F$ with $E_x \neq \{0\}$, $F_x \neq \{0\}$ for every $x \in \Lambda$ and such that there are positive constants $K, \lambda$ satisfying $$\begin{aligned}
\label{eq:def-dom-split}
    \|DX_t|_{E_x}\|\cdot\|DX_{-t}|_{F_{X_t(x)}}\|<Ke^{-{\lambda}t}, \ \textrm{for all} \ x \in \Lambda, \ \textrm{and
      all} \,\,t> 0.
  \end{aligned}$$

A compact invariant set $\Lambda$ is said to be *partially hyperbolic* if it exhibits a dominated splitting $T_{\Lambda}M = E \oplus F$ such that subbundle $E$ is uniformly contracted. In this case $F$ is the *central subbundle* of $\Lambda$.

A compact invariant set $\Lambda$ is said to be *singular-hyperbolic* if it is partially hyperbolic and the action of the tangent cocycle expands volume along the central subbundle, i.e., $$\begin{aligned}
\label{eq:def-vol-exp}
      \vert \det (DX_t\vert_{F_x}) \vert > C e^{{\lambda}t},
      \forall t>0, \ \forall \ x \in \Lambda.
    \end{aligned}$$

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

\[def:sec-exp\] A *sectional hyperbolic set* is a singular hyperbolic one such that for every two-dimensional linear subspace $L_x \subset F_x$ one has $$\begin{aligned}
\label{eq:def-sec-exp}
      \vert \det (DX_t \vert_{L_x})\vert > C e^{{\lambda}t},
      \forall t>0.
    \end{aligned}$$

In [@Salg2016], this author give another definition of singular hyperbolicity encompassing two previous as follow.

\[new-def-singhyp\] A compact invariant set $\Lambda \subset M$ is *$p$-sectional hyperbolic or singular hyperbolic of order $p$* for $X$ if all singularities in $\Lambda$ are hyperbolic, there exists a partially hyperbolic splitting of the tangent bundle on $T_{\Lambda}M = E \oplus F$ and constants $C,\lambda > 0$ such that for every $x
  \in \Lambda$ and every $t>0$ we have

1.  $\Vert DX_t\vert_{E_x} \Vert \leq C e^{- \lambda t}$;

2.  $\vert \wedge^p DX_t \vert_{L_x}\vert > C^{-1} e^{{\lambda}t}$, for every $p$-dimensional linear subspace $L_x \subset
      F_x$.

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 $2 \leq p \leq \dim{F}$.

From now on, we consider $M$ a connected compact finite dimensional riemannian manifold and all singularities of $X$ (if they exist) are hyperbolic.

Fields of quadratic forms {#sec:fields-quadrat-forms}
-------------------------

From now, we introduce the quadratic forms and its properties.

Let ${\EuScript{J}}:E_U\to{{\mathbb R}}$ be a continuous family of quadratic forms ${\EuScript{J}}_x:E_x\to{{\mathbb R}}$ which are non-degenerate and have index $0<q<\dim(E)=n$, where $U\subset M$ is an open set such that $X_t(U) \subset \overline{U}, \forall t \geq 0,$ for a vector field $X$. We also assume that $({\EuScript{J}}_x)_{x\in U}$ is continuously differentiable along the flow.

The continuity assumption on ${\EuScript{J}}$ just means that for every continuous section $Z$ of $E_U$ the map $U\to{{\mathbb R}}$ given by $x\mapsto {\EuScript{J}}(Z(x))$ is continuous. The $C^1$ assumption on ${\EuScript{J}}$ along the flow means that the map $x\mapsto
{\EuScript{J}}_{X_t(x)} (Z(X_t(x)))$ is continuously differentiable for all $x\in U$ and each $C^1$ section $Z$ of $E_U$.

The assumption that $M$ is a compact manifold enables us to globally define an inner product in $E$ with respect to which we can find the an orthonormal basis associated to ${\EuScript{J}}_x$ for each $x$, as follows. Fixing an orthonormal basis on $E_x$ we can define the linear operator $$\begin{aligned}
  J_x:E_x\to E_x \quad\text{such that}\quad {\EuScript{J}}_x(v)=<J_x
  v,v> \quad \text{for all}\quad v\in T_xM,\end{aligned}$$ where $<,>=<,>_x$ is the inner product at $E_x$. Since we can always replace $J_x$ by $(J_x+J_x^*)/2$ without changing the last identity, where $J_x^*$ is the adjoint of $J_x$ with respect to $<,>$, we can assume that $J_x$ is self-adjoint without loss of generality. Hence, we represent ${\EuScript{J}}(v)$ by a non-degenerate symmetric bilinear form $<{\EuScript{J}}_x v,v>_x$. Now we use Lagrange’s method to diagonalize this bilinear form, obtaining a base $\{u_1,\dots,u_n\}$ of $E_x$ such that $$\begin{aligned}
  {\EuScript{J}}_x(\sum_{i}\alpha_iu_i)=\sum_{i=1}^q -\lambda_i\alpha_i^2 +
  \sum_{j=q+1}^n \lambda_j\alpha_j^2, \quad
  (\alpha_1,\dots,\alpha_n)\in{{\mathbb R}}^n.\end{aligned}$$ Replacing each element of this base according to $v_i=|\lambda_i|^{1/2}u_i$ we deduce that $$\begin{aligned}
{\EuScript{J}}_x(\sum_{i}\alpha_iv_i)=\sum_{i=1}^q -\alpha_i^2 +
  \sum_{j=q+1}^n \alpha_j^2, \quad
  (\alpha_1,\dots,\alpha_n)\in{{\mathbb R}}^n.\end{aligned}$$ Finally, we can redefine $<,>$ so that the base $\{v_1,\dots, v_n\}$ is orthonormal. This can be done smoothly in a neighborhood of $x$ in $M$ 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 [@Maltsev63].

In this adapted inner product we have that $J_x$ has entries from $\{-1,0,1\}$ only, $J_x^*=J_x$ and also that $J_x^2=J_x$.

Having fixed the orthonormal frame as above, the *standard negative subspace* at $x$ is the one spanned by $v_{1},\dots, v_{q}$ and the *standard positive subspace* at $x$ is the one spanned $v_{q+1},\dots,v_n$.

### Positive and negative cones {#sec:positive-negative-co}

Let ${\EuScript{C}}_\pm=\{C_\pm(x)\}_{x\in U}$ be the family of positive and negative cones $$\begin{aligned}
  C_\pm(x):=\{0\}\cup\{v\in E_x: \pm{\EuScript{J}}_x(v)>0\}  \quad x\in U\end{aligned}$$ and also let ${\EuScript{C}}_0=\{C_0(x)\}_{x\in U}$ be the corresponding family of zero vectors $C_0(x)={\EuScript{J}}_x^{-1}(\{0\})$ for all $x\in U$. In the adapted coordinates obtained above we have $$\begin{aligned}
  C_0(x)=\{v=\sum_{i}\alpha_iv_i\in E_x :
  \sum_{j=q+1}^n \alpha_j^2 = \sum_{i=1}^q
  \alpha_i^2\}\end{aligned}$$ is the set of *extreme points* of $C_\pm(x)$.

The following definitions are fundamental to state our main result.

\[def:J-separated\] Given a continuous field of non-degenerate quadratic forms ${\EuScript{J}}$ with constant index on the trapping region $U$ for the flow $X_t$, we say that the flow is

-   ${\EuScript{J}}$-*separated* if $DX_t(x)(C_+(x))\subset
      C_+(X_t(x))$, for all $t>0$ and $x\in U$;

-   *strictly ${\EuScript{J}}$-separated* if $DX_t(x)(C_+(x)\cup
      C_0(x))\subset C_+(X_t(x))$, for all $t>0$ and $x\in U$;

-   ${\EuScript{J}}$-*monotone* if ${\EuScript{J}}_{X_t(x)}(DX_t(x)v)\ge {\EuScript{J}}_x(v)$, for each $v\in
      T_xM\setminus\{0\}$ and $t>0$;

-   *strictly ${\EuScript{J}}$-monotone* if $\partial_t\big({\EuScript{J}}_{X_t(x)}(DX_t(x)v)\big)\mid_{t=0}>0$, for all $v\in T_xM\setminus\{0\}$, $t>0$ and $x\in U$;

-   ${\EuScript{J}}$-*isometry* if ${\EuScript{J}}_{X_t(x)}(DX_t(x)v) = {\EuScript{J}}_x(v)$, for each $v\in T_xM$ and $x\in U$.

Thus, ${\EuScript{J}}$-separation corresponds to simple cone invariance and strict ${\EuScript{J}}$-separation corresponds to strict cone invariance under the action of $DX_t(x)$.

\[rmk:J-separated-C-\] If a flow is strictly ${\EuScript{J}}$-separated, then for $v\in T_xM$ such that ${\EuScript{J}}_x(v)\le0$ we have ${\EuScript{J}}_{X_{-t}(x)}(DX_{-t}(v))<0$ for all $t>0$ and $x$ such that $X_{-s}(x)\in U$ for every $s\in[-t,0]$. Indeed, otherwise ${\EuScript{J}}_{X_{-t}(x)}(DX_{-t}(v))\ge0$ would imply ${\EuScript{J}}_x(v)={\EuScript{J}}_x\big(DX_t(DX_{-t}(v))\big)>0$, contradicting the assumption that $v$ was a non-positive vector.

This means that a flow $X_t$ is strictly ${\EuScript{J}}$-separated if, and only if, its time reversal $X_{-t}$ is strictly $(-{\EuScript{J}})$-separated.

A vector field $X$ is ${\EuScript{J}}$-*non-negative* on $U$ if ${\EuScript{J}}(X(x))\ge0$ for all $x\in U$, and ${\EuScript{J}}$-*non-positive* on $U$ if ${\EuScript{J}}(X(x))\leq 0$ for all $x\in U$. When the quadratic form used in the context is clear, we will simply say that $X$ is non-negative or non-positive.

We apply this notion to the linear Poincaré flow defined on regular orbits of $X_t$ as follows.

Suppose that the vector field $X$ is non-negative on $U$. Then, the span $E^X_x$ of $X(x)\neq 0$ is a ${\EuScript{J}}$-non-degenerate subspace.

According to item (1) of Proposition \[pr:propbilinear\], we have that $T_xM=E_x^X\oplus N_x$, where $N_x$ is the pseudo-orthogonal complement of $E^X_x$ with respect to the bilinear form ${\EuScript{J}}$, and $N_x$ is also non-degenerate. Moreover, by the definition, the index of ${\EuScript{J}}$ restricted to $N_x$ is the same as the index of ${\EuScript{J}}$. Thus, we can define on $N_x$ the positive and negative cones with core $N_x^+$ and $N_x^-$, respectively.

Define the Linear Poincaré Flow $P^{\, t}$ of $X_t$ along the orbit of $x$, by projecting $DX_t$ orthogonally (with respect to ${\EuScript{J}}$) over $N_{X_t(x)}$ for each $t\in{{\mathbb R}}$: $$\begin{aligned}
  P^{\, t} v := \Pi_{X_t(x)}DX_t v ,
  \quad
  v\in T_x M, t\in{{\mathbb R}}, X(x)\neq 0,\end{aligned}$$ where $\Pi_{X_t(x)}:T_{X_t(x)}M\to N_{X_t(x)}$ is the projection on $N_{X_t(x)}$ parallel to $X(X_t(x))$. We remark that the definition of $\Pi_x$ depends on $X(x)$ and ${\EuScript{J}}_X$ only. The linear Poincaré flow $P^{\,t}$ is a linear multiplicative cocycle over $X_t$ on the set $U$ with the exclusion of the singularities of $X$.

In this setting we can say that the linear Poincaré flow is (strictly) ${\EuScript{J}}$-separated and (strictly) ${\EuScript{J}}$-monotonous using the non-degenerate bilinear form ${\EuScript{J}}$ restricted to $N_x$ for a regular $x\in U$. More precisely: $P^t$ is ${\EuScript{J}}$-monotonous if $\partial_t{\EuScript{J}}(P^tv)\mid_{t=0}\ge0$, for each $x\in U, v\in T_xM\setminus\{0\}$ and $t>0$, and strictly ${\EuScript{J}}$-monotonous if $\partial_t{\EuScript{J}}(P^tv)\mid_{t=0}>0$, for all $v\in T_xM\setminus\{0\}$, $t>0$ and $x\in U$.

\[pr:J-separated-spectrum\] Let $L:V\to V$ be a ${\EuScript{J}}$-separated linear operator. Then

1.  $L$ can be uniquely represented by $L=RU$, where $U$ is a ${\EuScript{J}}$-isometry and $R$ is ${\EuScript{J}}$-symmetric (or ${\EuScript{J}}$-pseudo-adjoint; see Proposition \[pr:propbilinear\]) with positive spectrum.

2.  the operator $R$ can be diagonalized by a ${\EuScript{J}}$-isometry. Moreover the eigenvalues of $R$ satisfy $$\begin{aligned}
          0<r_-^q\le\dots\le r_-^1=r_-\le r_+=r_1^+\le\dots\le r_+^p.
        \end{aligned}$$

3.  the operator $L$ is (strictly) ${\EuScript{J}}$-monotonous if, and only if, $r_-\le (<) 1$ and $r_+\ge (>) 1$.

${\EuScript{J}}$-separated linear maps {#sec:j-separat-linear}
--------------------------------------

### ${\EuScript{J}}$-symmetrical matrixes and ${\EuScript{J}}$-selfadjoint operators {#sec:j-symmetr-matrix}

The symmetrical bilinear form defined by $(v,w)=\langle J_x
v,w\rangle$, $v,w\in E_x$ for $x\in M$ endows $E_x$ with a pseudo-Euclidean structure. Since ${\EuScript{J}}_x$ is non-degenerate, then the form $(\cdot,\cdot)$ is likewise non-degenerate and many properties of inner products are shared with symmetrical non-degenerate bilinear forms. We state some of them below.

We recall that $E^\perp:=\{v\in V: (v,w)=0
    \quad\text{for all}\quad w\in E\}$, the pseudo-orthogonal space of $E$, is defined using the bilinear form.

\[pr:propbilinear\] Let $(\cdot,\cdot):V\times V \to{{\mathbb R}}$ be a real symmetric non-degenerate bilinear form on the real finite dimensional vector space $V$.

1.  $E$ is a subspace of $V$ for which $(\cdot,\cdot)$ is non-degenerate if, and only if, $V=E\oplus E^\perp$.

2.  Every base $\{v_1,\dots,v_n\}$ of $V$ can be orthogonalized by the usual Gram-Schmidt process of Euclidean spaces, that is, there are linear combinations of the basis vectors $\{w_1,\dots, w_n\}$ such that they form a basis of $V$ and $(w_i,w_j)=0$ for $i\neq j$. Then this last base can be pseudo-normalized: letting $u_i=|(w_i,w_i)|^{-1/2}w_i$ we get $(u_i,u_j)=\pm\delta_{ij}, i,j=1,\dots,n$.

3.  There exists a maximal dimension $p$ for a subspace $P_+$ of ${\EuScript{J}}$-positive vectors and a maximal dimension $q$ for a subspace $P_-$ of ${\EuScript{J}}$-negative vectors; we have $p+q=\dim V$ and $q$ is known as the *index* of ${\EuScript{J}}$.

4.  For every linear map $L:V\to{{\mathbb R}}$ there exists a unique $v\in V$ such that $L(w)=(v,w)$ for each $w\in V$.

5.  For each $L:V\to V$ linear there exists a unique linear operator $L^+:V\to V$ (the pseudo-adjoint) such that $(L(v),w)=(v,L^+(w))$ for every $v,w\in V$.

6.  Every pseudo-self-adjoint $L:V\to V$, that is, such that $L=L^+$, satisfies

    1.  eigenspaces corresponding to distinct eigenvalues are pseudo-orthogonal;

    2.  if a subspace $E$ is $L$-invariant, then $E^\perp$ is also $L$-invariant.

The proofs are rather standard and can be found in [@Maltsev63].

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

\[le:kuhne\] Let $V$ be a real finite dimensional vector space endowed with a non-positive definite and non-degenerate quadratic form ${\EuScript{J}}:V\to{{\mathbb R}}$.

If a symmetric bilinear form $F:V\times V\to{{\mathbb R}}$ is non-negative on $C_0$ then $$\begin{aligned}
  r_+=\inf_{v\in C_+} \frac{F(v,v)}{\langle Jv,v\rangle}
  \ge \sup_{u\in C_-}\frac{F(u,u)}{\langle Ju,u\rangle}=r_-\end{aligned}$$ and for every $r$ in $[r_-,r_+]$ we have $F(v,v)\ge r\langle Jv,v\rangle$ for each vector $v$.

In addition, if $F(\cdot,\cdot)$ is positive on $C_0\setminus\{0\}$, then $r_-<r_+$ and $F(v,v)>
r\langle Jv,v\rangle$ for all vectors $v$ and $r\in(r_-,r_+)$.

\[rmk:Jseparated\] Lemma \[le:kuhne\] shows that if $F(v,w)=\langle \tilde J
  v,w\rangle$ for some self-adjoint operator $\tilde J$ and $F(v,v)\ge0$ for all $v$ such that $\langle J v,
  v\rangle=0$, then we can find $a\in{{\mathbb R}}$ such that $\tilde J \ge a J$. This means precisely that $\langle
  \tilde J v,v\rangle\ge a\langle Jv, v\rangle$ for all $v$.

If, in addition, we have $F(v,v)>0$ for all $v$ such that $\langle J v, v\rangle=0$, then we obtain a strict inequality $\tilde J > a J$ for some $a\in{{\mathbb R}}$ since the infimum in the statement of Lemma \[le:kuhne\] is strictly bigger than the supremum.

The (longer) proofs of the following results can be found in [@Wojtk01] or in [@Pota79]; see also [@Wojtk09].

For a ${\EuScript{J}}$-separated operator $L:V\to V$ and a $d$-dimensional subspace $F_+\subset C_+$, the subspaces $F_+$ and $L(F_+)\subset C_+$ have an inner product given by ${\EuScript{J}}$. Thus both subspaces are endowed with volume elements. Let $\alpha_d(L;F_+)$ be the rate of expansion of volume of $L\mid_{F_+}$ and $\sigma_d(L)$ be the infimum of $\alpha_d(L;F_+)$ over all $d$-dimensional subspaces $F_+$ of $C_+$.

\[pr:product-vol-exp\] We have $\sigma_d(L)=r_+^1 \cdots r_+^d$, where $r^i_+$ are given by Proposition \[pr:J-separated-spectrum\](2).

Moreover, if $L_1,L_2$ are ${\EuScript{J}}$-separated, then $\sigma_d(L_1L_2)\ge\sigma_d(L_1)\sigma_d(L_2)$.

The following corollary is very useful.

\[cor:compos-max-exp\] For ${\EuScript{J}}$-separated operators $L_1,L_2:V\to V$ we have $$\begin{aligned}
    r_+^1(L_1L_2)\ge r_+^1(L_1) r_+^1(L_2) \quad\text{and}\quad
    r_-^1(L_1L_2)\le r_-^1(L_1)r_-^1(L_2).
  \end{aligned}$$ Moreover, if the operators are strictly ${\EuScript{J}}$-separated, then the inequalities are strict.

\[rmk:J-mon-spec\] Another important property about the singular values of a ${\EuScript{J}}$-separated operator $L$ is that $$r_+^1 = r_+ \ge 1 (> 1) \quad\text{and}\quad r_-^1 = r_- \le 1 (< 1)$$ if, and only if, $L$ is (strictly) ${\EuScript{J}}$-monotone.

This property will be used a lot of times in our proofs.

### Lyapunov exponents

By Oseledec’s Ergodic Theorem [@Os68], there exist a full probability set $X$ such that for every $x \in Y$ there is an invariant decomposition $$\begin{aligned}
T_xM = \langle X\rangle \oplus E_{1}(x) \oplus \cdots \oplus E_{l(x)}(x)\end{aligned}$$ and numbers $\chi_1 < \cdots < \chi_l$ corresponding to the limits $$\begin{aligned}
  \chi_j = \lim\limits_{t \to +\infty} \frac{1}{t} \log \Vert DX_t(x) \cdot v\Vert,\end{aligned}$$ for every $v \in E_i(x)\setminus \{0\}, i = 1, \cdots, l(x)$.

In this setting, Wojtkowski [@Wojtk01] proved that the logarithm of the pseudo-Euclidean singular values $0 \leq r_q^- \leq \cdots \leq r_1^- \leq r_1^+ \leq \cdots \leq r_p^+$ of $DX_t$ are $\mu$-integrable, and obtained estimates of the Lyapunov exponents related to the singular eigenvalues of strictly ${\EuScript{J}}$-separated maps.

[@Wojtk01 Corollary 3.7] \[thm:lyap-exp-sing-val\] For $1 \leq k_1 \leq p$ and $1 \leq k_2 \leq q$

$$\begin{aligned}
\chi^-_1 + \cdots + \chi^-_{k_1} \leq \sum_{i=1}^{k_1} \int \log r^-_i d\mu \ \textrm{and} \ \chi^+_1 + \cdots + \chi^+_{k_2} \geq \sum_{i=1}^{k_2} \int \log r^+_i d\mu.\end{aligned}$$

This result will be very useful in proof of Theorem \[mthm:strong-homog-to-quad\].

Some applications
=================

In this section, we present some applications of this theory related to some kind of hyperbolicities, as partial and singular ones. In particular, we provide another proof of [@GanLiWen2005 Theorem A].

### Some results about partial and sectional hyperbolicity from ${\EuScript{J}}$-separation

The author, together with V. Araújo, proved in [@ArSal2012] the following useful theorem which relates partial hyperbolicity and ${\EuScript{J}}$-separated sets for a flow.

[@ArSal2012 Theorem A] \[mthm:Jseparated-parthyp\] A maximal invariant subset $\Lambda$ of a trapping region $U$ whose singularities are hyperbolic is a partially hyperbolic set for a flow $X_t$ if, and only if, there is a $C^1$ field ${\EuScript{J}}$ of non-degenerate quadratic forms with constant index, equal to the dimension of the stable subspace of $\Lambda$, such that $X_t$ is a non-negative strictly ${\EuScript{J}}$-separated flow on $U$.

This result will be useful in our applications of Theorem \[mthm:strong-homog-equiv\].

In the sequence, we can give another proof of next result from [@GanLiWen2005].

[@GanLiWen2005 Theorem A] Let $X \in {\mathfrak{X}^{1}(M)}$, and $\Lambda$ be a robustly transitive singular set of $X$ that is strongly homogeneous of index ${\operatorname{ind}}$. If every singularity $\sigma$ of $X$ is hyperbolic of index ${\operatorname{ind}}(\sigma) > {\operatorname{ind}}$, then $\Lambda$ has a partially hyperbolic splitting of contracting dimension Ind. Likewise, if every singularity $\sigma$ of $X$ is hyperbolic of index ${\operatorname{ind}}(\sigma) \leq {\operatorname{ind}}$, then $\Lambda$ has a partially hyperbolic splitting of expanding dimension $n - 1 - Ind$.

We are going to deal with the case ${\operatorname{ind}}(\sigma) > {\operatorname{ind}}$, the other case is analogous.

Since $\Lambda$ is strongly homogeneous and ${\operatorname{ind}}(\sigma) > {\operatorname{ind}}$, by [@GanLiWen2005 Lemma 4.1] there is a dominated splitting $T_{\sigma}M = E_{\sigma} \oplus F_{\sigma}$ such that $\dim(E_{\sigma}) = {\operatorname{ind}}$. Hence, Theorem \[mthm:strong-homog-to-quad\] implies that there exists a field of non-degenerate quadratic forms ${\EuScript{J}}$ on $\Lambda$ with index ${\operatorname{ind}}({\EuScript{J}}) = {\operatorname{ind}}(\Lambda)$ for which $X$ is strictly ${\EuScript{J}}$-separated and the associated linear Poincaré flow $P^t$ is strictly ${\EuScript{J}}$-monotone on every compact invariant subset $\gamma$ of $\Lambda^*$. Therefore, Theorem \[mthm:Jseparated-parthyp\] completes 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 [@MeMor06] and [@ArbMo2013].

\[mcor:2-sec-exp-J-monot\] Let $X\in{\mathfrak{X}^{1}(M)}, \dim(M) \geq 4$ with a nontrivial transitive compact invariant set $\Lambda$ whose singularities, if any, are hyperbolic.

Then the following conditions are equivalent:

1.  There exists a family ${\EuScript{J}}$ of smooth non-degenerate indefinite quadratic forms with constant index ${\operatorname{ind}}({\EuScript{J}})$ on $\Lambda$ such that $X$ is a non-negative strictly ${\EuScript{J}}$-separated vector field, for which the linear Poincaré flow is strictly ${\EuScript{J}}$-monotonous on every compact invariant set in $\Lambda_X(U)^*=\Lambda_X(U)\setminus {\mathrm{Sing}}(X)$

2.  The set $\Lambda$ is a sectional-hyperbolic subset for $X$ with constant index ${\operatorname{ind}}({\EuScript{O}})={\operatorname{ind}}({\EuScript{J}})$ for all periodic orbits ${\EuScript{O}}$ of $\Lambda$ and ${\operatorname{ind}}(\sigma)=Ind({\EuScript{J}})+1$ for all singularities $\sigma\in\Lambda\cap {\mathrm{Sing}}(X)$.

For the next statement, we recall that a hyperbolic singularity $\sigma$ is said to be of codimension one if its index satisfies either ${\operatorname{ind}}(\sigma) = 1$ or ${\operatorname{ind}}(\sigma) = n - 1$, where $n = \dim(M)$.

\[obs-lyap-est\] Every attracting set is Lyapunov stable.

\[thm:lyap-est-sec-hyp\] Let $\Lambda \subset M^n, n \geq 4$, 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.  $\Lambda$ is sectional-hyperbolic with $1 \leq dim(E^s) = {\operatorname{ind}}({\EuScript{J}}) \leq n-2$;

2.  There exists a field of non-degenerate quadratic forms with constant index $1 \leq {\operatorname{ind}}({\EuScript{J}}) \leq n-2$ such that $X$ is non-negative strictly ${\EuScript{J}}$-separated on $\Lambda$ and every compact invariant subset $\Gamma \subset \Lambda$ is strictly ${\EuScript{J}}$-monotone for linear Poincaré flow associated to $X$.

Proof of Corollaries \[mcor:2-sec-exp-J-monot\] and \[thm:lyap-est-sec-hyp\]
----------------------------------------------------------------------------

Indeed, suppose that $(2)$ is true. Then, $X$ is strongly homogeneous on $\Lambda$. By [@ArbMo2013 Corollary 8], this is a sectional hyperbolic set for $X$. To prove the converse statement, we need just use [@ArSal2012 Theorem D].

The next proof needs the following lemma.

Let $\Lambda$ be a compact invariant set for a flow $X$ of a $C^1$ vector field $X$ on $M$.

[@AraArbSal Lemma 5.1] \[le:flow-center\] Given a continuous splitting $T_\Lambda M = E\oplus F$ such that $E$ is uniformly contracted, then $X(x)\in F_x$ for all $x\in \Lambda$.

Suppose that $\Lambda$ is sectional-hyperbolic with decomposition $E \oplus F$. So, it is clearly strongly homogeneous. Once the subbundles are non-trivial and $E$ is uniformly contracting, we must have $1 \leq \dim(E) := {\operatorname{ind}}({\EuScript{J}}) \leq n-2$, because by Lemma \[le:flow-center\], $\langle X \rangle \subset F$.

By Theorem \[mthm:Jseparated-parthyp\], there exists a field ${\EuScript{J}}$ of differentiable quadratic forms with constant index equal to the dimension of $E$ with the required properties.

Reciprocally, the existence of such a field ${\EuScript{J}}$ implies, by Theorem \[mthm:strong-homog-equiv\], that $\Lambda$ is strongly homogeneous of index ${\operatorname{ind}}({\EuScript{J}})$. Thus, once the singularities are hyperbolic of codimension one, it is enough to use Lemma [@ArbMo2013 Corollary 9].

Proof of Theorems
=================

Now, we prove our mains results.

To prove the Theorem \[cor:star-flow\] we use the following result from [@ArSal2012].

[@ArSal2012 item 3,Theorem 2.23] \[prop:J-hyperbolic\] Let $\Gamma$ be a compact invariant set for $X$ with a dominated splitting $T_{\Gamma}M = E \oplus F$. Let ${\EuScript{J}}$ be a $C^1$ field of indefinite quadratic forms such that $DX_t$ is strictly ${\EuScript{J}}$-separated. Then, $E \oplus F$ is uniformly hyperbolic if, and only if, there is an equivalent field ${\EuScript{J}}$ of quadratic forms on a neighborhood of $\Gamma$ such that ${\EuScript{J}}'(v) > 0$, for all $v \in T_{\Gamma}M$ and all $x \in \Gamma$.

If $X$ is a star flow, then each singular point $\sigma$ is hyperbolic and it is well known that its hyperbolic decomposition $E^s_{\sigma} \oplus E^u_{\sigma}$ is a dominated one. So, by using adapted metrics (see [@Goum07]) we construct the desired quadratic form $J_\sigma$ such that $X$ is strictly separated (see [@ArSal2012]) and, by Proposition \[prop:J-hyperbolic\] ${\EuScript{J}}'(v)>0$ for all $v \in T_{\Gamma}M$.

Analogously, for every periodic orbit $\gamma$ of $X$, consider the hyperbolic splitting $T_\gamma M = E^s \oplus E^X \oplus E^u$. Again, considering $E^s \oplus (E^X \oplus E^u)$ as a dominated splitting we obtain a quadratic form ${\EuScript{J}}$ for which $X$ is strictly separated on $\gamma$. By construction of the adapted metrics, we have that ${\EuScript{J}}$ is $C^ 1$ along the flow (see [@Goum07] for details about the construction of such a adapted metric). In addition, the linear Poincaré flow associated to $X$, $P^t$ is hyperbolic and then ${\EuScript{J}}$-monotone on $\gamma$.

If $\gamma$ is a sink (respectively, a source) the splitting $E^s \oplus E^X$ (respectively, $E^u \oplus E^X$) is a dominated one and we proceed constructing the cones the same way, however the core of the nonnegative cone is the field direction.

Reciprocally, take a small neighborhood $U$ of $Per_*(X)$ such that there is a $C^1$ neighborhood $\mathcal{V}$ of $X$ for which $Per_*(Y) \subset U$ and suppose that such a field of quadratic forms is defined on $U$. By Proposition \[prop:J-hyperbolic\], every singularity $\sigma \in U$ is hyperbolic. The case of periodic orbits is analogous.

Shrinking $U$, if necessary, we may suppose that, for each preperiodic orbit and each singularity in $U$ of each $Y \in \mathcal{V}$, we have quadratic forms (still denoted ${\EuScript{J}}$) with the same features as before. Indeed, since the quadratic form on each periodic orbit is $C^1$ along the flow, for any $Y \in \mathcal{V}$, shrinking $\mathcal{V}$ if necessary, we must have that any preperiodic orbit of $X$ present stricly montonicity for the linear Poincaré flow.

If, for some $Y \in \mathcal{V}$, another periodic orbit is created, by $C^1$-closeness it is ${\EuScript{J}}$-monotone for the linear Poincaré flow $P^t_Y$ associated to $Y$, since it comes from a preperiodic one of $X$ which is ${\EuScript{J}}$-monotone for the linear Poincaré flow $P^t_X$, by hyphotesis. See [@ArSal2012 Section 2.5.4].

Hence, every periodic orbit for any $Y \in \mathcal{V}$ is hyperbolic. Therefore, $X$ is a star flow.

Now, it is proved the second main result.

Since the linear Poincaré flow is strictly ${\EuScript{J}}$-monotone on each preperiodic orbit $\gamma \subset \Lambda^*$ implies that, if $\gamma$ is a closed orbit, then it is a hyperbolic subset of $\Lambda$, with a constant index, which we denote ${\operatorname{ind}}({\EuScript{J}})$. Moreover, taking a small enough neighborhood $U$ of $Per_*(X\vert_\Lambda)$ there exists some neighborhood $\mathcal{V}$ of $X$ such that $Per_*(Y\vert_{\Lambda_Y})\subset U$, $\forall Y \in \mathcal{V}$. If some periodic orbit $\gamma_Y$ is created by a small $C^1$ perturbation of $X$, it comes from a preperiodic orbit of $X$. Thus, $\gamma_Y$ is a hyperbolic closed orbit of $Y$, and must have index equal to ${\operatorname{ind}}({\EuScript{J}})$.

Hence, ${\operatorname{ind}}({\EuScript{J}})$ does not change by small differentiable perturbations of $X$ on a neighborhood of $\Lambda$, so the index of hyperbolic periodic orbits also does not change. Therefore, $\Lambda$ is strongly homogeneous for $X$.

Reciprocally, if $\Lambda$ is strongly homogeneous of index ${\operatorname{ind}}$, then cannot be there a non-hyperbolic periodic orbit. Otherwise, we can create two periodic orbits with different indices, by Frank’s Lemma. Moreover, $X$ is a star flow in a neighborhood of $\Lambda$. Hence, by Theorem \[cor:star-flow\] we can define the desired field of quadratic forms ${\EuScript{J}}$, with fixed index ${\operatorname{ind}}({\EuScript{J}}) = {\operatorname{ind}}$, defined on a neighborhood $U$ of $\Lambda$, where $U$ is the neighborhood for which $Per_*(Y\vert_{\Lambda_Y}) \subset U$ for any $Y$ close enough to $X$.

Note that Corollary \[mcor:strong-homog\] follows from Theorem \[mthm:strong-homog-equiv\], just observing now that if any singularity $\sigma$ is accumulated by regular orbits, it cannot present ${\operatorname{ind}}(\sigma) < {\operatorname{ind}}{{\EuScript{J}}}$, once $X \in {\mathfrak{X}^{1}(M)}$, ${\EuScript{J}}$ is a continuous field of quadratic forms and $X$ is ${\EuScript{J}}$- monotonic over any compact invariant nonsingular set $\Gamma$.

Now, we prove our last main result.

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

Let $Z$ be a compact metric space and denote $\mathcal{M}(Z)$ the set of probabilities measures on the Borel $\sigma$-algebra of $Z$. If $T: Z \to Z$ is a measurable map, we say that a probability measure $\mu$ is an invariant measure of $T$, if $\mu(T^{-1}(A)) = \mu(A)$, for every measurable set $A \subset Z$. We say that $\mu$ is an invariant measure of $X$ if it is an invariant measure of $X_t$ for every $t \in \mathbb{R}$. We will denote by $\mathcal{M}_X$ the set of all invariant measures of $X$. A subset $Y\subset Z$ has *total probability* if for every $\mu\in \mathcal{M}_X$ we have $\mu(Y)=1$ (see [@Man82]). The support of a measure $\mu$, denoted by $supp(\mu)$, is the set of points for which the measure is non-zero. An invariant measure is said to be *atomic* if its support is either a closed orbit or a singularity.

A probability measure $\mu$ is an *ergodic measure* if for every invariant set $A$ we have $\mu(A) = 1$ or $\mu(A) = 0$. Finally, a certain property is said to be valid in *$\mu$-almost every point* if it is valid in the whole Z except, possibly, in a set of null measure.

We recall the definition of $\delta$-closable points of [@Man82]. We say that a point $x \in M \setminus Sing(X)$ is $\delta$-closable if, for any $C^1$ neighborhood ${{\mathcal U}}\subset {\mathfrak{X}^{1}(M)}$ of $X$, there exists a vector field $Z \in {{\mathcal U}}$, a point $z \in M$ and $T > 0$ such that:

1.  $Z_T(z) = z$,

2.  $Z = X$ on $M \setminus B_{\delta} (X_{[0, T]}(x))$ and

3.  $dist(Z_t(z), X_t(x)) < \delta, \forall 0 \leq t \leq T$.

We denote by $\Sigma(X)$ the set of points of $M$ which are $\delta$-closable for any $\delta$ sufficiently small.

If $\Lambda$ is a strongly homogeneous set for $X$ with singularities all of them hyperbolic, then $X$ is a star flow in $\Lambda$.

By Ergodic Closing Lemma, the $\delta$-closable set of $X$ has total probability.

If $x \in \Lambda$ is a regular $\delta$-closable point, then it is a pre-periodic point of index ${\operatorname{ind}}(\Lambda)$.

According the proof of [@GanLiWen2005 Lemma 5.3], we have a dominated splitting $E_x \oplus F_x$ of index ${\operatorname{ind}}(\Lambda)$ in $T_xM$, for all $x$ .

By Theorem \[thm:lyap-exp-sing-val\], we have $$\begin{aligned}
\chi^-_1 + \cdots + \chi^-_{k_1} \leq \sum_{i=1}^{k_1} \int \log r^-_i d\nu \ \textrm{and} \ \chi^+_1 + \cdots + \chi^+_{k_2} \geq \sum_{i=1}^{k_2} \int \log r^+_i d\nu,\end{aligned}$$ for any $k_1 \leq q, k_2 \leq p$.

Also according the proof of [@GanLiWen2005 Lemma 5.3], the ergodic probability measures are not atomic.

Now, Birkhoff’s ergodic theorem and Corollary \[cor:compos-max-exp\] imply that the Lyapunov exponents on $E$ are negative and the sectional Lyapunov exponents are positive, in a total probability subset of $\Lambda$.

Moreover, for singularities $\sigma \in {\mathrm{Sing}}(\Lambda)$ we have two possibilities:

First case: $\sigma$ is accumulated by recurrent orbits (including periodic orbits), then since ${\operatorname{ind}}(\sigma) \geq {\operatorname{ind}}(\Lambda)$, by [@GanLiWen2005 Lemma 4.1] there is a dominated splitting $T_{\sigma}M = E_{\sigma} \oplus F_{\sigma}$, where $dim (E) = {\operatorname{ind}}(\Lambda)$.

Second case: Either there exists a dominated splitting on $T_{\sigma}M = E_{\sigma} \oplus F_{\sigma}$ with $dim (E) = {\operatorname{ind}}(\Lambda)$, which guarantees the definition of ${\EuScript{J}}$ such that $X$ is stricly ${\EuScript{J}}$-separated. Or, otherwise, since $\sigma$ is an isolated hyperbolic singularity with ${\operatorname{ind}}(\sigma) \geq {\operatorname{ind}}(\Lambda)$, we have an invariant splitting for which we only guarantee that ${\EuScript{J}}$ such that $X$ is (not strictly) ${\EuScript{J}}$-separated.

So, we have an invariant splitting $T_{\Lambda}M = E_{\Lambda} \oplus F_{\Lambda}$ which has uniformly angle bounded away from zero and $T_{\sigma}M = E_{\sigma} \oplus F_{\sigma}$ is dominated for every $\sigma \in {\mathrm{Sing}}(X)$.

Now, [@AraArbSal Theorem C] implies that the corresponding decomposition $T_{\Lambda}M = E \oplus F$ is dominated of index ${\operatorname{ind}}(\Lambda)$.

By using the adapted metric for dominated splitting [@Goum07], we obtain a field of $C^1$ non-degenerated quadratic forms ${\EuScript{J}}$ such that $X$ strictly ${\EuScript{J}}$-separated over $\Lambda$, as in [@ArSal2012].

Now, to prove the ${\EuScript{J}}$-monotonicity, take a compact invariant set $\Gamma$ in $\Lambda^*$. Since $X$ is a star flow and $\Gamma$ is nonsingular, by [@GanWen2006 Theorem A], this set must be a hyperbolic one. So, by well known results, the linear Poincaré flow associated to $X$ is strictly ${\EuScript{J}}$-monotone on any compact invariant set $\Gamma \in \Lambda^*$.

Acknowledgements {#acknowledgements .unnumbered}
================

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

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[^1]: L. S. address - Universidade Federal da Bahia, Instituto de Matemática - Avenida Adhemar de Barros, Ondina, Zip code 40170-110, Salvador, Bahia, Brazil. email: lsalgado@ufba.br, lsalgado@im.ufrj.br.
