We prove that each invariant measure in a non-uniformly hyperbolic system can be approximated by atomic measures on hyperbolic periodic orbits. This contributes to our main result that the mean angle (Definition 1.10), independence number (Definition 1.6) and Oseledec splitting for an ergodic hyperbolic measure with simple spectrum can be approximated by those for atomic measures on hyperbolic periodic orbits, respectively. Combining this result with the approximation property of Lyapunov exponents by Wang and Sun, 2005 (Theorem 1.9), we strengthen Katok's closing lemma (1980) by presenting more extensive information not only about the state system but also its linearization. In the present paper, we also study an ergodic theorem and a variational principle for mean angle, independence number and Liao's style number (Definition 1.3) which are bases for discussing the approximation properties in the main result.

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Approximation properties on invariant measure and Oseledec splitting in non-uniformly hyperbolic systems

The specification property for flows from the robust and generic viewpoint

In the paper, the author addresses the Lyapunov characteristic spectrum of an ergodic autonomous ordinary differential system on a complete riemannian manifold of finite dimension such as the d-dimensional euclidean space ℝ d , not necessarily compact, by Liaowise spectral theorems that give integral expressions of Lyapunov exponents. In the context of smooth linear skew-product flows with Polish driving systems, the results are still valid. This paper seems to be an interesting contribution to the stability theory of ordinary differential systems with non-compact phase spaces.

Integral expressions of Lyapunov exponents for autonomous ordinary differential systems

We introduce an equivalent definition of the entropy for flows, and by using the equivalent definition we answer a problem raised by Liao concerning Liao hyperbolic systems, i.e. the state flow and its orthonormal n-frame flow having the same entropy.

Entropy of orthonormal n-frame flows

We prove that any vector field on a three-dimensional compact manifold can be approximated in the C1-topology by one which is singular hyperbolic or by one which exhibits a homoclinic tangency associated to a regular hyperbolic periodic orbit. This answers a conjecture by Palis. During the proof we obtain several other results with independent interest: a compactification of the rescaled sectional Poincar\'e flow and a generalization of Ma\~n\'e-Pujals-Sambarino theorem for three-dimensional C2 vector fields with singularities.

Homoclinic tangencies and singular hyperbolicity for three-dimensional vector fields

We present in the paper a Borel subset of a compact smooth manifold Ma to satisfy the requirements of the multiplicative ergodic theorem for a C1 tangent vector field on M(?). This set is more restrictive but might be more useful than other ones for the same purpose appearing previously in the literature.

On Characteristic Exponents Construction of a New Borel Set for the Multiplicative Ergodic Theorem for Vector Fields

Let M

 n
 be an n-dimensional compact C
∞-differentiable manifold, n ≥ 2, and let S be a C
1-differential system on M

 n
 . The system induces a one-parameter C
1 transformation group φ
 t
 (−∞ < t < ∞) over M

 n
 and, thus, naturally induces a one-parameter transformation group of the tangent bundle of M

 n
 . The aim of this paper, in essence, is to study certain ergodic properties of this latter transformation group. Among various results established in the paper, we mention here only the following, which might describe quite well the nature of our study. (A) Let M be the set of regular points in M

 n
 of the differential system S. With respect to a given C
∞ Riemannian metric of M

 n
 , we consider the bundle ${{\mathcal{L}}}^{\sharp }$

 of all (n−2) spheres Q

 x
 

n−2, x∈M, where Q

 x
 

n−2 for each x consists of all unit tangent vectors of M

 n
 orthogonal to the trajectory through x. Then, the differential system S gives rise naturally to a one-parameter transformation group ψ
 t
 #
 (−∞<t<∞) of ${{\mathcal{L}}}^{\sharp }$

 . For an l-frame α = (u
1, u
2,⋯, u

 l
 ) of M

 n
 at a point x in M, 1 ≥ l ≥ n−1, each u

 i
 being in ${{\mathcal{L}}}^{\sharp }$

 , we shall denote the volume of the parallelotope in the tangent space of M

 n
 at x with edges u
1, u
2,⋯, u

 l
 by υ(α), and let $\eta ^{*}_{\alpha } {\left( t \right)} = v {\big( {\psi ^{\sharp }_{t} {\left( {\mathbf{u}_{1} } \right)},\psi ^{\sharp }_{t} {\left( {\mathbf{u}_{2} } \right)}, \cdots ,\psi ^{\sharp }_{t} {\left( {\mathbf{u}_{l} } \right)}} \big)}$

 . This is a continuous real function of t. Let
 $$ I^{*}_{+} (\alpha) = \mathop {\overline\lim }\limits_{T \to \infty} \frac{1}{T}\int_{0}^{T} \eta^{*}_{\alpha} (t)\mbox{d}t,\quad I^{*}_{-} (\alpha) = \mathop {\overline\lim }\limits_{T \to - \infty}\frac{1}{T}\int_{0}^{T} \eta ^{*}_{\alpha}(t)\mbox{d}t.$$

 
α is said to be positively linearly independent of the mean if I
+
*(α) > 0. Similarly, α is said to be negatively linearly independent of the mean if I
−
*(α) > 0. A point x of M is said to possess positive generic index κ = κ
+
*(x) if, at x, there is a κ-frame $\alpha = {\left( {\mathbf{u}_{1},\;\mathbf{u}_{2},\; \cdots ,\;\mathbf{u}_{\kappa } } \right)}$

 , $\mathbf{u}_{i} \in {{\mathcal{L}}}^{\sharp }$

 , of M

 n
 having the property of being positively linearly independent in the mean, but at x, every l-frame $\beta = {\left( {\mathbf{v}_{1} ,\mathbf{v}_{2} , \cdots ,\mathbf{v}_{l} } \right)},\mathbf{v}_{i} \in {{\mathcal{L}}}^{\sharp }$

 , of M

 n
 with l >
κ does not have the same property. Similarly, we define the negative generic index κ
−
*(x) of x. For a nonempty closed subset F of M

 n
 consisting of regular points of S, invariant under φ

 t
 (−∞ < t < ∞), let the (positive and negative) generic indices of F be defined by
 $$ \kappa ^{*}_{ + } {\left( F \right)} = {\mathop {\max }\limits_{x \in F} }{\left\{ {\kappa ^{*}_{ + } {\left( x \right)}} \right\}},\quad \kappa ^{*}_{ - } {\left( F \right)} = {\mathop {\max }\limits_{x \in F} }{\left\{ {\kappa ^{*}_{ - } {\left( x \right)}} \right\}}. $$

 
Theorem
κ

 +
 
*(F)=κ

 −
 
*(F).
 (B) We consider a nonempty compact metric space x and a one-parameter transformation group ϕ

 t
 (−∞ < t < ∞) over X. For a given positive integer l ≥ 2, we assume that, to each x∈X, there are associated l-positive real continuous functions
 
$$h_{x1} (t), h_{x2} (t), \cdots , h_{xl} (t)$$

 of −∞ < t < ∞. Assume further that these functions possess the following properties, namely, for each of k = 1, 2,⋯, l,
 for each x∈X, each −∞ < s < ∞, and each −∞ < t < ∞. 
Theorem
With X, etc., given above, let μ
be a normal measure of X that is ergodic and invariant under ϕ

 t
 (−∞
< t < ∞). Then, for a certain permutation k→p(k) of k= 1, 2,⋯, l, the set W of points x of X such that all the inequalities (I
 k
 )
 
$$\frac{1}{T}\int_0^T {\frac{{\min \{ h_{xp(1)} (t),h_{xp(2)} (t), \cdots ,h_{xp(k - 1)} (t)\} }}{{\max \{ h_{xp(k)} (t),h_{xp(k + 1)} (t), \cdots ,h_{xp(l)} (t)\} }}dt} > 0,$$

 (II
 k
 )
 
$$\overline {\mathop {\lim }\limits_{T \to - \infty } } \frac{1}{T}\int_0^T {\frac{{\min \{ h_{xp(k)} (t), h_{xp(k + 1)} (t), \cdots , h_{xp(l)} (t)\} }}{{\max \{ h_{xp(1)} (t), h_{xp(2)} (t), \cdots , h_{xp(k - 1)} (t)\} }}dt > 0} $$

 (k=2, 3,⋯, l) hold is invariant under ϕ

 t
 (−∞
< t < ∞) and is μ-measurable with μ-measure1. In practice, the functions h

 xk
 (t) will be taken as length functions of certain tangent vectors of M

 n
 . This theory, established such as in this paper, is expected to be used in the study of structurally stable differential systems on M

 n
 .

Certain Ergodic Properties of a Differential System on a Compact Differentiable Manifold

The study of linear and global properties of linear dynamical systems on vector bundles appeared rather extensive already in the past. Presently we propose to study perturbations of this linear dynamics. The perturbed dynamical system which we shall consider is no longer linear, while the properties to be studied will be still global in general. Moreover, we are intersted in the nonuniformly hyperbolic properties. In this paper, we set an appropriate definition for such perturbations. Though it appears somewhat not quite usual, yet has deeper root in standard systems of differential equations in the theory of differentiable dynamical systems. The general problem is to see which property of the original given by the dynamical system is persistent when a perturbation takes place. The whole content of the paper is devoted to establishing a theorem of this sort.

Notes on a study of vector bundle dynamical systems (ii) part 1

This paper is a continuation of a previous one. We still emphasize the discussion on the relation between the dynamics on the base space of a vector bundle and that on each associated bundle of frames.

Notes on a study of vector bundle dynamical systems (i)

In the part 2, theorem 3.1 studied in part 1[15] is proved first. The proof is obtained via a way of changing variables to reduce the original system of differential equations to a form concerning standard systems of equations in the theory of differentiable dynamical systems. Then by using theorem 3.1 together with the preliminary theorem 2.1, the main theorem of this paper announced in part 1 is proved. The definition of admissible perturbation is contained in the appendix of part 2. The meanings of the main theorem is described in the introduction of part 1.

Liao Shantao

Papers

On Characteristic Exponents Construction of a New Borel Set for the Multiplicative Ergodic Theorem for Vector Fields

Certain Ergodic Properties of a Differential System on a Compact Differentiable Manifold

Notes on a study of vector bundle dynamical systems (ii) part 1

Notes on a study of vector bundle dynamical systems (i)

Notes on a study of vector bundle dynamical systems (ii)─part 2