Frontiers of Mathematics in China

, Volume 1, Issue 1, pp 1–52

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

  • Liao Shan-tao 
Research Article

DOI: 10.1007/s11464-005-0020-4

Cite this article as:
Liao, S. Front. Math. China (2006) 1: 1. doi:10.1007/s11464-005-0020-4
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Abstract

Let Mn be an n-dimensional compact C-differentiable manifold, n ≥ 2, and let S be a C1-differential system on Mn. The system induces a one-parameter C1 transformation group φt(−∞ < t < ∞) over Mn and, thus, naturally induces a one-parameter transformation group of the tangent bundle of Mn. 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 Mn of the differential system S. With respect to a given C Riemannian metric of Mn, we consider the bundle \({{\mathcal{L}}}^{\sharp }\) of all (n−2) spheres Qxn−2, xM, where Qxn−2 for each x consists of all unit tangent vectors of Mn orthogonal to the trajectory through x. Then, the differential system S gives rise naturally to a one-parameter transformation group ψt#(−∞<t<∞) of \({{\mathcal{L}}}^{\sharp }\). For an l-frame α = (u1, u2,⋯, ul) of Mn at a point x in M, 1 ≥ ln−1, each ui being in \({{\mathcal{L}}}^{\sharp }\), we shall denote the volume of the parallelotope in the tangent space of Mn at x with edges u1, u2,⋯, ul 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 Mn 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 Mn 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 Mn consisting of regular points of S, invariant under φt(−∞ < t < ∞), let the (positive and negative) generic indices of F be defined by
$$ \kappa ^{*}_{ + } {\left( F \right)} = {\mathop {\max }\limits_{x \in F} }{\left\{ {\kappa ^{*}_{ + } {\left( x \right)}} \right\}},\quad \kappa ^{*}_{ - } {\left( F \right)} = {\mathop {\max }\limits_{x \in F} }{\left\{ {\kappa ^{*}_{ - } {\left( x \right)}} \right\}}. $$

Theoremκ+*(F)=κ*(F).

(B) We consider a nonempty compact metric space x and a one-parameter transformation group ϕt(−∞ < t < ∞) over X. For a given positive integer l ≥ 2, we assume that, to each xX, 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,
  1. (i*)

    hk(x, t) = hxk(t) is a continuous function of the Cartesian product X×(−∞, ∞).

     
  2. (ii*)

    \(h_{{\varphi {}_{s} }} {}_{{{\left( x \right)}k}} {\left( t \right)} = \frac{{h_{{xk}} {\left( {s + t} \right)}}} {{h_{{xk}} {\left( s \right)}}}\)

     
for each xX, each −∞ < s < ∞, and each −∞ < t < ∞.

TheoremWith X, etc., given above, let μbe a normal measure of X that is ergodic and invariant under ϕt(−< 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

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

(k=2, 3,, l) hold is invariant under ϕt(−< t < ∞) and is μ-measurable with μ-measure1.

In practice, the functions hxk(t) will be taken as length functions of certain tangent vectors of Mn. This theory, established such as in this paper, is expected to be used in the study of structurally stable differential systems on Mn.

Keywords

ergodic theory style number differential dynamical system 

AMS (2004) Subject Classification

37B38 37D99 

Copyright information

© Higher Education Press and Springer-Verlag 2006

Authors and Affiliations

  • Liao Shan-tao 
    • 1
  1. 1.Department of MathematicsPeking UniversityBeijingChina

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