Certain Ergodic Properties of a Differential System on a Compact Differentiable Manifold
 Shantao Liao
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Let M ^{ n } be an ndimensional compact C ^{∞}differentiable manifold, n ≥ 2, and let S be a C ^{1}differential system on M ^{ n }. The system induces a oneparameter C ^{1} transformation group φ_{ t }(−∞ < t < ∞) over M ^{ n } and, thus, naturally induces a oneparameter 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 oneparameter transformation group ψ_{ t } ^{ # }(−∞<t<∞) of \({{\mathcal{L}}}^{\sharp }\). For an lframe 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
A point x of M is said to possess positive generic index κ = κ _{+} *(x) if, at x, there is a κframe \(\alpha = {\left( {\mathbf{u}_{1},\;\mathbf{u}_{2},\; \cdots ,\;\mathbf{u}_{\kappa } } \right)}\), \(\mathbf{u}_{i} \in {{\mathcal{L}}}^{\sharp }\), of M ^{ n } having the property of being positively linearly independent in the mean, but at x, every lframe \(\beta = {\left( {\mathbf{v}_{1} ,\mathbf{v}_{2} , \cdots ,\mathbf{v}_{l} } \right)},\mathbf{v}_{i} \in {{\mathcal{L}}}^{\sharp }\), 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
Theorem κ _{ + } *(F)=κ _{ − } *(F).
(B) We consider a nonempty compact metric space x and a oneparameter transformation group ϕ _{ t }(−∞ < t < ∞) over X. For a given positive integer l ≥ 2, we assume that, to each x∈X, there are associated lpositive real continuous functions
 (i*)
h _{ k }(x, t) = h _{ xk }(t) is a continuous function of the Cartesian product X×(−∞, ∞).
 (ii*)
\(h_{{\varphi {}_{s} }} {}_{{{\left( x \right)}k}} {\left( t \right)} = \frac{{h_{{xk}} {\left( {s + t} \right)}}} {{h_{{xk}} {\left( s \right)}}}\)
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 })
(II_{ k })
(k=2, 3,⋯, l) hold is invariant under ϕ _{ t }(−∞ < t < ∞) and is μmeasurable with μmeasure1.
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 }.
 Title
 Certain Ergodic Properties of a Differential System on a Compact Differentiable Manifold
 Journal

Frontiers of Mathematics in China
Volume 1, Issue 1 , pp 152
 Cover Date
 200601
 DOI
 10.1007/s1146400500204
 Print ISSN
 16733452
 Online ISSN
 16733576
 Publisher
 Higher Education Press
 Additional Links
 Topics
 Keywords

 ergodic theory
 style number
 differential dynamical system
 37B38
 37D99
 Authors

 Shantao Liao ^{(1)}
 Author Affiliations

 1. Department of Mathematics, Peking University, Beijing, 100871, China