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

Abstract

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,

• (i*) h _k (x, t) = h _xk (t) is a continuous function of the Cartesian product X×(−∞, ∞).

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

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.