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

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 φ_t(−∞ < 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 Q_x ⁿ⁻², x∈M, where Q_x ⁿ⁻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 ψ_t ^#(−∞<t<∞) of ${{\mathcal{L}}}^{\sharp }$. For an l-frame α = (u ₁, u ₂,⋯, u _l) of Mⁿ 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 Mⁿ 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 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 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\}}. $

Theoremκ₊*(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,

1.	hk(x, t) = hxk(t) is a continuous function of the Cartesian product X×(−∞, ∞).
2.	$h_{{\varphi {}_{s} }} {}_{{{\left( x \right)}k}} {\left( t \right)} = \frac{{h_{{xk}} {\left( {s + t} \right)}}} {{h_{{xk}} {\left( s \right)}}}$

for each x∈X, each −∞ < s < ∞, and each −∞ < t < ∞.

TheoremWith X, etc., given above, let μbe 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 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 / 37B38 / 37D99

References

[1] Peixoto M. Structural stability on two-dimensional manifolds. Topology, 1962, 1: 101-120.

[2] Nemytskii B. B., Stepanov B. B. Qualitative Theory of Differential Equations, 1960, New Jersey: Princeton University Press.

[3] Steenrod N. The Topology of Fibre Bundles, 1951, Princeton: Princeton University Press.