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 Ø_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 £^# 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 £^#. For an l-frame α= (u₁, u₂, ... , u_l) of Mⁿ at a point x in M, 1 > l > n-1, each u_i being in £^#, we shall denote the volume of the parallelotope in the tangent space of Mⁿ at x with edges u₁, u₂, ... , u_l by v(α), and let η^_*α(t) =v(ψ_t^#(u₁),ψ_t^#(u₂),...,ψ_t^#(u_l)).This is a continuous real function of t. Let 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 = (u₁, u₂, ... , u_κ) , u_i " £^#, of Mⁿ having the property of being positively linearly independent in the mean, but at x, every l-frame = (v₁, v₂, ... , v_l), v_i " £^#, 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 Ø_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 one-parameter transformation group φ_t(-" < t < ") over X. For a given positive integer l "e 2, we assume that, to each x2X, there are associated l-positive real continuous functions h_x1(t), h_x2(t), ... , 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_φs(x)k(t)=

h_xk(s+t)

h_xk(s)