Frontiers of Mathematics in China >
Certain Ergodic Properties of a Differential System on a Compact Differentiable Manifold
Published date: 05 Mar 2006
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 Øt(-" < 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 Qxn-2, x"M, 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 £#. For an l-frame α= (u1, u2, ... , ul) of Mn at a point x in M, 1 > l > n-1, each ui being in £#, we shall denote the volume of the parallelotope in the tangent space of Mn at x with edges u1, u2, ... , ul by v(α), and let η*α(t) =v(ψt#(u1),ψt#(u2),...,ψt#(ul)).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, ... , uκ) , ui " £#, of Mn having the property of being positively linearly independent in the mean, but at x, every l-frame = (v1, v2, ... , vl), vi " £#, 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 Øt(-" < 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 φt(-" < 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 hx1(t), hx2(t), ... , hxl(t) of (-" < t < "). Assume further that these functions possess the following properties, namely, for each of k = 1, 2, ... , l, (i*) hk(x, t) = hxk(t) is a continuous function of the Cartesian product X×(-","). (ii*) hφs(x)k(t)=hxk(s+t) hxk(s)
LIAO Shan-tao, LIAO Shan-tao . Certain Ergodic Properties of a Differential System on a Compact Differentiable Manifold[J]. Frontiers of Mathematics in China, 2006 , 1(1) : 1 -52 . DOI: 10.1007/s11464-005-0020-4
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