LIAO STYLE NUMBERS OF DIFFERENTIAL SYSTEMS

For any C1 differential system S on a compact Riemannian manifold M of dimension d with d≥2, this paper studies the Liao style numbers, κ(S) (or respectively, κ*(S)) of S from the view-point of ergodic theory. Here κ(S) (κ*(S)) is the largest number of moving vectors (or respectively, conjugate-) of the differential system S that are mean linearly independent. For any ergodic measure ν of S, two positive integers κ*(ν) and κ(ν), called the reduced and non-reduced style number of ν respectively, are introduced. The connection between the style numbers of the system (M,S) and ones of the ergodic system (M,S; ν) are discovered by the variational principle of style number proved in the paper. Several characterization theorems with respect to the style numbers κ*(S), κ*(ν), κ(S) and κ(ν) are presented respectively.