ON THE CONTINUITY OF LIAO QUALITATIVE FUNCTIONS OF DIFFERENTIAL SYSTEMS AND APPLICATIONS

Abstract

Let 𝔛r(M), r ≥ 1, denote the space of all Cr vector fields over a compact, smooth and boundaryless Riemannian manifold M of finite dimension; let [Formula: see text], 1 ≤ ℓ ≤ dim M, be the bundle of orthonormal ℓ-frames of the tangent space TM of M. For any V ∈ 𝔛r(M), Liao defined functions [Formula: see text], k = 1, …, ℓ, on [Formula: see text], which are qualitatively equivalent to the Lyapunov exponents of the differential system V. In this paper, the author shows that every [Formula: see text] depends Cr-1-continuously upon [Formula: see text] and Cr-continuously on [Formula: see text] for any given V. In addition, applying the qualitative functions, the author generalizes Liao's global linearization along a given orbit of V and considers the stochastic stability of Lyapunov spectra of linear skew-product flows based on a given ergodic system.