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

Let 픛r(M), r ≥ 1, denote the space of all Cr vector fields over a compact, smooth and boundaryless Riemannian manifold M of finite dimension; let $\mathcal F_{\ell}^{\sharp}, 1 ≤ ℓ ≤ dim M, be the bundle of orthonormal ℓ-frames of the tangent space TM of M. For any V ∈ 픛r(M), Liao defined functions $\omega_k(\vec{\gamma}_x, V), k = 1, ..., ℓ, on $\mathcal F_{\ell}^{\sharp}$, which are qualitatively equivalent to the Lyapunov exponents of the differential system V. In this paper, the author shows that every $\omega_k(\vec{\gamma}_x, V)$ depends Cr-1-continuously upon $(\vec{\gamma}_x, V)\in\mathcal F_{\ell}^{\sharp}\times\mathfrak X^{r}(M)$ and Cr-continuously on $\vec{\gamma}_x$ 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. [ABSTRACT FROM AUTHOR]