LIAOWISE REDUCED REORDERING AND SPECTRUM THEOREMS FOR DIFFERENTIAL SYSTEMS ON COMPACT MANIFOLDS AND APPLICATIONS

Abstract

For any differential system $\vec{V}$ of class $C^1$ on an $n$-dimensional compact, smooth, and boundaryless riemannian manifold $M$, we consider the Liao frame skew-product flow on the reduced orthonormal frame bundle ${\mathcal C}^{\sharp}(M,\vec{V})$ naturally induced by $\vec{V}$ and, using some technical ideas due to S.~Liao, we prove a `reduced reordering' theorem and a `reduced spectrum' theorem. As consequences, we also provide a reordering lemma for the natural skew-product flow $({\mathcal F}(k),\{\mathcal V_t\})$ on the flag bundles ${\mathcal F}(k)$ of the tangent bundle ${TM}$, and give two characteristic spectra for parallelepiped. In addition, we obtain the uniformity of some non-uniformly expanding (resp.~contracting) sets.