## Taiwanese Journal of Mathematics

### 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.

#### Article information

Source
Taiwanese J. Math., Volume 12, Number 5 (2008), 1211-1237.

Dates
First available in Project Euclid: 20 July 2017

https://projecteuclid.org/euclid.twjm/1500574259

Digital Object Identifier
doi:10.11650/twjm/1500574259

Mathematical Reviews number (MathSciNet)
MR2431891

Zentralblatt MATH identifier
1169.37006

#### Citation

Dai, Xiongping; Sun, Wenxiang. LIAOWISE REDUCED REORDERING AND SPECTRUM THEOREMS FOR DIFFERENTIAL SYSTEMS ON COMPACT MANIFOLDS AND APPLICATIONS. Taiwanese J. Math. 12 (2008), no. 5, 1211--1237. doi:10.11650/twjm/1500574259. https://projecteuclid.org/euclid.twjm/1500574259

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