Taiwanese Journal of Mathematics

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

Xiongping Dai and Wenxiang Sun

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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

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500574259

Digital Object Identifier
doi:10.11650/twjm/1500574259

Mathematical Reviews number (MathSciNet)
MR2431891

Zentralblatt MATH identifier
1169.37006

Subjects
Primary: 37C10: Vector fields, flows, ordinary differential equations 37H15: Multiplicative ergodic theory, Lyapunov exponents [See also 34D08, 37Axx, 37Cxx, 37Dxx] 37C40: Smooth ergodic theory, invariant measures [See also 37Dxx]

Keywords
Liao theory Liao function $\omega_k^{\ast}$ Lyapunov exponent Liao reordering and spectrum theorem non-uniformly expanding and contracting

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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