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Hyperbolicity and integral expression of the Lyapunov exponents for linear cocycles
- Source :
-
Journal of Differential Equations . Nov2007, Vol. 242 Issue 1, p121-170. 50p. - Publication Year :
- 2007
-
Abstract
- Abstract: Consider in this paper a linear skew-product system where or , and is a topological dynamical system on a compact metrizable space W, and where satisfies the cocycle condition based on θ and is continuously differentiable in t if . We show that ‘semi λ-exponential dichotomy’ of implies ‘λ-exponential dichotomy.’ Precisely, if Θ has no Lyapunov exponent λ and is almost uniformly λ-contracting along the λ-stable direction and if is constant a.e., then Θ is almost λ-exponentially dichotomous. To prove this, we first use Liao''s spectrum theorem, which gives integral expression of the Lyapunov exponents, and then use the semi-uniform ergodic theorem by Sturman and Stark, which allows one to derive uniform estimates from nonuniform ones. As a consequence, we obtain the open-and-dense hyperbolicity of eventual -cocycles based on a uniquely ergodic endomorphism, and of -cocycles based on a uniquely ergodic equi-continuous endomorphism, respectively. On the other hand, in the sense of -topology we obtain the density of -cocycles having positive Lyapunov exponent based on a minimal subshift satisfying the Boshernitzan condition. [Copyright &y& Elsevier]
- Subjects :
- *DIFFERENTIAL equations
*BESSEL functions
*CALCULUS
*MATHEMATICAL analysis
Subjects
Details
- Language :
- English
- ISSN :
- 00220396
- Volume :
- 242
- Issue :
- 1
- Database :
- Academic Search Index
- Journal :
- Journal of Differential Equations
- Publication Type :
- Academic Journal
- Accession number :
- 26822750
- Full Text :
- https://doi.org/10.1016/j.jde.2007.07.007