کد مقاله کد نشریه سال انتشار مقاله انگلیسی نسخه تمام متن
4612157 1338663 2007 50 صفحه PDF دانلود رایگان
عنوان انگلیسی مقاله ISI
Hyperbolicity and integral expression of the Lyapunov exponents for linear cocycles
کلمات کلیدی
موضوعات مرتبط
مهندسی و علوم پایه ریاضیات آنالیز ریاضی
پیش نمایش صفحه اول مقاله
Hyperbolicity and integral expression of the Lyapunov exponents for linear cocycles
چکیده انگلیسی

Consider in this paper a linear skew-product system(θ,Θ):T×W×Rn→W×Rn;(t,w,x)↦(t⋅w,Θ(t,w)⋅x) where T=RT=R or ZZ, and θ:(t,w)↦t⋅w is a topological dynamical system on a compact metrizable space W  , and where Θ(t,w)∈GL(n,R)Θ(t,w)∈GL(n,R) satisfies the cocycle condition based on θ and is continuously differentiable in t   if T=RT=R. We show that ‘semi λ  -exponential dichotomy’ of (θ,Θ)(θ,Θ) implies ‘λ-exponential dichotomy.’ Precisely, if Θ has no Lyapunov exponent λ and is almost uniformly λ-contracting along the λ  -stable direction Es(w;λ)Es(w;λ) and if dimEs(w;λ)dimEs(w;λ) 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 GL+(2,R)GL+(2,R)-cocycles based on a uniquely ergodic endomorphism, and of GL(2,R)GL(2,R)-cocycles based on a uniquely ergodic equi-continuous endomorphism, respectively.On the other hand, in the sense of C0C0-topology we obtain the density of SL(2,R)SL(2,R)-cocycles having positive Lyapunov exponent based on a minimal subshift satisfying the Boshernitzan condition.

ناشر
Database: Elsevier - ScienceDirect (ساینس دایرکت)
Journal: Journal of Differential Equations - Volume 242, Issue 1, 1 November 2007, Pages 121–170
نویسندگان
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