TAILIEUCHUNG - Đề tài " Almost all cocycles over any hyperbolic system have nonvanishing Lyapunov exponents "

We prove that for any s 0 the majority of C s linear cocycles over any hyperbolic (uniformly or not) ergodic transformation exhibit some nonzero Lyapunov exponent: this is true for an open dense subset of cocycles and, actually, vanishing Lyapunov exponents correspond to codimension-∞. This open dense subset is described in terms of a geometric condition involving the behavior of the cocycle over certain heteroclinic orbits of the transformation. | Annals of Mathematics Almost all cocycles over any hyperbolic system have nonvanishing Lyapunov exponents By Marcelo Viana Annals of Mathematics 167 2008 643 680 Almost all cocycles over any hyperbolic system have nonvanishing Lyapunov exponents By Marcelo Viana Abstract We prove that for any s 0 the majority of Cs linear cocycles over any hyperbolic uniformly or not ergodic transformation exhibit some nonzero Lyapunov exponent this is true for an open dense subset of cocycles and actually vanishing Lyapunov exponents correspond to codimension-1. This open dense subset is described in terms of a geometric condition involving the behavior of the cocycle over certain heteroclinic orbits of the transformation. 1. Introduction In its simplest form a linear cocycle consists of a dynamical system f M M together with a matrix valued function A M SL d C one considers the associated morphism F x v f x A x v on the trivial vector bundle M X Cd. More generally a linear cocycle is just a vector bundle morphism over the dynamical system. Linear cocycles arise in many domains of mathematics and its applications from dynamics or foliation theory to spectral theory or mathematical economics. One important special case is when f is differentiable and the cocycle corresponds to its derivative we call this a derivative cocycle. Here the main object of interest is the asymptotic behavior of the products of A along the orbits of the transformation f An x A fn 1 x A f x A x especially the exponential growth rate largest Lyapunov exponent A A x lim log An x . nn n Research carried out while visiting the College de France the Université de Paris-Sud Orsay and the Institut de Mathématiques de Jussieu. The author is partially supported by CNPq Faperj and PRONEX. 644 MARCELO VIANA The limit exists -almost everywhere relative to any f-invariant probability measure ụ. on M for which the function log II Ak is integrable as a consequence of the subadditive ergodic theorem of Kingman 21 . We .

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