The Approximation Of Lyapunov Exponents And Its Applications

In this paper,we study the approximation of Lyapunov exponents,including approximated by the Lyapunov exponents at periodic points and by the Lyapunov exponents at the invariant measures supported on horseshoes.We also study their applications on a non-uniformly hyperbolic version of the Livsic theorem and on the continuity of spectral radius.First of all,we investigates the periodic approximation of Lyapunov exponents for matrices cocycles.We consider the completely non-invertible cocycles,that is,both the dynamical system f:X? X and the cocycle A are non-invertible.It's proved in this paper that if the dynamical system f has enough hyperbolicity,then the Lyapunov exponents of an ergodic measure can be approximated by the exponents at periodic points.Secondly,we give an application of our first result:the Livsic theorem for matrices cocycles over non-uniformly hyperbolic systems.More precisely,let f:M ? M be a non-uniformly hyperbolic system,preserving a hyperbolic measure ?:If the cocycle A:M ?Md(R)satisfies A(fn-1p)…A(fp)A(p)= Id,(?)p?M,n ? 1,? fnp=p.then the cohomological equation A = C o f · C-1 has a measurable solution,that is,there exits a measurable map C:M ? GL(d,R)such that A(x)= C(fx)C(x)-1 for?-a.e.x ? M.It's rather remarkable that ? is not assumed to be ergodic.We also apply the periodic approximation to the continuity of joint spectral ra-dius and generalized spectral radius.That is,is the system has enough hyperbolic-ity(satisfying the closing property),then the joint spectral radius function is continuous on the space of Holder continuous cocycles.To prove the continuity of the generalized spectral radius,we generalize the Berger-Wang formula to our dynamical setting,i.e.the generalized spectral radius equals the joint spectral radius.We note that the clas-sical Berger-Wang formula considers the cocycle driven by full-shift system,while we consider a more general dynamical system,and in the definition of the joint spectral radius,the norm of matrices is replaced by the singular value function of matrices.For Banach cocycles,it is studied in this paper the periodic approximation of Lyapunov exponents for quasi-compact cocycles.For a more general Banach cocycle,Kalinin and Sadovskaya gave a counter-example of the periodic approximation in[1].At last,considering a non-uniformly hyperbolic system with positive entropy,and a quasi-compact Banach cocycle,we show that there exist horseshoes such that the Lyapunov exponents of an ergodic hyperbolic measure can be approximated by the ergodic measures supported on the horseshoes.More precisely,we prove the existence of the horseshoes whose topological entropy close to the given measure-theoretic entropy,meanwhile,their exist a dominated splitting for cocycles on the horseshoe such that the growing rates of vectors in the subspaces are close to the corresponding Lyapunov exponents of the given hyperbolic measure.