A Sufficient Condition For The Existence Of Uniform Lyapunov Exponent On Matrix Cocycles

In this paper,we consider the existence of uniform Lyapunov exponents on reversible matrix cocycles.This problem is closely related to the problems in fractal geometry and dynamical systems.Professor D.J.Feng of the Chinese University of Hong Kong gave two necessary and sufficient conditions for the existence of uniform Lyapunov exponent of matrix cocycle[1].And ask the question that on what condition the dynaminc system have uniform Lyapunov exponent?In our paper,we consider sy mbolic dynamic systems,we give a sufficient condition for the Lyapnnov exponent on compact reversible matrix cocycle,including locally constant and Holder continous situation.Further more we extend similar results to singular value functions which have the similar sub multiplicative properties as matrix norni,and get more general results.At lastwe use our work to analysis the properties of pressure.To solve the prolbem,we use the tools of basic hyperbolic theorem in dynamic system.We also use tools of outer product of matricx cocycle.We use the result of the relative product bound on hyperbolic automorphism,which studied by Bochi.We also use the result of lower bound estimation on the case of finite upper bound of the sequence of subadditive functions,which has been studied by Morris and the periodic approximation of Lyapunov exponent,which has been studied by Kalinin.In this thesis,we prove that:(1)For a compact reversible inreduc.ible matrix cocycle A.let A(x)be a locally constant function.If all the periodic points have the same maximal Lyapunov exponent,then A have the uniform Lyapunov exponent.(2)For a reversible ineducible matrix cocycle.Let ∑I be symbolic space,let A(x):∑I→gL(d,R)be a Holder continuous function.If the matrix cocycle satisfies the strong fiber bunched condition,and all the periodic points have the same maximal Lyapunov exponent,then the matrix cocycle has the uniform Lyapunov exponent.(3)For a reversible matrix cocycle,and for any constant s>0,let φs be the singular value function,if the cocycle satisfies the strong fiber bunched condition,and η≤θ/d,the matrix cocycle induced by outer product is irreducible,and all the corresponding Lyapunov exponent are same on all periodic points,then the singular value function has the similar estimation.For any constant s>0,there exist a constant C,such that for(?)n ∈N,x ∈ ∑I,the following holds:C-1ρ(A)ns ≤φs(A(x,n))≤Cρ(A)ns,s≤1;C-1ρ(A∧([s]+1)),n(s-[s])ρ(A∧[s])n([s]+1-s)≤φs(A(x,n))≤Cp(A∧([s]+1))n(s-[s])ρ(A∧[s])n([s]+1-s,s≥1.This property is called uniformly bounded singular value propert.In particular,we have C-1ρs(A)n≤ρs(A(x,n))≤Cρs(A)n,s≥0.The ρ(A)represents the joint spectral radius of A.(4)For a typical cocycle,if the matrix cocycle is Holder continous and satisfies the fiber bunched condition,then the cocycle with uniform singular value being the pressure is linear on every interval[k,k+1],0≤k ≤d-1.