Entropy Formula For Infnite Dimensional Random Dynamical Systems

Measure-theoretic entropy measures the complexity of a dynamical system in a globalway, while Lyapunov exponents give a local and geometrical description of the sensitivedependence on initial conditions. These quantities are related closely to each other bythe famous Ruelle inequality and Pensin's entropy formula.The signifcance of Pesin'sentropy formula (or Ledrappier-Young's entropy formula for SRB measures) classicallylies in its characterizing SRB measures by their Lyapunov exponents and entropy.Precisely, let f be a C2(or C1+α) diffeomorphism of a (fnite dimensional) com-pact Riemannian manifold M preserving a probability measureμ. According to Os-eledec Theorem [27], there exist Lyapunov exponentsλ1(x) >···>λr(x)(x) withmultiplicities m1(x),···, mr(x)(x) atμ-a.e. x∈M . Denote by hμ(f ) the measure-theoretic entropy ofμwith respect to f . In general the so called Ruelle inequalityholdswhich only provides us partial information of the measure-theoretic entropy.This relation was frst noted by Ruelle [32] for C1diffeomorphism , Pesin[28] further showed that when the above diffeomorphism is C1+αand measureμissmooth i.e.μis absolutely continuous with respect to the Lebesgue measure of M ),the above inequality turns into an equality(i.e. Pesin's entropy formula)Later, Ledrappier, Strelcyn and Young [13, 14, 16] showed that entropy formulaholds if and only ifμis SRB in the sense that it has absolutely continuous conditionalmeasures on unstable manifolds.In this thesis, we extent the above theories to infnit dimensional random dynami-cal systems. We frst introduce in Chapter One some notation,related backgrounds andmain results about our work. Then Ruelle inequality for Banach system is given in Chapter Two , and fnally in Chapter Three we prove Pensin's formula for general forHilbert system.