Sacker-Sell Spectrum And Lyapunov Spectrum For Random Dynamical Systems

Random dynamical system (RDS) is one kind of skew-product system. Because of more effectively describing the real world, it obtains more and more attentions. This thesis is devoted to the theories of exponential dichotomy, Sacker-Sell spectrum, Lyapunov exponents and random attractors. Exponential dichotomy describes a kind of hyperbolicity of the system. That is, the state space can be decomposed into a direct sum of two subspaces, which are continuous and invariant. And with the time evolving forward, the system is exponentially attracting in one subspace and exponentially expanding in the other subspace. Sacker-Sell spectrum is a definition in terms of exponential dichotomy. Sacker and Sell built the theory of Sacker-Sell spectrum. The theory has been developed into infinite dimensional dynamical system by many scholars, such as Magalh(a|~)es, Sacker and Sell, Chicone and Latushkin, Chow and Leiva. Also, Cong and Siegmund discussed some kind of dynamical systems in some random sense. Lyapunov exponents, one of the fundamental tools studying the asymptotic behaviors of dynamical system, reflects the average variance rate of the system with the development of time. Oseledec's Multiplicative Ergodic Theorem (MET) does not only solve the existence of Lyapunov exponents, but also show much more information of the dynamical structure. And henceforth, MET has been one of the most fundamental theorems in the theory of dynamical systems. MET has been developed by many people such as Ruelle , Ma(n|~)é, Thieullen , Zeng Lian and Kening Lu and so on. For finite dimensional dynamical system, Johnson, Palmer and Sell discussed the relations between the Sacker-Sell spectrum and the Lyapunov spectrum. They proved in particular that the Lyapunov spectrum is a subset of the Sacker-Sell spectrum, while the boudary points of the Sacker-Sell spectrum correspond to some Lyapunov exponents. They also proved that the Oseledec subbundles are the refined version of the Sacker-Sell subbundles. Attractor is one very important definition in the theory of differential equations and dynamical systems. In this thesis, we are interested in the following problem: when is a compact invariant subset a attractor? In the determined system, Ashwin, using normal Lyapunov exponents, discussed this problem. And then, he studied again the problem for one concrete example in the framework of random dynamical system.This thesis is mainly concerned with the theory of Sacker-Sell spectrum, the mul- tiplicative ergodic theorems and random attractors for RDS. Comparing to the deterministic case, the base space of RDS is one probability space without any topology, which is one of the essential difficulties from the deterministic case to random case. We conquer the difficulty in this paper. In terms of a new exponential dichotomy for the finite dimensional RDS, we define the Sacker-Sell spectrum for RDS and show Sacker-Sell Spectrum Decomposition Theorem (SDT). Basing the SDT, we study the relations between Sacker-Sell spectrum and Lyapunov spectrum and establish the Spectrum Relationship Theorem in the framework of finite dimensional RDS. The relations between the two spectrum for infinite dimensional RDS are also obtained in this paper. In infinite dimensional RDS, different to finite case, the cocycle can be usually considered only in the positive half-time line. With this in mind, we define a backward continuation for the semi-dynamical system such that it can be extended to the whole time line. Besides this, there is another difficulty in the infinite dimensional case that a bounded and closed subset in the infinite dimensional state space is not necessarily compact. Having in mind this, we are concerned with the RDS with random uniformly completely continuous and random uniformlyα- contraction, respectively. In addition, MET for a very general infinite dimensional RDS is obtained in this paper. We take the ideas from Mane and Thieullen. In the final part, we discuss two problems in the theory of nonuniform hyperbolicity: the relation between random nonuniformly exponential dichotomy and random uniformly exponential dichotomy, the relation between a random compact and invariant subset and a random attractor. Under some conditions, we prove that random nonuniformly exponential dichotomy actually imply random uniformly exponential dichotomy. And, using the normal Lyapunov exponents, we also obtain a sufficient condition under which a random compact and invariant subset is a random forward attractor. This is a generalization of the results of Ashwin's.Professor Yongluo Cao's two theorems, one concerning random continuous functions with sub-additivity and the other concerning nonuniform hyperbolicity, play an important role in this thesis. In the first theorem, Cao proved that the maximal growth rate of one random continuous function with sub-additivity can be achieved by some ergodic measure. In the second theorem, Cao, Luzzatto and Rios proved that a priori very weak nonuniform hyperbolicity conditions (caused by Lyapunov exponents) actually imply uniform hyperbolicity.