Ergodic Theorem And The Existence Problem Of Invariant Measure In A Number Of Issues

Ergodic theory is large and rapidly developing mathematical subject, having its origins in the analysis of statistical properties of Hamiltonian dynamics. Its aim is to study the long-term average behavior of systems. Ergodic theory has important and profound applications in many mathematical branches such as probability theory and stochastic processes, mathematical physics, functional analysis, number theory and etc[21 ][41].Ergodic theorems and the existence of invariant measures are two major topics in Ergodic theory. The general study on Ergodic theorems originated in works on the proof of the results , which are known as the Mean Ergodic Theorem and the Pointwise(Individual) Ergodic Theorem, respectively, done by Neumann and Birkhoff both in 1931. Generalizations and improvements of their results have been appearing continually since 1932. In particular Bourgain[10][11][12] blend number theory, Fourier analysis and ergodic theory to prove that the sequence of squares and the sequence of the n-th prime are both Lp(p > 1) - good, which is one of great achievement in ergodic theory since Birhoff proved that the natural numbers is pointwise ergodic in 1931. In the area, we refer the reader to Rosenblatt and Wierdl's excellent survey articles[79] and the reference therein.The study of the problems on the existence of the invariant measures appeared in the Kryloff and Bogoliouboffs works in [67] on compact topological dynamics as early as 1937. The study of Ergodic theorems in particular the point ergodic theorems is developed on theassumption that there exists invariant measures in the most cases. It playa also an important role in the study of Markov processes theory that there exists invariant measures. For the studying works on the problems of the existence of the invariant measures , see Foguel 's [36] and Lasota and Mackey 's[70] respectively.This paper is to discuss several problems of the ergodic theorems and the existence of the invariant measures in the ergodic theory. It is divided into two parts. In the first part we discuss the some problems in ergodic theorems, which contain two chapters. In Chapter I, we study subsequential ergodic theorems with stochastic weights; In Chapter 2, we study ergodic theorems and ergodic decomposition of uniform Lipschitz mappings. In the second parts, which contain two chapters, we study the problem of the existence of invariant measures for continuous linear operators. In Chapter 3, we study the existence and ergodicity of invariant Gaussian measure for continuous linear operators. In Chapter 4, we study the existence of invariant measures for weighted composition operators.Chapter 1 Let (X, ) be a probability space, Lp = Lp(X, , ) (1 < p < ) , T: Lp Lp. If a continuous linear operator. Let a = be a sequence of complex number a strictly increasing sequence of nonnegative integers. Consider the convergence in the almost everywhere or in the norm of weighted average for T with a weight a = (an }r/ /etf (0.1.1)and subsequential average for T along {kn}-r TV /€//> (0.1.2)^* n=lSuch weighted and subsequential ergodic theorems have been examined by many authorsxuincluding Baxte and Olsen[4], Below and Losert[6], and Bougain[8][9][10][ll][12]. In particular Furstenberg proved in [24] that k= {?} which have zero density is good for mean L2 in the applications of van der Corput' lemma for vectors in Hilbert space formulated by Bergelson, i.e.Theorem A.1.3[40. Theorem 2.2.] Let (X, ) be a measure preserving(dynamical)system, thenconverge in L2-norm; where T is isometric operator induced by measures preserving transformation r on L2 , Tf(x) = f().Bougain proved in [10] that subsequence k= {n2} is L2 - good ;i.e for every operator T on Z2 induced by preserving measures transformations , the limit of (0.1.3) exists almost everyewhere. Applying Bougain's method, Rosenblatt and Wield[79] also obtain Theorem E. 1.1 .In addition, Bougain dis...