遍历序列在动力系统平均方法定量研究中的应用

主要研究了遍历序列及其应用.在遍历函数的基础上,引入了遍历序列的概念.然后,从两个方面来研究离散情况下的遍历问题,从中得到了一些遍历序列的性质,并利用平均方法得到了离散情况下的动力系统的解的定量性质.     

一类非齐次马氏链的绝对平均强遍历性

引用马氏链绝对平均强遍历的概念,首先给出齐次马氏链绝对平均强遍历与强遍历的等价性,其次通过引进另一个强遍历的非齐次马氏链,给出一个非齐次马氏链绝对平均强遍历的充分条件.     

Operator Ergodic Theorems for Actions of Free Semigroups and Groups

New ergodic theorems are obtained for measure-preserving actions of free semigroups and groups. These theorems are derived from ergodic theorems for Markov operators. This approach also allows one to obtain ergodic theorems for some classes of Markov semigroups. Results of the paper generalize classical ergodic theorems of Kakutani, Oseledets, and Guivarc'h, and recent ergodic theorems of Grigorchuk, Nevo, and Nevo and Stein.     

二重非齐次马氏链的遍历性及其应用

给出二重非齐次马氏链的强遍历性,绝对平均强遍历性,Cesaro平均收敛的概念．利用二维马氏链的遍历性和C-K方程,建立了二维马氏链与二重非齐次马氏链遍历性的关系．并讨论了齐次二重马氏链绝对平均强遍历与强遍历的等价性．最后给出Cesaro平均收敛在马氏决策过程和信息论中应用．     

连续时间非时齐马氏过程的广义Dobrushin系数的估计

本文研究了非时齐马氏过程的广义Dobrushin系数的估计问题.在将经典Dobrushin遍历系数推广为加权的遍历系数的基础上,利用了矩阵拆分的方法,得到了对这种广义遍历系数的估计方法,推广了时齐马氏过程关于遍历系数的估计结果,借此可进一步得到有关遍历性的判定结论.     

Ergodic isospectral theory of the Lax pairs of Euler equations with harmonic analysis flavor

Isospectral theory of the Lax pairs of both 3D and 2D Euler equations of inviscid fluids is developed. Eigenfunctions are represented through an ergodic integral. The Koopman group and mean ergodic theorem are utilized. Further harmonic analysis results on the ergodic integral are introduced. The ergodic integral is a limit of the oscillatory integral of the first kind.

     

Mixing Limit Theorems for Ergodic Transformations

We show that distributional and weak functional limit theorems for ergodic processes often hold for arbitrary absolutely continuous initial distributions. This principle is illustrated in the setup of ergodic sums, renewal-theoretic variables, and hitting times for ergodic measure preserving transformations.     

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In this paper, we introduce the definitions of geometric strongly ergodic, strongly ergodic and weakly ergodic for continuous-state Markov chains, then we give a primary proof of equivalence of the ergodicities for continuous-state Markov chains.     

Ergodic theorems for noncommutative dynamical systems

Recently, E.C. Lance extended the pointwise ergodic theorem to actions of the group of integers on von Neumann algebras. Our purpose is to extend other pointwise ergodic theorems to von Neumann algebra context: the Dunford-Schwartz-Zygmund pointwise ergodic theorem, the pointwise ergodic theorem for connected amenable locally compact groups, the Wiener's local ergodic theorem for ₊ ^d and for general Lie groups.     

Ergodic BSDEs and related PDEs with Neumann boundary conditions

We study a new class of ergodic backward stochastic differential equations (EBSDEs for short) which is linked with semi-linear Neumann type boundary value problems related to ergodic phenomena. The particularity of these problems is that the ergodic constant appears in Neumann boundary conditions. We study the existence and uniqueness of solutions to EBSDEs and the link with partial differential equations. Then we apply these results to optimal ergodic control problems.     

Maximal ergodic theorems for some group actions

We prove maximal ergodic inequalities for a sequence of operators and for their averages in the noncommutative Lp-space. We also obtain the corresponding individual ergodic theorems. Applying these results to actions of a free group on a von Neumann algebra, we get noncommutative analogues of maximal ergodic inequalities and pointwise ergodic theorems of Nevo-Stein.     

Computability of the ergodic decomposition

The study of ergodic theorems from the viewpoint of computable analysis is a rich field of investigation. Interactions between algorithmic randomness, computability theory and ergodic theory have recently been examined by several authors. It has been observed that ergodic measures have better computability properties than non-ergodic ones. In a previous paper we studied the extent to which non-ergodic measures inherit the computability properties of ergodic ones, and introduced the notion of an effectively decomposable measure. We asked the following question: if the ergodic decomposition of a stationary measure is finite, is this decomposition effective? In this paper we answer the question in the negative.     

有限域上遍历矩阵的特性研究

对有限域上遍历矩阵的性质进行了分析,给出了有限域上遍历矩阵的计数定理,并对遍历矩阵序对(A,B)关于矩阵M的双侧幂乘集〈A〉M〈B〉的秩及基数进行了全面分析.给出了R_k(A,B)集的构成及其基数的有关定理,所得到的结论对利用遍历矩阵实现有关的公钥密码具有理论上的指导意义.     

Invariant measures and asymptotics for some skew products

For certain group extensions of uniquely ergodic transformations, we identify all locally finite, ergodic, invariant measures. These are Maharam type measures. We also establish the asymptotic behaviour for these group extensions proving logarithmic ergodic theorems, and bounded rational ergodicity.     

拓扑遍历与拓扑双重遍历

令X为紧致度量空间,f:X→X为连续映射,U,V为X的任意非空开集,若{n>0|fn(U)∩V≠ )为正上密度集,则称f拓扑遍历.f拓扑双重遍历意味着f×f拓扑遍历.本文在[2]的基础上进一步讨论拓扑遍历与拓扑双重遍历映射的性质.     

Realisation of measured dynamics as uniquely ergodic minimal homeomorphisms on manifolds

We prove that the family of measured dynamical systems which can be realised as uniquely ergodic minimal homeomorphisms on a given manifold (of dimension at least two) is stable under measured extension. As a corollary, any ergodic system with an irrational eigenvalue is isomorphic to a uniquely ergodic minimal homeomorphism on the two-torus. The proof uses the following improvement of Weiss relative version of Jewett–Krieger theorem: any extension between two ergodic systems is isomorphic to a skew-product on Cantor sets.     

Isometries of ergodic Hardy spaces

The isometries between ergodic Hardy spaces are identified. As a consequence it is shown that the isometry class of an ergodic Hardy space associated with an ergodic, measure preserving flow constitutes an essentially complete set of conjugacy invariants for the flow. This research was supported in part by a grant from the National Science Foundation.     

用耦合方法研究马氏链f-指数遍历

该文在一般状态空间下研究马氏链指数遍历性,指数遍历马氏链,增加条件π(f~p)<∞, p> 1,利用耦合方法得到了存在满的吸收集,使得马氏链在其上是f-指数遍历的.     

Random walks on compact groups and the existence of cocycles

We show that certain skew products in ergodic theory are isomorphic to the shifts defined by random walks. We conclude the existence of cocycles for any finite measure preserving ergodic automorphism or flow, taking values in an arbitrary compact group, which determine ergodic skew products.     

The structure of ergodic measures for compact group extensions

In this paper, the structure of the set of invariant measures on a transformation group which is a free compact abelian group extension of another transformation group is studied from both the geometric and analytic viewpoints. It is shown in general that genuine ergodic decompositions are obtained in the non-metric setting for measures that project onto an ergodic measure. In addition, when all the spaces involved are metric, there is a structure theorem for all ergodic measures in terms of the ergodic measures on the base and naturally defined subgroups.