Ergodic averages on circles

LetT ₁ andT ₂ be commuting invertible ergodic measure preserving flows on a probability space (X, A, μ). For t = (u,ν) ∈ ℝ², letT ^t =T ₁ ^u T ₂ ^v . LetS ¹ denote the unit circle in ℝ² and σ the rotation invariant unit measure on it. Then, forf∈Lp(X) withp>2, the averagesA _t f(x) = ∫ _s ¹ f(T ^ts x)σ(ds) conver the integral off for a. e.x, ast tends to 0 or infinity. This extends a result of R. Jones [J], who treated the case of three or more dimensions.     

Ergodic averages with vector-valued Besicovitch weights

For a von Neumann algebra ${\mathcal M}$ , we introduce ${\mathcal M}$ -valued Besicovitch sequences and study the norm and individual convergences of the corresponding weighted ergodic averages. The limits of the averages are examined under the condition that the contraction in question is weakly mixing.     

Ergodic Theorems and Ergodic Decomposition for Markov Chains

This paper considers Markov chains on a locally compact separable metricspace, which have an invariant probability measure but with no otherassumption on the transition kernel. Within this context, the limit providedby several ergodic theorems is explicitly identified in terms of the limitof the expected occupation measures. We also extend Yosidasergodic decomposition for Feller-like kernels to arbitrarykernels, and present ergodic results for empirical occupation measures, aswell as for additive-noise systems.     

Galois averages

In this paper, we introduce a notion of “Galois average” which allows us to give a suitable answer to the question: how can one extend a finite Galois extension E/F by a prime degree extension N/E to get a Galois extension N/F? Here, N/E is not necessarily a Kummer extension.     

Ergodic potential

Potential Theory for ergodic Markov chains (with a discrete state spare and a continuous parameter) is developed in terms of the fundamental matrix of a chain.A notion of an ergodic potential for a chain is introduced and a form of Riesz decomposition theorem for measures is proved. Ergodic potentials of charges (with total charge zero) are shown to play the role of Green potentials for transient chains.     

Ergodic decomposition

Ergodic systems, being indecomposable are important part of the study of dynamical systems but if a system is not ergodic, it is natural to ask the following question:

Is it possible to split it into ergodic systems in such a way that the study of the former reduces to the study of latter ones?

Also, it will be interesting to see if the latter ones inherit some properties of the former one. This document answers this question for measurable maps defined on complete separable metric spaces with Borel probability measure, using the Rokhlin Disintegration Theorem.     

关于遍历测度乘积的遍历性

本文利用拟特征标序列收敛的零一律与遍历测度的关系,讨论了遍历测度乘积的遍历性．     

Rademacher averages on noncommutative symmetric spaces

Let E be a separable (or the dual of a separable) symmetric function space, let M be a semifinite von Neumann algebra and let E(M) be the associated noncommutative function space. Let (εk)_k?1 be a Rademacher sequence, on some probability space Ω. For finite sequences (xk)_k?1 of E(M), we consider the Rademacher averages k_∑εk⊗xk as elements of the noncommutative function space and study estimates for their norms ‖k_∑εk⊗xkE_‖ calculated in that space. We establish general Khintchine type inequalities in this context. Then we show that if E is 2-concave, ‖k_∑εk⊗xkE_‖ is equivalent to the infimum of over all yk, zk in E(M) such that xk=yk+zk for any k?1. Dual estimates are given when E is 2-convex and has a nontrivial upper Boyd index. In this case, ‖k_∑εk⊗xkE_‖ is equivalent to . We also study Rademacher averages _∑i,jεi⊗εj⊗xij for doubly indexed families (xij)_i,j of E(M).     

Kahane-Khinchin type averages

We prove a Kahane-Khinchin type result with a few random vectors, which are distributed independently with respect to an arbitrary log-concave probability measure on . This is an application of a small ball estimate and Chernoff's method, that has been recently used in the context of Asymptotic Geometric Analysis.

     

Ergodic Behaviour of Nonconventional Ergodic Averages for Commuting Transformations

Based on T. Tao’s celebrated result on the norm convergence of multiple ergodic averages for commuting transformations, we find that there is a subsequence which converges almost everywhere. Meanwhile, we obtain the ergodic behaviour of diagonal measures, which indicates the time average equals the space average. According to the classification of transformations, we also give several different results. Additionally, on the torus Td with special rotation, we prove the pointwise convergence in Td, and get a result for ergodic behaviour.     

Combinations of multivariate averages

Rate of approximation of combinations of averages on the spheres is shown to be equivalent to K-functionals yielding higher degree of smoothness. Results relating combinations of averages on rims of caps of spheres are also achieved.     

Ergodic averaging sequences

We consider generalizations of the pointwise and mean ergodic theorems to ergodic theorems averaging along different subsequences of the integers or real numbers. The Birkhoff and Von Neumann ergodic theorems give conclusions about convergence of average measurements of systems when the measurements are made at integer times. We consider the case when the measurements are made at timesa(n) or ([a(n)]) where the functiona(x) is taken from a class of functions called a Hardy field, and we also assume that |a(x)| goes to infinity more slowly than some positive power ofx. A special, well-known Hardy field is Hardy’s class of logarithmico-exponential functions. The main theme of the paper is to point out that for a functiona(x) as described above, a complete characterization of the ergodic averaging behavior of the sequence ([a(n)]) is possible in terms of the distance ofa(x) from (certain) polynomials. This research was supported by grants from the NSF.     

拓扑遍历映射的一些性质

本文研究拓扑遍历映射.指出对于由不可约方阵所决定的符号空间有限型子转移而言,或紧致交换群的仿射变换及线段上连续自映射而言,拓扑遍历与拓扑可迁这两个概念是一致的.同时还通过例子,指出拓扑遍历是不同于拓扑可迁与拓扑混合的概念.