A mean ergodic theorem on vector spaces

A mean ergodic theorem for a matrix is first proved from which a mean ergodic theorem for affine operators on a vector space without any topological structure is obtained.     

Applications of the ergodic iteration theorem

I prove several natural preservation theorems for the countable support iteration. This solves a question of Ros?anowski regarding the preservation of localization properties and greatly simplifies the proofs in the area (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Individual ergodic theorem for unitary maps of random matrices

Using simple techniques of finite von Neumann algebras, we prove a limit theorem for random matrices.

     

Multiparameter ratio ergodic theorems for semigroups

After one-parameter treatment of ratio ergodic theorems for semigroups, we formulate the Sucheston a.e. convergence principle of continuous parameter type. This principle plays an effective role in proving some multiparameter generalizations of Chacon?s type continuous ratio ergodic theorems for semigroups and of Jacobs? type continuous random ratio ergodic theorems for quasi-semigroups. In addition, a continuous analogue of the Brunel–Dunford–Schwartz ergodic theorem is given of sectorially restricted averages for a commutative family of semigroups. We also formulate a local a.e. convergence principle of Sucheston?s type. The local convergence principle is effective in proving multiparameter local ergodic theorems. In fact, a multiparameter generalization of Akcoglu–Chacon?s local ratio ergodic theorem for semigroups of positive linear contractions on _L1$L_1$ is proved. Moreover, some multiparameter martingale theorems are obtained as applications of convergence principles.     

A dominated ergodic theorem for some bilinear averages

Let T be a positive invertible linear operator with positive inverse on some Lp(μ), 1?p<∞, where μ is a σ-finite measure. We study the convergence in the Lp(μ)-norm and the almost everywhere convergence of the bilinear operators

     

A Poisson limit theorem for a strongly ergodic non-homogeneous Markov chain

A strongly ergodic non-homogeneous Markov chain is considered in the paper. As an analog of the Poisson limit theorem for a homogeneous Markov chain recurring to small cylindrical sets, a Poisson limit theorem is given for the non-homogeneous Markov chain. Meanwhile, some interesting results about approximation independence and probabilities of small cylindrical sets are given.     

A central limit theorem for the sojourn times of strongly ergodic Markov chains

Let X_n be an irreducible aperiodic recurrent Markov chain with countable state space I and with the mean recurrence times having second moments. There is proved a global central limit theorem for the properly normalized sojourn times. More precisely, if $\text{t}{}^{\mathrm{(n}}\text{)}{}_{\mathrm{i}}\text{=Σ}{}^{\mathrm{n}}{}_{\mathrm{k=1}}\text{i}\text{?}{}_{\mathrm{i}}\text{(X}{}_{\mathrm{k}}\text{)}$, then the probability measures induced by {t⁽ⁿ⁾_i/√n?√nπ_i}_i?I(π_i being the ergotic distribution) on the Hilbert-space of square summable I-sequences converge weakly in this space to a Gaussian measure determined by a certain weak potential operator.     

A non-singular dynamical system without maximal ergodic inequality

It is shown that there is a non-singular dynamical system for which the maximal ergodic inequality does not hold. We further discuss the connection between a non-singular dynamical systems and the pointwise convergence of the Furstenberg ergodic averages.     

Subadditive ergodic theorems for random sets in infinite dimensions

We prove pointwise and mean versions of the subadditive ergodic theorem for superstationary families of compact, convex random subsets of a real Banach space, extending previously known results that were obtained in finite dimensions or with additional hypotheses on the random sets. We also show how the techniques can be used to obtain the strong law of large numbers for pairwise independent random sets, as well as results in the weak topology.     

The spherical ergodic theorem revisited

In this paper, we give a new proof of a result of R. Jones showing almost everywhere convergence of spherical means of actions of R^d${\mathbb{R}}^d$ on L^p(X)$L^p(X)$-spaces are convergent for d?3$d?3$ and p>d/(d-1)$p>d/(d-1)$.     

Semi-uniform sub-additive ergodic theorems for skew-product quasi-flows

The aim of this paper is to extend the semi-uniform ergodic theorem and semi-uniform sub-additive ergodic theorem to skew-product quasi-flows. Furthermore, more strict inequalities about these two theorems are established. By making use of these results, it is feasible to get uniform estimation of the Lyapunov exponent of some special systems even under non-uniform hypotheses     

An abstract ergodic theorem and some inequalities for operators on Banach spaces

We prove an abstract mean ergodic theorem and use it to show that if is a sequence of commuting -dissipative (or normal) operators on a Banach space , then the intersection of their null spaces is orthogonal to the linear span of their ranges. It is also proved that the inequality holds for any -dissipative operator . These results either generalize or improve the corresponding results of Shaw, Mattila, and Crabb and Sinclair, respectively.

     

A conditionally sure ergodic theorem with an application to percolation

We prove an apparently new type of ergodic theorem, and apply it to the site percolation problem on sparse random sublattices of Z^d${\mathbb{Z}}^d$ (d≥2$d\ge 2$), called “lattices with large holes”. We show that for every such lattice the critical probability lies strictly between zero and one, and the number of the infinite clusters is at most two with probability one. Moreover for almost every such lattice, the infinite cluster, if it exists, is unique with probability one.     

A pointwise ergodic theorem for contractions in \mathbb{L}_p (ℋ)-spaces

We consider L _p -spaces of measurable functions ranging in a Hilbert space and single out a class of contractions on such spaces for which the pointwise ergodic theorem holds.     

Nonlinear random ergodic theorems for affine operators

Let (Ω,ß,μ) be a finite measure space and let (S,F,ν) be another probability measure space on which a measure preserving transformation φ is given. We introduce the so-called affine systems and prove a vector-valued nonlinear random ergodic theorem for the random affine system determined by a strongly F-measurable family of affine operators, where B is a reflexive Banach space, is a strongly F-measurable family of linear contractions on L₁(Ω,B) as well as on L_∞(Ω,B) and ξ is a function in (I−T)Lp(S×Ω,B) (1?p<∞) with the operator T defined by Tf(s,ω)=[Tsfφs](ω) which denotes the F⊗ß-measurable version of Tsfφs(ω). Moreover, some variant forms of the nonlinear random ergodic theorem are also obtained with some examples of affine systems for which the nonlinear ergodic theorems fail to hold.     

A martingale ergodic theorem

We prove the martingale ergodic theorem of Kachurovskii which unifies ergodic theorems and theorems on the convergence of martingales, without using the previously required additional integrability condition for the supremum of the process. This condition is replaced by the commutation condition on the conditional expectation and ergodic averaging operators, which for automorphisms is equivalent to the invariance condition on the filtration; meanwhile, the unification remains valid.