Pointwise ergodic theorems for the symmetric exclusion process

Summary Almost sure convergence theorems are proved for Cesaro averages of continous functions in the case of the symmetric exclsion processes in dimension d≧3. For the occupation time of a single site the same result is proved in all dimensions. Partially supported by CNPq     

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.     

On the pointwise ergodic theorem onL p for arithmetic sets

The purpose of this note is to show how the results of [B] on the pointwise ergodic theorem forL ²-functions may be extended toL ^p for certainp<2. More precisely, we give a proof of the almost sure convergence of the means (t≧1) given a dynamical system (Ω,B, μ, T) andf of classL ^p(Ω,μ),p>(√5+1)/2.     

Pointwise theorems for amenable groups

In this paper we prove the pointwise ergodic theorem for general locally compact amenable groups along F?lner sequences that obey some restrictions. These restrictions are mild enough so that such sequences exist for all amenable groups. We also prove a generalization of the Shannon-McMillan-Breiman theorem to all discrete amenable groups. --> Oblatum 10-I-2000 & 9-V-2001?Published online: 13 August 2001     

Multiple ergodic theorems

Given measure preserving transformationsT ₁,T ₂,...,T _s of a probability space (X,B, μ) we are interested in the asymptotic behaviour of ergodic averages of the form $$\frac{1}{N}\sum\limits_{n = 0}^{N - 1} {T_1^n f_1 \cdot T_2^n f_2 } \cdot \cdots \cdot T_s^n f_s $$ wheref ₁,f ₂,...,f _s ?L ^∞(X,B,μ). In the general case we study, mainly for commuting transformations, conditions under which the limit of (1) inL ²-norm is ∫ _x f ₁ dμ·∫ _x f ₂ dμ...∫ _x f _s dμ for anyf ₁,f ₂...,f _s ?L ^∞(X,B,μ). If the transformations are commuting epimorphisms of a compact abelian group, then this limit exists almost everywhere. A few results are also obtained for some classes of non-commuting epimorphisms of compact abelian groups, and for commuting epimorphisms of arbitrary compact groups.     

Pointwise inequalities for ergodic averages and reversed martingales

It is known that the ergodic averages A_n? in the context of the shift action on satisfy pointwise inequalities of the form

     

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.     

Subsequence ergodic theorems for amenable groups

For amenable groups that have a Følner sequence {A _n} satisfying $\overline {lim} \left| {A_n^{ - 1} A_n } \right|/\left| {A_n } \right|< + \infty $ we show that a subsequence ergodic theorem is valid for the visit times to a set of positive measure.     

Mean ergodic theorems for bi-continuous semigroups

In this paper we study the main properties of the Cesàro means of bi-continuous semigroups, introduced and studied by Kühnemund (Semigroup Forum 67:205–225, 2003). We also give some applications to Feller semigroups generated by second-order elliptic differential operators with unbounded coefficients in C _b (ℝ ^N ) and to evolution operators associated with nonautonomous second-order differential operators in C _b (ℝ ^N ) with time-periodic coefficients.     

Pointwise decomposable sets

We show that, under the conditionala<0, every recursively enumerable (r.e.) A bia has a pointwise decomposable complement. If A T^B, A and ¯B are r.e. co-retraceable sets, and f(x)=f^B(x), then there exists a r.e. co-retraceable C, such thatA(c),BT ^C , (A n) (f(n) <c _n), where ¯C=C ₀<C ₁<C ₂<....Translated from Matematicheskie Zametki, Vol. 13, No. 6, pp. 893–898, June, 1973.The author thanks A. N. Degtev for his interest in this work.     

Noncommutative maximal ergodic theorems

This paper is devoted to the study of various maximal ergodic theorems in noncommutative -spaces. In particular, we prove the noncommutative analogue of the classical Dunford-Schwartz maximal ergodic inequality for positive contractions on and the analogue of Stein's maximal inequality for symmetric positive contractions. We also obtain the corresponding individual ergodic theorems. We apply these results to a family of natural examples which frequently appear in von Neumann algebra theory and in quantum probability.

     

Weighted ergodic theorems for mean ergodic -contractions

It is shown that any bounded weight sequence which is good for all probability preserving transformations (a universally good weight) is also a good weight for any -contraction with mean ergodic (ME) modulus, and for any positive contraction of with . We extend the return times theorem by proving that if is a Dunford-Schwartz operator (not necessarily positive) on a Lebesgue space, then for any bounded measurable is a universally good weight for a.e. We prove that if a bounded sequence has "Fourier coefficents", then its weighted averages for any -contraction with mean ergodic modulus converge in -norm. In order to produce weights, good for weighted ergodic theorems for -contractions with quasi-ME modulus (i.e., so that the modulus has a positive fixed point supported on its conservative part), we show that the modulus of the tensor product of -contractions is the product of their moduli, and that the tensor product of positive quasi-ME -contractions is quasi-ME.

     

Pointwise ergodic theorem along the prime numbers

The pointwise ergodic theorem is proved for prime powers for functions inL ^p,p>1. This extends a result of Bourgain where he proved a similar theorem forp>(1+√3)/2. This paper is a part of the author’s Ph.D. thesis supervised by V. Bergelson.     

Pointwise convergence of ergodic averages along cubes

Let (X, B, μ, T) be a measure preserving system. We prove the pointwise convergence of ergodic averages along cubes of 2 ^k − 1 bounded and measurable functions for all k. We show that this result can be derived from estimates about bounded sequences of real numbers and apply these estimates to establish the pointwise convergence of some weighted ergodic averages and ergodic averages along cubes for not necessarily commuting measure preserving transformations.     

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     

Sub-additive ergodic theorems for countable amenable groups

In this paper we generalize Kingman's sub-additive ergodic theorem to a large class of infinite countable discrete amenable group actions.     

Category theorems for some ergodic multiplier properties

We prove (Baire) category theorems for ergodic multiplier properties stronger than weak mixing, and weaker than mild mixing.     

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.     

Noncommutative maximal ergodic theorems for positive contractions

Let M be a von Neumann algebra equipped with a normal semifinite faithful trace τ. Let T be a positive linear contraction on M such that τ○T?τ and such that the numerical range of T as an operator on L₂(M) is contained in a Stoltz region with vertex 1. We show that Junge and Xu's noncommutative Stein maximal ergodic inequality holds for the powers of T on Lp(M), 1<p?∞. We apply this result to obtain the noncommutative analogue of a recent result of Cohen concerning the iterates of the product of a finite number of conditional expectations.