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.     

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.     

Level sets of multiple ergodic averages

We propose to study multiple ergodic averages from multifractal analysis point of view. In some special cases in the symbolic dynamics, the Hausdorff dimensions of the level sets for the limit of these multiple ergodic averages are determined by using Riesz products.     

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.     

A Banach space-valued ergodic theorem for amenable groups and applications

In this paper, we study unimodular amenable groups. The first part of the paper is devoted to results on the existence of uniform families of ε-quasi tilings for these groups. First we extend constructions of Ornstein and Weiss by quantitative estimates for the covering properties of the corresponding decompositions. Then we apply the methods developed to obtain an abstract ergodic theorem for a class of functions mapping subsets of a countable amenable group into some Banach space. This result extends significantly and complements related results found in the literature. Further, using the Lindenstrauss ergodic theorem, we link our results to classical ergodic theory. We conclude with two important applications: uniform approximation of the integrated density of states on amenable Cayley graphs and almost-sure convergence of cluster densities in an amenable bond percolation model.     

On the almost everywhere convergence of ergodic averages for power-bounded operators on LP-subspaces

Let X be a closed subspace of L^P(), where is an arbitrary measure and 1A(n) and (n) denote the discrete ergodic averages and Hilbert transform truncates defined by U. We extend to this setting the -a. e. convergence criteria forA(n) and (n) which V. F. Gaposhkin and R. Jajte introduced for unitary operators on L²(). Our methods lift the setting from X to ^p, where classical harmonic analysis and interpolation can be applied to suitable square functions.     

Convergence of multiple ergodic averages along polynomials of several variables

LetT be an invertible measure preserving transformation of a probability measure spaceX. Generalizing a recent result of Host and Kra, we prove that the averages\(T^{p1(u)} f\) converge inL ² (X) for anyf ₁ ,…,f _r ?L ^∞ (X), any polynomialsp ₁ ,…,p _r :f ¹ ,...,:f ¹ ∈1\t8(X and and Følner sequence \s{\gF _r \s} _r ^\t8 in ? _d .     

Group structure and the pointwise ergodic theorem for connected amenable groups

Let G be a connected amenable group (thus, an extension of a connected normal solvable subgroup R by a connected compact group $\text{K =}\text{G}\text{R}$). We show how to explicitly construct sequences {U_n} of compacta in G in terms of the structural features of G which have the following property: For any “reasonable” action G × L^p(X, μ) ↓ L^p(X, μ) on an L^p space, 1 <p < ∞, and any f ∈ L^p(X, μ), the averages

$\text{A}{}_{\mathrm{n}}\text{f=}\text{1}\text{|U}{}_{\mathrm{n}}\text{|}\text{∫}\text{U}{}_{\mathrm{n}}\text{T}{}_{\mathrm{g}}{}^{\mathrm{?1f}}\text{dg (|E|=}\text{left Haar measure in}\text{G)}$

converge in L^p norm, and pointwise μ-a.e. on X, to G-invariant functions $\text{f1}$ in L^p(X, μ). A single sequence {U_n} in G works for all L^p actions of G. This result applies to many nonunimodular groups, which are not handled by previous attempts to produce noncommutative generalizations of the pointwise ergodic theorem.     

Convergence of polynomial ergodic averages

We prove theL ² convergence for an ergodic average of a product of functions evaluated along polynomial times in a totally ergodic system. For each set of polynomials, we show that there is a particular factor, which is an inverse limit of nilsystems, that controls the limit behavior of the average. For a general system, we prove the convergence for certain families of polynomials. Dedicated to Hillel Furstenberg upon his retirement The second author was partially supported by NSF grant DMS-0244994.     

Finite uniform generators for ergodic,finite entropy,free actions of amenable groups

Summary Let (X) be a Lebesgue space. We prove in the first part of this paper that any ergodic ²-action on (X) with finite entropy hLogk has generating partition P that is uniform and has k atoms. In the second part, we prove a similar result for any ergodic, free G-action with finite entropy hLog(k-2), for any discrete amenable group G.     

On ergodic averages for affine lattice actions on tori

We prove that for a ?^ν-action on a torus as a group of affine transformations all the ergodic averages exist if and only if all the eigenvalues of the automorphism parts of all the affine transformations are roots of unity.     

General theories unifying ergodic averages and martingales

The curious fact that the behavior of ergodic averages is analogous to the behavior of (reversed) martingales has long been known. For the last 60 years, at least five different approaches have been proposed to the construction of a unifying theory: by M. Jerison (1955), G.-C. Rota (1961), A. and C. Ionescu Tulcea (1963), A.M. Vershik (1960s), and A.G. Kachurovskii (1998). The aim of the present survey is to carry out a comparative analysis of all the approaches, with special focus being placed on the fifth approach, which is due to the present author and has not yet been published in full detail. Moreover, the comparative analysis carried out in the survey has led to a new, the sixth, approach.     

Multiple ergodic averages for three polynomials and applications

We find the smallest characteristic factor and a limit formula for the multiple ergodic averages associated to any family of three polynomials and polynomial families of the form . We then derive several multiple recurrence results and combinatorial implications, including an answer to a question of Brown, Graham, and Landman, and a generalization of the Polynomial Szemerédi Theorem of Bergelson and Leibman for families of three polynomials with not necessarily zero constant term. We also simplify and generalize a recent result of Bergelson, Host, and Kra, showing that for all and every subset of the integers the set

has bounded gaps for ``most' choices of integer polynomials .

     

Local ergodic theorems for K-spherical averages on the Heisenberg group

Given a Gelfand pair where is the Heisenberg group and K is a compact subgroup of the unitary group U(n) we consider the sphere and ball averages of certain K-invariant measures on . We prove local ergodic theorems for these measures when . We also consider averages over annuli in the case of reduced Heisenberg group and show that when the functions have zero mean value the maximal function associated to the annulus averages behave better than the spherical maximal function. We use square function arguments which require several properties of the K-spherical functions. Received September 1, 1998; in final form July 9, 1999     

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

     

Power convergence of Abel averages

Necessary and sufficient conditions are presented for the Abel averages of discrete and strongly continuous semigroups, T ^k and T _t , to be power convergent in the operator norm in a complex Banach space. These results cover also the case where T is unbounded and the corresponding Abel average is defined by means of the resolvent of T. They complement the classical results by Michael Lin establishing sufficient conditions for the corresponding convergence for a bounded T.     

On the mean ergodic theorem for weighted averages

Summary Let (, , P) be a probability space and let T be a measurable and measure preserving point transformation from into . Let f be a measurable and square integrable function on (, , P), and let a _N,k for N, K=0, 1, ... be such that for all N. The authors investigate conditions on the a _N,k 's such that the sequence converges in mean square for all (, , P, T) and f described above. The special cases T weakly mixing and T strongly mixing are also considered.Research partially sponsored by the Air Force Office of scientific Research, Office of Aerospace Research, United States Air Force, under Grant No. AFOSR-68-1394.Research done while this author held a National Science Foundation Traineeship at the University of Missouri, Columbia.