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.     

Théorèmes ergodiques aléatoires et suites de poids aléatoires régularisantsRandom ergodic theorems and regularizing random weights

In this work, we study the convergence of pointwise ergodic averages for random subsequences, in a universal framework. We give also results on the convergence of averages which are modulated by random weights. These results are obtained as a consequence of the study of the regularity of random trigonometric polynomials. The methods used in this work involve mainly Gaussian tools, random trigonometric polynomials and spectral theory. To cite this article: S. Durand, D. Schneider, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 375–378.     

Maximal functions and ergodic averages related to Waring’s problem

We study the arithmetic analogue of maximal functions on diagonal hypersurfaces. This paper is a natural step following the papers of [13], [14] and [16]. We combine more precise knowledge of oscillatory integrals and exponential sums to generalize the asymptotic formula in Waring’s problem to an approximation formula for the Fourier transform of the solution set of lattice points on hypersurfaces arising in Waring’s problem and apply this result to arithmetic maximal functions and ergodic averages. In sufficiently large dimensions, the approximation formula, ? ²-maximal theorems and ergodic theorems were previously known. Our contribution is in reducing the dimensional constraint in the approximation formula using recent bounds of Wooley, and improving the range of ? ^p spaces in the maximal and ergodic theorems. We also conjecture the expected range of spaces.     

Uniformity seminorms on ℓ<Superscript>∞</Superscript> and applications

A key tool in recent advances in understanding arithmetic progressions and other patterns in subsets of the integers is certain norms or seminorms. One example is the norms on ℤ/Nℤ introduced by Gowers in his proof of Szemerédi’s Theorem, used to detect uniformity of subsets of the integers. Another example is the seminorms on bounded functions in a measure preserving system (associated to the averages in Furstenberg’s proof of Szemerédi’s Theorem) defined by the authors. For each integer k ≥ 1, we define seminorms on ℓ^∞(ℤ) analogous to these norms and seminorms. We study the correlation of these norms with certain algebraically defined sequences, which arise from evaluating a continuous function on the homogeneous space of a nilpotent Lie group on a orbit (the nilsequences). Using these seminorms, we define a dual norm that acts as an upper bound for the correlation of a bounded sequence with a nilsequence. We also prove an inverse theorem for the seminorms, showing how a bounded sequence correlates with a nilsequence. As applications, we derive several ergodic theoretic results, including a nilsequence version of the Wiener-Wintner ergodic theorem, a nil version of a corollary to the spectral theorem, and a weighted multiple ergodic convergence theorem.     

DIFFERENCES OF ERGODIC AVERAGES FOR CESARO BOUNDED OPERATORS

We prove that the weighted differences of ergodic averages,induced by a Cesàro bounded, strongly continuous, one-parametergroup of positive, invertible, linear operators on L^p, 1 <p < , converge almost every where and in the L^p-norm. Weobtain first the boundedness of the ergodic maximal operatorand the convergence of the averages.     

The convergence of Cesaro averages for certain nonstationary Markov chains

If P is a stochastic matrix corresponding to a stationary, irreducible, positive persistent Markov chain of period d>1, the powers Pⁿ will not converge as n → ∞. However, the subsequences P^nd+k for k=0,1,...d-1, and hence Cesaro averages Σⁿ_k-1 P^k/n, will converge. In this paper we determine classes of nonstationary Markov chains for which the analogous subsequences and/or Cesaro averages converge and consider the rates of convergence. The results obtained are then applied to the analysis of expected average cost.     

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.     

Uniform estimates for spherical functions on complex semisimple Lie groups

We prove new estimates for spherical functions and their derivatives on complex semisimple Lie groups, establishing uniform polynomial decay in the spectral parameter. This improves the customary estimate arising from Harish-Chandra's series expansion, which gives only a polynomial growth estimate in the spectral parameter. In particular, for arbitrary positive-definite spherical functions on higher rank complex simple groups, we establish estimates for which are of the form in the spectral parameter and have uniform exponential decay in regular directions in the group variable a _t . Here is an explicit constant depending on G, and may be singular, for instance.?The uniformity of the estimates is the crucial ingredient needed in order to apply classical spectral methods and Littlewood—Paley—Stein square functions to the analysis of singular integrals in this context. To illustrate their utility, we prove maximal inequalities in L ^p for singular sphere averages on complex semisimple Lie groups for all p in . We use these to establish singular differentiation theorems and pointwise ergodic theorems in L ^p for the corresponding singular spherical averages on locally symmetric spaces, as well as for more general measure preserving actions. Submitted: May 2000, Revised version: October 2000.     

A Tauberian Theorem for Ergodic Averages,Spectral Decomposability,and the Dominated Ergodic Estimate for Positive Invertible Operators

Suppose that (,) is a -finite measure space, and 1 < p < . Let T:L^p( L ^p() be a bounded invertible linear operator such that T and T ^–1 are positive. Denote by _n(T) the nth two-sided ergodic average of T, taken in the form of the nth (C,1) mean of the sequence {T^j+T^–j}_{j =1} . Martín-Reyes and de la Torre have shown that the existence of a maximal ergodic estimate for T is characterized by either of the following two conditions: (a) the strong convergence of E_n(T)_n=1 ; (b) a uniform A _p p estimate in terms of discrete weights generated by the pointwise action on of certain measurable functions canonically associated with T. We show that strong convergence of the (C,2) means of {T^j+T^–j}_j=1 already implies (b). For this purpose the (C,2) means are used to set up an `averaged' variant of the requisite uniform A _p weight estimates in (b). This result, which can be viewed as a Tauberian-Type replacement of (C,1) means by (C,2) means in (a), leads to a spectral-theoretic characterization of the maximal ergodic estimate by application of a recent result of the authors establishing the strong convergence of the (C,2)-weighted ergodic means for all trigonometrically well-bounded operators. This application also serves to equate uniform boundedness of the rotated Hilbert averages of T with the uniform boundedness of the ergodic averages E_n(T)_{n = 1} .     

Convergence and Smoothness of Nonlinear Lane–Riesenfeld Algorithms in the Functional Setting

We investigate the Lane–Riesenfeld subdivision algorithm for uniform B-splines, when the arithmetic mean in the various steps of the algorithm is replaced by nonlinear, symmetric, binary averaging rules. The averaging rules may be different in different steps of the algorithm. We review the notion of a symmetric binary averaging rule, and we derive some of its relevant properties. We then provide sufficient conditions on the nonlinear binary averaging rules used in the Lane–Riesenfeld algorithm that ensure the convergence of the algorithm to a continuous function. We also show that, when the averaging rules are C ² with uniformly bounded second derivatives, then the limit is a C ¹ function. A canonical family of nonlinear, symmetric averaging rules, the p-averages, is presented, and the Lane–Riesenfeld algorithm with these averages is investigated.     

CONVERGENCE THEOREMS AND MAXIMAL INEQUALITIES FOR MARTINGALE ERGODIC PROCESSES

In this article, we study two types of martingale ergodic processes. We prove that a.e. convergence and L^p convergence as well as maximal inequalities, which are established both in ergodic theory and martingale setting, also hold well for these new sequences of random variables. Moreover, the corresponding theorems in the former two areas turn out to be degenerate cases of the martingale ergodic theorems proved here.     

von Neumann��s mean ergodic theorem on complete random inner product modules

We first prove two forms of von Neumann’s mean ergodic theorems under the framework of complete random inner product modules. As applications, we obtain two conditional mean ergodic convergence theorems for random isometric operators which are defined on L _ℱ ^p (ℰ, H) and generated by measure-preserving transformations on Ω, where H is a Hilbert space, L ^p (ℰ, H) (1 ⩽ p < ∞) the Banach space of equivalence classes of H-valued p-integrable random variables defined on a probability space (Ω, ℰ, P), F a sub σ-algebra of ℰ, and L _ℱ ^p (ℰ(E,H) the complete random normed module generated by L ^p (ℰ, H).     

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.     

Wiener-Wintner return-times ergodic theorem

We state a new ergodic theorem, combining the Wiener-Wintner theorem and Bourgain’s theorem concerning the convergence of ergodic averages along return-times sequences. We consider ergodic averages of the form $$\frac{1}{N}\sum\limits_{n = 0}^{N - 1} {e^{in\theta } \cdot f'(S^n y) \cdot f(T^n x)} $$ and we show that the behaviour of these averages characterizes an algebraC of functions, which contains the Kronecker algebra and has interesting properties, linked with multiple recurrence ergodic theorems.     

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.     

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.     

Rate of convergence to ergodic distribution for queue length in systems of the type <Emphasis Type="Italic">M</Emphasis><Superscript>θ</Superscript>/<Emphasis Type="Italic">G</Emphasis>/1/<Emphasis Type="Italic">N</Emphasis>

For finite-capacity queuing systems of the type M ^θ/G/1, convenient formulas for the ergodic distribution of the queue length are found, an estimate for the rate of convergence of the distribution of the queue length in the transient mode to the ergodic distribution is obtained, and computational algorithms for finding the rate of convergence are presented. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 9, pp. 1169–1178, September, 2007.     

Dominated convergence in measure on semifinite von Neumann algebras and arithmetic averages of measurable operators

Consider a von Neumann algebra M with a faithful normal semifinite trace τ. We prove that each order bounded sequence of τ-compact operators includes a subsequence whose arithmetic averages converge in τ. We also prove a noncommutative analog of Pratt’s lemma for L ₁(M, τ). The results are new even for the algebra M = B(H) of bounded linear operators with the canonical trace τ = tr on a Hilbert space H. We apply the main result to L _p (M, τ) with 0 < p ≤ 1 and present some examples that show the necessity of passing to the arithmetic averages as well as the necessity of τ-compactness of the dominant.     

Local-global principles for representations of quadratic forms

We prove a local-global principle for the problem of representations of quadratic forms by quadratic forms over ℤ, in codimension ≥5. The proof uses the ergodic theory of p-adic groups, together with a fairly general observation on the structure of orbits of an arithmetic group acting on integral points of a variety.     

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.