Mustafa A. Akcoglu

The strong sweeping out property for lacunary sequences, Riemann sums, convolution powers, and related matters

In this paper we establish conditions on a sequence of operators which imply divergence. In fact, we give conditions which imply that we can find a set B of measure as close to zero as we like, but such that the operators applied to the characteristic function of this set have a lim sup equal to 1 and a lim inf equal to 0 a.e. (strong sweeping out). The results include the fact that ergodic averages along lacunary sequences, certain convolution powers, and the Riemann sums considered by Rudin are all strong sweeping out. One of the criteria for strong sweeping out involves a condition on the Fourier transform of the sequence of measures, which is often easily checked. The second criterion for strong sweeping out involves showing that a sequence of numbers satisfies a property similar to the conclusion of Kroneckers lemma on sequences linearly independent over the rationals.

Convergence of averages of point transformations

converges a.e. for each E1 = L 1(X, 5f, [t). This raises the following question. What are the necessary conditions on the matrix (ani) so that the sequence fn(x) = ,ianif/(rZ x) converges a.e. for each / E LE and for each automorphism r? The answer is not known. Spectral considerations would suggest, however, the following conjecture. If (ani) is such that the sequence of functions pn(z) = X iani z is uniformly bounded and pointwise convergent on the unit circle lZj = 1, then /n converges a.e. In fact, recently an attempt has been made to prove this as a theorem [1]. In this note we would like to observe the following simple fact which shows that this conjecture is far from being correct. If r is a real number, let [r] denote the greatest integer which is less than or equal to r. Define a matrix (ani), n= 1, 2,*, as

On restricted weak type (1,1); The discrete case

In this paper we study the relationship between restricted weak type (1,1) and weak type (1,1) for convolution operators on ℝ.

Two examples of local ergodic divergence

This note contains the first example of a 1-parameter semigroup {Tpt≧0} of linear contractions inLp (1<p<∞) for which the assertion of the local ergodic theorem (t1∝0′Tsfds conv. a.e. ast → 0+0 for allf ∈Lp) fails to be true. The first example is a continuous semigroup of unitary operators inL2, the second a power-bounded continuous semigroup of positive operators inL1. This answers problems of Kubokawa, Fong and Sucheston.

Sweeping out properties of operator sequences

Let Lp ≥ Lp(X,ñ), 1 p 1, be the usual Banach Spaces of real valued functions on a complete non-atomic probability space. Let (T1, . . . , TK) be L2contractions. Let 0 Ú ¢ Ú é 1. Call a function f a é-spanning function if kfk2 ≥ 1 and if kTkf Qk 1Tkfk2 1⁄2 é for each k ≥ 1, . . . , K, where Q0 ≥ 0 and Qk is the orthogonal projection on the subspace spanned by (T1f , . . . , Tkf ). Call a function h a (é, ¢) -sweeping function if khk1 1, khk1 Ú ¢, and if max1 k K jTkhj Ù é ¢ on a set of measure greater than 1 ¢. The following is the main technical result, which is obtained by elementary estimates. There is an integer K ≥ K(¢, é) 1⁄2 1 such that if f is a é-spanning function, and if the joint distribution of (f , T1f , . . . , TK f ) is normal, then h ≥ (f^M)_( M) ÛM is a (é, ¢)-sweeping function, for some M Ù 0. Furthermore, if Tks are the averages of operators induced by the iterates of a measure preserving ergodic transformation, then a similar result is true without requiring that the joint distribution is normal. This gives the following theorem on a sequence (Ti) of these averages. Assume that for each K 1⁄2 1 there is a subsequence (Ti1 , . . . , TiK ) of length K, and a é-spanning function fK for this subsequence. Then for each ¢ Ù 0 there is a function h, 0 h 1, khk1 Ú ¢, such that lim supi Tih 1⁄2 é a.e.. Another application of the main result gives a refinement of a part of Bourgain’s “Entropy Theorem”, resulting in a different, self contained proof of that theorem. The first author was partially supported by an NSERC Grant. The second author was partially supported by an NSERC Grant. The third author was partially supported by an NSF Grant. Received by the editors May, 1995. AMS subject classification: Primary: 28D99; Secondary: 60F99.

A stochastic ergodic theorem for superadditive processes

An elementary proof is given of Krengels stochastic ergodic theorem in the setting of multiparameter superadditive processes.

Divergence of averages obtained by sampling a flow

In this paper we consider ergodic averages obtained by sampling at discrete times along a measure preserving ergodic flow. We show, in particular, that if U t is an aperiodic flow, then averages obtained by sampling at times n + t n satisfy the strong sweeping out property for any sequence t n → 0. We also show that there is a flow (which is periodic) and a sequence t n → 0 such that the Cesaro averages of samples at time n + t n do converge a.e. In fact, we show that every uniformly distributed sequence admits a perturbation that makes it a good Lebesgue sequence

A Solution to a Problem of A. Bellow

This chapter focuses on an elementary and self-contained solution to the problem of A. Bellow. It presents the application of an extension of a method of W. Rudin method to solve this problem. To find out the solution to the problem of A. Bellow, it is convenient to work with the interval X = (0, 1), rather than the real line R . If Q is an isometry of a compact metric space Y, H is an open subset of Y , and y 0 ∈ H , then there is a number K and an infinite sequence of integers { k i } such that for all j , 0 k j + 1 − k j K , and Q kj y 0 ∈ H .

Tail field representations and the zero-two law

The zero-two law was proved for a positiveL1-contractionT by Ornstein and Sucheston, and gives a condition which impliesTnf−Tn+1f → 0 for allf. Extensions of this result to the case of a positiveLp-contraction, 1≤p<∞, have been obtained by several authors. In the present paper we prove a theorem which is related to work of Wittmann.We will say that a positive contractionT contains a circle of lengthm if there is a nonzero functionf such that the iterated valuesf, T f,…,Tm-1f have disjoint support, whileTmf=f. Similarly, a contractionT contains a line if for everym there is a nonzero functionf (which may depend onm) such thatf,…,Tm-1f have disjoint support. Approximate forms of these conditions are defined, which are referred to as asymptotic circles and lines, respectively. We show (Theorem 3) that if the conclusionTnf−Tn+1f→0 of the zero-two law does not hold for allf inLp, then eitherT contains an asymptotic circle orT contains an asymptotic line. The point of this result is that any condition onT which excludes circles and lines must then imply the conclusion of the zero-two law.Theorem 3 is proved by means of the representation of a positiveLp-contraction in terms of anLp-isometry. Asymptotic circles and lines forT correspond to exact circles and lines for the isometry on tail-measurable functions, and exact circles and lines for the isometry are obtained using the Rohlin tower construction for point transformations.

An example of pointwise non-convergence of iterated conditional expectation operators

AbstractGiven ∈, we construct a sequence