On some properties of transformations of a logic
We extend the notion of Dobrushin coefficient of ergodicity to positive contractions defined on the L¹-space associated with a finite von Neumann algebra, and in terms of this coefficient we prove stability results for L¹-contractions.
It is pointed out that a strong law of large numbers for L-statistics established by van Zwet (1980) for i.i.d. sequences, remains valid for stationary ergodic data. When the underlying process is weakly Bernoulli, the result extends even to generalized L-statistics considered in Helmers et al. (1988).
We construct infinite-dimensional chains that are L¹ good for almost sure convergence, which settles a question raised in this journal [N]. We give some conditions for a coprime generated chain to be bad for L² or , using the entropy method. It follows that such a chain with positive lower density is bad for . There also exist such bad chains with zero density.
Let T be a finite entropy, aperiodic automorphism of a nonatomic probability space. We give an elementary proof of the existence of a finite entropy, countable generating partition for T.
Let T be a measure-preserving ergodic transformation of a measure space (X,,μ) and, for f ∈ L(X), let . In this paper we mainly investigate the question of whether (i) and whether (ii) for some a > 0. It is proved that (i) holds for every f ≥ 0. (ii) holds if f ≥ 0 and f log log (f + 3) ∈ L(X) or if μ(X) = 1 and the random variables are independent. Related inequalities are proved. Some examples and counterexamples are constructed. Several known results are obtained as corollaries.
Let (X,,μ,τ) be an ergodic dynamical system and φ be a measurable map from X to a locally compact second countable group G with left Haar measure . We consider the map defined on X × G by and the cocycle generated by φ. Using a characterization of the ergodic invariant measures for , we give the form of the ergodic decomposition of or more generally of the -invariant measures , where is χ∘φ-conformal for an exponential χ on G.
The Stein-Weiss theorem that the distribution function of the Hilbert transform of the characteristic function of E depends only on the measure of E is generalized to the ergodic Hilbert transform.
A generalization of the Avez method of construction of an invariant measure is presented.