We define the space of trajectories of a doubly stochastic operator on L¹(X,μ) as a shift space , where ν is a probability measure defined as in the Ionescu-Tulcea theorem and σ is the shift transformation. We study connections between the entropy of a doubly stochastic operator and the entropy of the shift on the space of trajectories of this operator.
Two types of weighted ergodic averages are studied. It is shown that if F = {Fₙ} is an admissible superadditive process relative to a measure preserving transformation, then a Wiener-Wintner type result holds for F. Using this result new good classes of weights generated by such processes are obtained. We also introduce another class of weights via the group of unitary functions, and study the convergence of the corresponding weighted averages. The limits of such weighted averages are also identified....
The purpose of this note is to prove various versions of the ergodic decomposition theorem for probability measures on standard Borel spaces which are quasi-invariant under a Borel action of a locally compact second countable group or a discrete nonsingular equivalence relation. In the process we obtain a simultaneous ergodic decomposition of all quasi-invariant probability measures with a prescribed Radon-Nikodym derivative, analogous to classical results about decomposition of invariant probability...
Let be a strongly continuous semigroup of linear contractions in , , of a -finite measure space. In this paper we prove that if there corresponds to each a positive linear contraction in such that for all , then there exists a strongly continuous semigroup of positive linear contractions in such that for all and . Using this and Akcoglu’s dominated ergodic theorem for positive linear contractions in , we also prove multiparameter pointwise ergodic and local ergodic theorems...
We provide sufficient conditions for sums of two unbounded operators on a Banach space to be (pre-)generators of contraction semigroups. Necessary conditions and applications to positive emigroups on Banach lattices are also presented.
Under different compactness assumptions pointwise and mean ergodic theorems for subadditive superstationary families of random sets whose values are weakly (or strongly) compact convex subsets of a separable Banach space are presented. The results generalize those of [14], where random sets in are considered. Techniques used here are inspired by [3].
Junge and Xu (2007), employing the technique of noncommutative interpolation, established a maximal ergodic theorem in noncommutative -spaces, 1 < p < ∞, and derived corresponding maximal ergodic inequalities and individual ergodic theorems. In this article, we derive maximal ergodic inequalities in noncommutative -spaces directly from the results of Yeadon (1977) and apply them to prove corresponding individual and Besicovitch weighted ergodic theorems. Then we extend these results to noncommutative...
Jones and Rosenblatt started the study of an ergodic transform which is analogous to the martingale transform. In this paper we present a unified treatment of the ergodic transforms associated to positive groups induced by nonsingular flows and to general means which include the usual averages, Cesàro-α averages and Abel means. We prove the boundedness in , 1 < p < ∞, of the maximal ergodic transforms assuming that the semigroup is Cesàro bounded in . For p = 1 we find that the maximal ergodic...
We give lower and upper estimates of the capacity of self-similar measures generated by iterated function systems where are bi-lipschitzean transformations.