In the paper we investigate a new natural notion of ergodic infinite permutations, that is, infinite permutations which can be defined by equidistributed sequences. We show that, unlike for permutations in general, the minimal complexity of an ergodic permutation α is pα(n) = n. The class of ergodic permutations of minimal complexity coincides with the class of so-called Sturmian permutations, ...

Let S = {ki,ki, ...} be an increasing sequence of positive integers. We call S strongly ergodic if for every measure preserving transformation T on a probability space (Cl, J, P) and every / £ Li(f2) we have limn-»oo(l/n) J^^j f(TkiuJ) = Pf(w) a.e. where Pf is the appropriate limit guaranteed by the individual ergodic theorem. We give sufficient conditions for a sequence S to be strongly ergodi...

In this article we study weighted polynomial multiple ergodic averages. A sequence of weights is called universally good if any polynomial multiple ergodic average with this sequence of weights converges in L. We find a necessary condition and show that for any bounded measurable function φ on an ergodic system, the sequence φ(Tnx) is universally good for almost every x. The linear case was cov...

converge to ET (x) in A for any x ∈ A, as n → ∞, here ET is an projection of A to the fixed point subspace of T . Note that when T is a completely positive mapping then ET is a conditional expectation. We provide an example of uniquely ergodic dynamical system relative its fixed point subspace, for which ET is not a conditional expectation. It is also constructed a uniquely ergodic entangled Ma...

Necessary and sufficient conditions are given for the uniform convergence over an arbitrary index set in von Neumann’s mean and Birkhoff’s pointwise ergodic theorem. Three different types of conditions already known from probability theory are investigated. Firstly it is shown that the property of being eventually totally bounded in the mean is necessary and sufficient. This condition involves ...

In the thirty or so years since H. Furstenberg reproved Szemerédi’s theorem using methods from ergodic theory, many striking discoveries have been made in the area now known as Ergodic Ramsey theory. Perhaps the most surprising of these is the discovery that recurrence results can be obtained for polynomial sets, meaning sets of values of polynomials. The following pretty theorem, a special cas...