Basic ergodic properties of the ELF class of automorphisms, i.e. of the class of ergodic automorphisms whose weak closure of measures supported on the graphs of iterates of T consists of ergodic self-joinings are investigated. Disjointness of the ELF class with: 2-fold simple automorphisms, interval exchange transformations given by a special type permutations and time-one maps of measurable flows is discussed. All ergodic Poisson suspension automorphisms as well as dynamical systems determined...

If the ergodic transformations S, T generate a free ${\mathbb{Z}}^2$ action on a finite non-atomic measure space (X,S,µ) then for any $c_1,c_2\in ℝ$ there exists a measurable function f on X for which ${\left(N+1\right)}^{-1}{\sum }_{j=0}^{N}f\left(S^{j}x\right)\to c_1$ and ${(N+1)}^{-1}{\sum }_{j=0}^{N}f\left(T^{j}x\right)\to c_2\mu$-almost everywhere as N → ∞. In the special case when S, T are rationally independent rotations of the circle this result answers a question of M. Laczkovich.

We study the convergence of the ergodic averages $\frac{1}{N}{\sum }_{k=0}^{N-1}\theta \left(k\right)f\circ T^{u_k}$ where ${\left(\theta \left(k\right)\right)}_{k\in \mathbb{N}}$ is a bounded sequence and ${\left(u_k\right)}_{k\in \mathbb{N}}$ a strictly increasing sequence of integers such that ${\mathrm{Sup}}_{\alpha \in ℝ}\left|{\sum }_{k=0}^{N-1}\theta \left(k\right)\exp \left(2i\pi \alpha u_k\right)\right|=O\left(N^{\delta }\right)$ for some $\delta \<1$. Moreover we give explicit such sequences $\theta$ and $u$ and we investigate in particular the case where $\theta$ is a $q$-multiplicative sequence.

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...

We define a class of discrete Abelian group extensions of rank-one transformations and establish necessary and sufficient conditions for these extensions to be power weakly mixing. We show that all members of this class are multiply recurrent. We then study conditions sufficient for showing that Cartesian products of transformations are conservative for a class of invertible infinite measure-preserving transformations and provide examples of these transformations.

We construct a natural invariant measure concentrated on the set of square-free numbers, and invariant under the shift. We prove that the corresponding dynamical system is isomorphic to a translation on a compact, Abelian group. This implies that this system is not weakly mixing and has zero measure-theoretical entropy.

Recently, T. Tao gave a finitary proof of a convergence theorem for multiple averages with several commuting transformations, and soon thereafter T. Austin gave an ergodic proof of the same result. Although we give here another proof of the same theorem, this is not the main goal of this paper. Our main concern is to provide tools for the case of several commuting transformations, similar to the tools successfully used in the case of a single transformation, with the idea that they may be used in...

The aim of this short note is to present in terse style the meaning and consequences of the "filling scheme" approach for a probability measure preserving transformation. A cohomological equation encapsulates the argument. We complete and simplify Woś' study (1986) of the reversibility of the ergodic limits when integrability is not assumed. We give short and unified proofs of well known results about the behaviour of ergodic averages, like Kesten's lemma (1975). The strikingly simple proof of the...

We determine the number and properties of the invariant measures under the projective flow defined by a family of one-dimensional Jacobi operators. We calculate the derivative of the Floquet coefficient on the absolutely continuous spectrum and deduce the existence of the non-tangential limit of Weyl m-functions in the $L^1$-topology.

A unified introduction to the dynamics of interval exchange maps and related topics, such as the geometry of translation surfaces, renormalization operators, and Teichmüller flows, starting from the basic definitions and culminating with the proof that almost every interval exchange map is uniquely ergodic. Great emphasis is put on examples and geometric interpretations of the main ideas.

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 $L^p$, 1 < p < ∞, of the maximal ergodic transforms assuming that the semigroup is Cesàro bounded in $L^p$. For p = 1 we find that the maximal ergodic...