Using the ratio ergodic theorem for a measure preserving transformation in a σ-finite measure space we give a straightforward proof of Derriennic’s reverse maximal inequality for the supremum of ergodic ratios.

For a general class of unitary quantum maps, whose underlying classical phase space is divided into ergodic and non-ergodic components, we prove analogues of Weyl’s law for the distribution of eigenphases, and the Schnirelman-Zelditch-Colin de Verdière Theorem on the equidistribution of eigenfunctions with respect to the ergodic components of the classical map (quantum ergodicity). We apply our...

We prove the L convergence for the linear multiple ergodic averages of commuting transformations T1, . . . , Tl, assuming that each map Ti and each pair TiT −1 j is ergodic for i 6= j. The limiting behavior of such averages is controlled by a particular factor, which is an inverse limit of nilsystems. As a corollary we show that the limiting behavior of linear multiple ergodic averages is the s...

We classify n-point extensions of ergodic Z-actions up to relative orbit equivalence and establish criteria under which one n-point extension of an ergodic Z-action can be sped up to be relatively isomorphic to an n-point extension of another ergodic Z-action. Both results are characterized in terms of an algebraic object associated to each n-point extension which is a conjugacy class of subgro...

An updated discussion on physical and mathematical aspects of the ergodic hypothesis in classical equilibrium statistical mechanics is presented. Then a practical attitude for the justification of the microcanonical ensemble is indicated. It is also remarked that the difficulty in proving the ergodic hypothesis should be expected.

We discuss classical results of Ramsey theory together with their dynamical counterparts, survey recent developments and formulate some natural open questions and conjectures. Mathematics Subject Classification (2000). Primary 28D15, 05D10; Secondary 37A45, 37B10.

 In this paper by use of mathematical modeling of an observer [14,15] the notion of relative information functional for relative dynamical systemson compact metric spaces is presented. We extract the information function ofan ergodic dynamical system (X,T) from the relative information of T fromthe view point of observer χX, where X denotes the base space of the system.We also generalize the in...