A sequence (sn) of integers is good for the mean ergodic theorem if for each invertible measure preserving system (X, B, µ, T) and any bounded measurable function f , the averages 1 N P N n=1 f (T sn x) converge in the L 2 (µ) norm. We construct a sequence (sn) that is good for the mean ergodic theorem, but the sequence (s 2 n) is not. Furthermore, we show that for any set of bad exponents B, t...

For general quantum systems the semiclassical behaviour of eigenfunctions in relation to the ergodic properties of the underlying classical system is quite difficult to understand. The Wignerfunctions of eigenstates converge weakly to invariant measures of the classical system, the so called quantum limits, and one would like to understand which invariant measures can occur that way, thereby cl...

Ye, X.D., Coexistence of uniquely ergodic subsystems of interval mappings, Theoretical Computer Science 123 (1994) 1677181. We prove that if (1.f) has a uniquely ergodic subsystem (X,f] x, x), then fo$ every sqh, (I,f) has a uniquely ergodic subsystem (Y,f],, p) such thatfs=s, where l=[O, l].f~C(l,l).

We investigate ergodic theory of Poisson suspensions. In the process, we establish close connections between finite and infinite measure preserving ergodic theory. Poisson suspensions thus provide a new approach to infinite measure ergodic theory. Fields investigated here are mixing properties, spectral theory, joinings. We also compare Poisson suspensions to the apparently similar looking Gaus...

We study nonlinear Neumann type boundary value problems related to ergodic phenomenas. The particularity of these problems is that the ergodic constant appears in the (possibly nonlinear) Neumann boundary conditions. We provide, for bounded domains, several results on the existence, uniqueness and properties of this ergodic constant.

We discuss a family of estimators for the entropy rate of a stationary ergodic process and prove their pointwise and mean consistency under a Doeblin-type mixing condition. The estimators are Cesàro averages of longest match-lengths, and their consistency follows from a generalized ergodic theorem due to Maker. We provide examples of their performance on English text, and we generalize our resu...

We consider the problem of Lp-consistent density estimation from the initial segments of strongly dependent processes. It is shown that no procedure can consistently estimate the one-dimensional marginal density of every stationary ergodic process for which such a density exists. A similar result is established for the problem of estimating the support of the marginal distribution of an ergodic...

Let f be a real-valued function defined on the phase space of a dynamical system. Ergodic optimization is the study of those orbits, or invariant probability measures, whose ergodic f -average is as large as possible. In these notes we establish some basic aspects of the theory: equivalent definitions of the maximum ergodic average, existence and generic uniqueness of maximizing measures, and t...

In cryptography and coding theory, it is important to study the pseudo-random sequences and the ergodic transformations. We already have the 1-Lipshitz ergodic theory over Z2 established by V. Anashin and others. In this paper we present an ergodic theory over F2[[T ]] and some ideas which might be very useful in applications.

We study a stability property of probability laws with respect to small violations of algorithmic randomness. A sufficient condition of stability is presented in terms of Schnorr tests of algorithmic randomness. Most probability laws, like the strong law of large numbers, the law of iterated logarithm, and even Birkhoff’s pointwise ergodic theorem for ergodic transformations, are stable in this...