Every ergodic transformation (X, 7, :~,/z) has an isomorphic system (Y, U, ~, v) which is uniquely ergodic and topologically mixing.

We construct a stationary ergodic process X1,X2, . . . such that each Xt has the uniform distribution on the unit square and the length Ln of the shortest path through the points X1,X2, . . . ,Xn is not asymptotic to a constant times the square root of n. In other words, we show that the Beardwood, Halton, and Hammersley [Proc. Cambridge Philos. Soc. 55 (1959) 299–327] theorem does not extend f...

Let G = SL(n, R) (or, more generally, let G be a connected, noncompact, simple Lie group). For any compact Lie group K, it is easy to find a compact manifold M , such that there is a volume-preserving, connection-preserving, ergodic action of G on some smooth, principal K-bundle P over M. Can M can be chosen independent of K? We show that if M = H / A is a homogeneous space, and the action of G...

We study ergodic Sym(N)-invariant probability measures on the space of L-structures with domain N. We call such measures “ergodic structures”. In particular, we are interested in the properly ergodic case, in which no isomorphism class has measure 1. A Morley–Scott analysis shows that proper ergodicity can always be explained by a splitting of measure over continuum-many types in a countable fr...

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...