This paper gives a survey of recent results on generalized Jackson networks , where classical exponential or i.i.d. assumptions on services and routings are replaced by stationary and ergodic assumptions. We rst show that the most basic features of the network may exhibit unexpected behavior. Several probabilistic properties are then discussed, including a strong law of large numbers for the nu...

Ergodic theory involves the study of transformations on measure spaces. Interchanging the words “measurable function” and “probability density function” translates many results from real analysis to results in probability theory. Ergodic theory is no exception. Ergodic theory has fundamental applications in probability theory, starting from areas that are very well understood, such as finite st...

Weak conditions are provided under which society’s long-run distribution of wealth is independent of initial asset holdings.

We show that for any co-amenable compact quantum group A = C(G) there exists a unique compact Hausdorff topology on the set EA(G) of isomorphism classes of ergodic actions of G such that the following holds: for any continuous field of ergodic actions of G over a locally compact Hausdorff space T the map T → EA(G) sending each t in T to the isomorphism class of the fibre at t is continuous if a...

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...

We study ergodic quadratic optimal stochastic control problems for an affine state equation with state and control dependent noise and with stochastic coefficients. We assume stationarity of the coefficients and a finite cost condition. We first treat the stationary case and we show that the optimal cost corresponding to this ergodic control problem coincides with the one corresponding to a sui...

It has been conjectured that the stably ergodic diieomorphisms are open and dense in the space of volume-preserving, partially hyperbolic diieomorphisms of a compact manifold. In this paper we deal with two recalcitrant examples; the standard map cross Anosov and the ergodic automorphisms of the four torus. In both cases we show that they may be approximated by stably ergodic diieomorphisms whi...

We review recent results which relate spectral theory of discrete one-dimensional Schrödinger operators over strictly ergodic systems to uniform existence of the Lyapunov exponent. In combination with suitable ergodic theorems this allows one to establish Cantor spectrum of Lebesgue measure zero for a large class of quasicrystal Schrödinger operators. The results can also be used to study non-u...

where ̺ is a positive measure with compact support in (0,∞). Hence V is symmetric and nonconvex in general. While for strictly convex V ’s the translation-invariant, ergodic gradient Gibbs measures are completely characterized by their tilt, a non-convex potential as above may lead to several ergodic gradient Gibbs measures with zero tilt. Still, every ergodic, zero-tilt gradient Gibbs measure f...

We consider models of random Schrr odinger operators which exist thanks to a cohomological theorem in ergodic theory. Examples are ergodic Schrr odinger operators with random magnetic uxes on discrete two-dimensional lattices or non-periodic situations like Penrose lattices.