We prove that certain Gibbs measures on subshifts of finite type are nonsingular and ergodic for certain countable equivalence relations, including the orbit relation of the adic transformation (the same as equality after a permutation of finitely many coordinates). The relations we consider are defined by cocycles taking values in groups, including some nonabelian ones. This generalizes (half of) the identification of the invariant ergodic probability measures for the Pascal adic transformation as exactly the Bernoulli measures---a version of de Finetti's Theorem. Generalizing the other half, we characterize the measures on subshifts of finite type that are invariant under both the adic and the shift as the Gibbs measures whose potential functions depend on only a single coordinate. There are connections with and implications for exchangeability, ratio limit theorems for transient Markov chains, interval splitting procedures, `canonical' Gibbs states, and the triviality of remote sigma-fields finer than the usual tail field.

/pdf/symmetric-gibbs-measures-1u9rr6sybb.pdf

Symmetric Gibbs measures

The theory of dynamical systems with invariant measure, or ergodic theory, is one of those domains of mathematics whose form changed radically in the last 15-20 years. This has to do both with the internal problems of ergodic theory End with its connections with other parts of mathematics. In ergodic theory itself, there arose the theory of entropy of dynamical systems, whose origin was in the papers of A. N. Kolmogorov. Recently, remarkable progress has been made by Ornstein and his collaborators in the problem of metric isomorphism of Bernoulli automorphisms and K-automorphisms, i.e., dynamical systems with very strong mixing properties. Another important event of recent times is a new, profound connection of ergodic theory with statistical mechanics, not only enriching ergodic theory itself, but also leading to new progress in the mathematical problems of statistical mechanics. Both of the circles of problems mentioned occupy a significant place in this survey. On the other hand, a series of applications of ergodic theory is intentionally excluded from our survey. This has to do in the first place with physics papers, which do not contain strictly mathematical results. Also, we shall not dwell on many mathematical papers connected in one way or another with ergodic theory, but not relating directly to it. Such, for example, are the papers of G. A. Margulis and Mostow on quasiconformal mappings of manifolds of negative curvature, in which the ergodicity of flows on such manifolds is used, or the papers of Glimm and Jaffe, which can be partially interpreted as investigations of the mixing properties of some dynamical systems which arise in quantum field theory.

Theory of dynamical systems and general transformation groups with invariant measure

The intention of the effort presented is to characterize the solution to a general class of preventive maintenance optimization problems. Successive lengths of operating periods and the successive repair times for a piece of equipment are each assumed to be i.i.d. Weibull random variables. The equipment is subjected to an age replacement preventive maintenance policy and the successive preventive maintenance times are assumed to be i.i.d. Weibull random variables. An optimal age replacement policy is a policy, T/sub age/*, that maximizes A/sub avg/(T/sub use/), the average availability of the equipment over its useful life (T/sub use/</spl infin/). The relationship between T/sub age/* and the parameters of the previously mentioned Weibull distributions is approximated by a first-order polynomial function. The model is formulated through regression analysis of data obtained through simulation-optimization techniques. Model validation is presented. In general, the model predicts an optimal T/sub age/ close to the true optimal. The model, however, tends to prescribe less frequent preventive maintenance than the true optimal policy and is not useful in some in situations where the true optimal T/sub age/ is very small.

http://ieeexplore.ieee.org/iel5/5250/14226/00653714.pdf

Characterization of optimal age-replacement policies

We discuss some recent work on various constructions that accumulate or remove information within dynamical systems: tail fields, numeration systems and formal languages (especially of s-shifts), and factor mappings between symbolic or tiling dynamical systems.

Information Compression and Retention in Dynamical Processes

It is an important problem to determine the spectral type of automorphisms or flows on a probability measure space. We shall deal with a unitary operator U and a 1-parameter group of unitary operators { U t } on a separable Hilbert space H, and discuss their spectral types, although U and { U t } are not necessarily supposed to be derived from an automorphism or a flow respectively.

/pdf/spectral-type-of-one-parameter-group-of-unitary-operators-1wyd5uu86n.pdf

Spectral Type of One-Parameter Group of Unitary Operators with Transversal Group

Preventive maintenance of a 1-unit system with two types of repair

/pdf/on-a-complete-metric-invariant-for-group-automorphisms-on-1h2v7w5li6.pdf

Masasi Kowada

Papers

Spectral Type of One-Parameter Group of Unitary Operators with Transversal Group

Preventive maintenance of a 1-unit system with two types of repair

On a complete metric invariant for group automorphisms on the torus