Journal ArticleDOI
Finitely determined implies very weak Bernoulli
TLDR
In this article, it was shown that if a process is finitely determined then it is very weak Bernoulli (VWB) and combined with known results, it is shown that a process can be isomorphic to a Bernoullis shift if and only if it satisfies an asymptotic independence condition.Abstract:
It is shown that if a process is finitely determined then it is very weak Bernoulli (VWB). Combined with known results this says that a process is isomorphic to a Bernoulli shift if and only if it satisfies an asymptotic independence condition, namely that of being VWB.read more
Citations
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Journal ArticleDOI
Basic properties of strong mixing conditions. A survey and some open questions
TL;DR: In this paper, an update of, and a supplement to, a 1986 survey paper by the author on basic properties of strong mixing conditions is presented, which is an extension to the survey paper.
Book ChapterDOI
Basic Properties of Strong Mixing Conditions
TL;DR: A survey of the properties of strong mixing conditions for sequences of random variables can be found in this paper, where the focus will be on the structural properties of these conditions, and not at all on limit theory.
Journal ArticleDOI
NewK-automorphisms and a problem of Kakutani
TL;DR: In this paper, a property called loose Bernoulliness (LB) was introduced for 1-1 measure-preserving transformations of probability spaces, which is invariant under taking factors, inducing, and tower-building.
Journal ArticleDOI
Statistical properties of chaotic systems
TL;DR: The theory of smooth dynamical systems and the theory of abstract dynamical system (ergodic theory) have for many years developed quite independently of one another as discussed by the authors, and these theories have now matured to the point where they can be combined to shed light on the nature of chaotic behavior
Journal ArticleDOI
The structure of skew products with ergodic group automorphisms
TL;DR: In this article, it was shown that ergodic automorphisms of compact groups are Bernoulli shifts, and that skew products with such automomorphisms are isomorphic to direct products, and the set of possible values for entropy is one of two alternatives depending on the answer to an open problem in algebraic number theory.
References
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Journal ArticleDOI
Disjointness in ergodic theory, minimal sets, and a problem in diophantine approximation
TL;DR: The objects of ergodic theory -measure spaces with measure-preserving transformation groups- will be called processes, those of topological dynamics-compact metric spaces with groups of homeomorphisms-will be called flows, and what may be termed the "arithmetic" of these classes of objects is concerned.
Journal ArticleDOI
Geodesic flows are Bernoullian
TL;DR: In this article, a geometric method is developed for proving that transformations are isomorphic to Bernoulli shifts on surfaces of negative curvature, and the method is applied to the geodesic flows.
Journal ArticleDOI
Convolution of invariant measures, maximal entropy
TL;DR: The behavior of the entropy h as a function of/z is studied, and it is shown that h achieves its maximum value for Haar measure m, if T is ergodic with respect to m, and if h(m) is finite, then m is uniquely characterized as that measure which maximizes the entropy.
Journal ArticleDOI
Subsequences of normal sequences
TL;DR: In this paper, the authors characterized a set of indices τ = τ(0)<τ(1)<…} such that for any normal sequence (α, α(1),…) of a certain type, the subsequence (α(τ(0)), α(τ (1)),…) is a normal sequence of the same type.