Bernoulli Schemes of the Same Entropy are Finitarily Isomorphic

Aspects of Ergodic, Qualitative and Statistical Theory of Motion

“…It was followed by several deep explicit constructions of the codes that realize the isomorphisms. The first one, due to Monroy and Russo, [MR75], concerned a very special case, but it introduced, in that case, some new ideas: such a construction has been improved by the constructions of Keane and Smorodinski, who, with the use of new and deep ideas, explicitly realized the code of the isomorphism between two isentropic Bernoulli schemes, see [KS79]. Such codes are "constructive" in the sense that it is possible to construct an arbitrarily prefixed number of values of the elements of the sequence σ ′ image, in the isomorphism in question, of a sequence σ by making use of an algorithm that can be implemented on a computer, so that it requires a finite time for almost all the sequences σ (randomly chosen with respect to the measure of one of the two Bernoulli schemes).…”

Finitary Codes, a short survey

“…, b} Z , the product σ-algebra is B, product measure is ν = q Z and the left shift transformation is T , entropy of T ish = h(q). In 1969 Ornstein (see [20]) proved that Bernoulli shifts of the same entropies were isomorphic and also showed the following: Before we continue, let us mention that the isomorphism result [11] relies upon a beautiful refinement and improvement of the methods developed for unequal entropies case. An excellent exposition of the finitary isomorphism result (which has become standard in ergodic theory) appeared in Petersen's book [24] (see also [4] or [23]).…”

“…The marker method of [11] and [12] was extended by Smorodinsky ([31]) to prove that m-dependent processes of equal entropy were finitarily isomorphic. Let us recall here that a stationary process is called m-dependent if its past and future become independent, if separated by m units of time.…”

“…We say that dynamical systems are finitarily isomorphic with finite expected code times (fect) if both the finitary isomorphism φ and its inverse φ A systematic study of finitary coding had begun with the works of Keane and Smorodinsky ([10], [11], [12]), and Denker and Keane ([5], [6]). In their 1977 paper [10], and in subsequent articles [11], [12], Keane and Smorodinsky developed a theory and methodology of finitary coding, creating a new area of research, which has been (and is still being) extended in a multitude of ways. In this note we want to recall the marker methods of [10], and later discuss the existence of finitary homomorphism with finite expected code length or a finitary isomorphism with fect.…”

Book Review: Entropy in dynamical systems

“…In other words, there is an efficient digitalization. Further research has shown that Bernoulli shifts with the same entropy over finite alphabets are finitarily isomorphic, which is to say there exists an almost-continuous measure-conjugacy Ψ with respect to the product topology [KS79,Se06]. In other words, a Bernoulli shift can be effectively recoded into an arbitrary second Bernoulli shift of the same entropy.…”