It is well known from Ornstein's isomorphism theory [FO, O] that entropy is
a complete invariant of measure-theoretic isomorphism for mixing Markov chains.
In that result the isomorphism is a two-sided isomorphism, meaning that the iso-
morphism and its inverse are allowed to use the entire future and the entire past
of a point in order to determine the zeroth coordinate of its image. In a one-sided
isomorphism, the isomorphism and its inverse are allowed to use the entire past,
but none of the future. In [AMT] Markov chains were completely classified up to
one-sided measure-theoretic isomorphism, via an effective algorithmic procedure.
An intermediate notion of isomorphism is regular isomorphism. Here, the iso-
morphism and its inverse are allowed to use the entire past and a uniformly bounded
amount of the future. Regular isomorphisms were introduced in [FP] and further
studied by Parry and others (see [PT3, BT] and their references). One of our re-
sults will give necessary and sufficient conditions for a Markov chain to be regularly
isomorphic to a Bernoulli shift.
Ideas emanating from symbolic dynamics, in particular right-resolving and
right-closing maps of shifts of finite type (see, for example, [LM]), have a strong
bearing on one-sided isomorphism and regular isomorphism of Markov chains: If a
right-closing factor map is one-to-one almost everywhere then it defines a (rather
concrete) regular isomorphism between the shifts of finite type with respect to
their measures of maximal entropy. More generally, if the shifts of finite type are
endowed by Markov measures and the map preserves measure, then we obtain a
regular isomorphism from one of the Markov chains to the other. It was shown in
[BT] that two Markov chains are regularly isomorphic if and only if there exists a
Markov chain which factors onto each of them by a right-closing factor map which is
one-to-one almost everywhere. Thus, the study of regular isomorphism for Markov
chains largely reduces to the study of right-closing factor maps between Markov
Likewise, two Markov chains are one-sidedly isomorphic if and only if there
exists a Markov chain which factors onto each of them by a right-resolving map
which is one-to-one almost everywhere. This fact serves as the first step of the
classification of [AMT]. Right-closing maps form the closure of right-resolving
ones under block isomorphism, that is, measure-preserving topological conjugacy.
In this paper, we will completely answer the following questions for an arbitrary
Markov chain and a Bernoulli shift.