It is known that two Markov subshifts with the same entropy are "almost isomorphic" (up to a subset of measure 0) if the entropy is a logarithm of an integer (see R. L. Adler, L. W. Goodwyn, and B.Weiss. Equivalence of topological markov shifts.
Israel J. Math, 27(1):49--63, 1977). Is it true (known) if the entropy is not a logarithm of an integer? 


 **Update**  Basically I want to know if the result of AGW is true without the integer assumption. I would appreciate an answer of the form "yes plus reference" or "no plus reference" or "it is still unknown". 



Subshifts with the same entropy 
Look at Section 9 of
 <http://www-users.math.umd.edu/users/mmb/papers/openfinalsub3nov2007.pdf>


