TITLE: Subshifts with the same entropy 
QUESTION [6 upvotes]: 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".

REPLY [2 votes]: The main theorem in Adler, Roy L.; Marcus, Brian
"Topological entropy and equivalence of dynamical systems"
doesn't use the fact that the entropy is log of a natural number. However it states that two ergodically supported topological Markov shifts are almost topologically conjugated if and only if them have the same topological entropy and the same ergodic period.
There is a work of Wenxiang Sun which extends this result for general expansive, ergodically supprted maps that have the shadowing property:
http://iopscience.iop.org/0951-7715/13/3/309