  In 1970, Donald Ornstein proved a landmark result in dynamical systems, viz.,
two Bernoulli systems with the same entropy are isomorphic except for a measure
0 set. Keane and Smorodinsky gave a finitary proof of this result. They also
indicated how one can generalize the result to mixing Markov Shifts. We adapt
their construction to show that if two computable mixing Markov systems have
the same entropy, then there is a layerwise computable isomorphism defined on
all Martin-Lof random points in the system. Since the set of Martin-Lof random
points forms a measure 1 set, it implies the classical result for such systems.
  This result uses several recent developments in computable analysis and
algorithmic randomness. Following the work by Braverman, Nandakumar, and Hoyrup
and Rojas introduced discontinuous functions into the study of algorithmic
randomness. We utilize Hoyrup and Rojas' elegant notion of layerwise computable
functions to produce the test of randomness in our result. Further, we use the
recent result of the effective Shannon-McMillan-Breiman theorem, independently
established by Hochman and Hoyrup to prove the properties of our construction.
  We show that the result cannot be improved to include all points in the
systems - only trivial computable isomorphisms exist between systems with the
same entropy.
