In 1968 Ornstein presented an argument showing that any two Bernoulli
shifts of the same entropy were isomorphic [01]. This settled a number of
long standing questions. Perhaps more importantly, though, it presented a
scheme by which a measurable isomorphism could be constructed. In 1975
Feldman, Weiss, Ornstein, and independently Katok developed a similar scheme
for characterizing the Kakutani equivalence class of the Bernoulli shifts and
irrational rotations J0,R,W], iFe], iKa]. Later it also became evident that
Dye's theorem ID], that any two ergodic finite measure preserving maps are
orbit equivalent, could also be proven by such a scheme. Fieldsteel in fact
used such a method iFiJ. With one last piece of evidence a general pattern
becomes evident. Del Junco has shown that even Kakutani equivalence can be
described as an orbit equivalence of a certain restricted type lDel.,Rj.
Thus in all three cases we are considering a certain equivalence relation
between ergodic finite measure preserving maps expressed as the existence of
isomorphic copies of the maps on the same orbit space (an orbit equivalence)
of a certain restricted type. For isomorphism we are simply saying the two
maps order the orbits in the same way. For even Kakutani equivalence we use
any of the del Junco characterizations. Orbit equivalence, of course, has
no restriction. Connected with each equivalence relation, we have a distin-
guished class of maps, finitely determined or very weakly Bernoulli for
isomorphism, finitely fixed or loosely Bernoulli for even Kakutani equival^*
ence, and ergodic for orbit equivalence. Within this distinguished class we
have a complete invariant for equivalence. In the case of isomorphism and
even Kakutani equivalence the invariant is entropy. For orbit equivalence
there is apparently no need for such an invariant.
What we will show here is that these three equivalence relations are
Received by the editors April 16, 198k.
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