An invariant of monotone equivalence determining the quotients of automorphisms monotonely equivalent to a Bernoulli shift

Two ergodic automorphisms of a Lebesgue space are called monotonely equivalent if they have metrically isomorphic induced automorphisms. We formulate properties of an automorphism of a Lebesgue space, similar to very weak Bernoulli and finitely determined. The difference is that instead of the Hamming metric on the space of words, we use a weaker metric $\rho^M$. These properties describe the class of quotient automorphisms of automorphisms monotonely equivalent to Bernoulli shifts.
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