A K Counterexample Machine

We present a general method for constructing families of measure preserving transformations which are K and loosely Bernoulli with various ergodic theoretical properties. For example, we construct two K transformations which are weakly isomorphic but not isomorphic, and a K transformation with no roots. Ornstein’s isomorphism theorem says families of Bernoulli shifts cannot have these properties [6]. The construction uses a combination of properties from maps constructed by Ornstein and Shields, and Rudolph, and reduces the question of isomorphism of two transformations to the conjugacy of two related permutations [10][13].