Weakly mixing transformation that is Shannon orbit equivalent to a given ergodic transformation

We prove that every ergodic transformation is Shannon orbit equivalent to a weakly mixing transformation. The proof is based on the techniques introduced by Fieldsteel and Friedman to show that there is a mixing transformation for a given ergodic transformation T that is, for all a ≥ 1 a≥ 1 a greater than or equals 1 , weak- a -equivalent to T and, for all b ∈ ( 0 , 1 ) b∈ (0,1) b element of left parenthesis 0 comma 1 right parenthesis , strong- b -equivalent to T . In particular, we adapt the construction of Fieldsteel and Friedman by which they permute the columns of each Rokhlin tower in a sequence of rapidly growing Rokhlin towers so that the corresponding cocycles converge to an orbit equivalence cocycle of T such that the resulting transformation and orbit equivalence have the desired properties. In addition to this, we demonstrate a flexible method for obtaining actions of Z 2 Z² double struck upper Z squared that are Shannon orbit equivalent to a given ergodic transformation.