2024/10/17 by O'Quinn, James
#37A05 #37A20 #37A25 #37A35 #Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.2410.13946
We prove that every ergodic transformation is Shannon orbit equivalent to a weak 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 which is, for all a≥1, weak-a-equivalent to T and, for all b∈(0,1), strong-b-equivalent to T. In particular, we will 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 will demonstrate a flexible method for obtaining actions of ℤ2 which are Shannon orbit equivalent to a given ergodic transformation.