Weak Mixing Transformation Which Is Shannon Orbit Equivalent to a Given Ergodic Transformation

TL;DR

This paper proves that every ergodic transformation can be Shannon orbit equivalent to a weak mixing transformation, introducing a flexible method to construct such transformations and actions of ^2.

Contribution

It establishes the existence of weak mixing transformations Shannon orbit equivalent to any ergodic transformation and develops a flexible construction method.

Findings

Every ergodic transformation is Shannon orbit equivalent to a weak mixing transformation.

A construction method based on Rokhlin towers is adapted for this purpose.

The method extends to actions of ^2 that are Shannon orbit equivalent to a given ergodic transformation.

Abstract

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 $a\geq1$, weak-$a$-equivalent to $T$ and, for all $b\in(0,1)$, 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 $\mathbb{Z}^{2} $ which are Shannon orbit equivalent to a given ergodic transformation.