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

James O'Quinn

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.

Weak Mixing Transformation Which Is 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 which is, for all , weak--equivalent to and, for all , strong--equivalent to . 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 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 which are Shannon orbit equivalent to a given ergodic transformation.

Paper Structure

This paper contains 7 sections, 29 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. Shannon Orbit Equivalence
  3. Column Permutations of Towers
  4. Actions of $\mathbb{Z}^{2}$ Induced by a Sequence of Column-to-Tile Maps
  5. Summary of the Construction from Fieldsteel and Friedman
  6. Fieldsteel-Friedman Orbit Equivalence Is Shannon
  7. Weak Mixing

Key Result

Theorem 1.3

If $S:X\to X$ and $T:Y\to Y$ are ergodic probability measure preserving transformations such that $S$ and $T$ are free or $X$ and $Y$ are atomless, then $S$ and $T$ are orbit equivalent.

Theorems & Definitions (80)

  • Definition 1.1
  • Remark 1.2
  • Theorem 1.3
  • Definition 1.4
  • Remark 1.5
  • Remark 1.6
  • Definition 1.7
  • Definition 1.8
  • Remark 1.9
  • Theorem 1.10
  • ...and 70 more