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arXiv · 2410.13946

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

James O'Quinn
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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.

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James O'Quinn. 2024-10-17. Weak Mixing Transformation Which Is Shannon Orbit Equivalent to a Given Ergodic Transformation. https://arxiv.org/abs/2410.13946

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