Weak Mixing Transformation Which Is Shannon Orbit Equivalent to a Given Ergodic Transformation
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866929549478985728 |
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| author | O'Quinn, James |
| author_facet | O'Quinn, James |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_13946 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Weak Mixing Transformation Which Is Shannon Orbit Equivalent to a Given Ergodic Transformation O'Quinn, James Dynamical Systems 37A05, 37A20, 37A25, 37A35 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. |
| title | Weak Mixing Transformation Which Is Shannon Orbit Equivalent to a Given Ergodic Transformation |
| topic | Dynamical Systems 37A05, 37A20, 37A25, 37A35 |
| url | https://arxiv.org/abs/2410.13946 |