A piecewise linear homeomorphism of the circle which is periodic under renormalization
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866914106635714560 |
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| author | Belk, James Hyde, James Moore, Justin Tatch |
| author_facet | Belk, James Hyde, James Moore, Justin Tatch |
| contents | We demonstrate the existence of a piecewise linear homeomorphism $f$ of $\mathbb{R}/\mathbb{Z}$ which maps rationals to rationals, whose slopes are powers of $\frac{2}{3}$, and whose rotation number is $\sqrt{2}-1$. This is achieved by showing that a renormalization procedure becomes periodic when applied to $f$. Our construction gives a negative answer to a question of D. Calegari. When combined with work of the 2nd and 3rd authors, our result also shows that $F_{\frac{2}{3}}$ does not embed into $F$, where $F_{\frac{2}{3}}$ is the subgroup of the Stein-Thompson group $F_{2,3}$ consisting of those elements whose slopes are powers of $\frac{2}{3}$. Finally, we produce some evidence suggesting a positive answer to a variation of Calegari's question and record a number of computational observations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2211_05825 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A piecewise linear homeomorphism of the circle which is periodic under renormalization Belk, James Hyde, James Moore, Justin Tatch Group Theory Mathematical Physics Dynamical Systems 20F38, 37E45, 37E10, 54H15 We demonstrate the existence of a piecewise linear homeomorphism $f$ of $\mathbb{R}/\mathbb{Z}$ which maps rationals to rationals, whose slopes are powers of $\frac{2}{3}$, and whose rotation number is $\sqrt{2}-1$. This is achieved by showing that a renormalization procedure becomes periodic when applied to $f$. Our construction gives a negative answer to a question of D. Calegari. When combined with work of the 2nd and 3rd authors, our result also shows that $F_{\frac{2}{3}}$ does not embed into $F$, where $F_{\frac{2}{3}}$ is the subgroup of the Stein-Thompson group $F_{2,3}$ consisting of those elements whose slopes are powers of $\frac{2}{3}$. Finally, we produce some evidence suggesting a positive answer to a variation of Calegari's question and record a number of computational observations. |
| title | A piecewise linear homeomorphism of the circle which is periodic under renormalization |
| topic | Group Theory Mathematical Physics Dynamical Systems 20F38, 37E45, 37E10, 54H15 |
| url | https://arxiv.org/abs/2211.05825 |