A piecewise linear homeomorphism of the circle preserving rational points and periodic under renormalization

Abstract

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 (2007). When combined with the results by J. Hyde and J. Tatch Moore (2023), 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.