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

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.