A piecewise linear homeomorphism of the circle which is periodic under renormalization

TL;DR

This paper constructs a specific piecewise linear homeomorphism of the circle with a rational rotation number and periodic renormalization, providing counterexamples to existing questions and insights into subgroup embeddings.

Contribution

It introduces a novel example of a piecewise linear circle homeomorphism with periodic renormalization, answering a question of Calegari and exploring subgroup embedding properties.

Findings

Constructed a homeomorphism with rotation number √2−1 and slopes as powers of 2/3.

Showed that the renormalization procedure becomes periodic for this homeomorphism.

Provided evidence on subgroup embedding and computational observations.

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. 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.