On roundness of rotation sets

Abstract

Motivated by the question whether a round disk can be realized as the rotation set of a torus diffeomorphism, we study the roundness of rotation sets of a parametric family of torus diffeomorphisms $F_{\ddot{I}}$, where the parameter $\ddot{I}$ ranges over irrational numbers in $(0,1)$. Each $F_{\ddot{I}}$ is a Kwapisz-like diffeomorphism with a 2-dimensional non-polygonal rotation set $$\hat{I}{›}_{\ddot{I}}^{\prime }=\mathrm{conv}\left(\left\{(\pm \frac{\lceil m\ddot{I}\rceil }{m+n+1},\pm \frac{\lceil n\ddot{I}\rceil }{m+n+1}):m,n\in {\mathbb{N}}_0,\lceil m\ddot{I}\rceil -m\ddot{I}<\ddot{I},\lceil n\ddot{I}\rceil -n\ddot{I}<\ddot{I}\right\}\right)$$ whose extreme point set contains exactly four (two-sided) accumulation points. We define the roundness of $\hat{I}{›}_{\ddot{I}}^{\prime }$ as the ratio $R_{\ddot{I}}=\frac{\mathrm{Area}(\hat{I}{›}_{\ddot{I}}^{\prime })}{\ddot{I}€\ddot{I}{}^2}$, and give its upper and lower bounds in terms of $\ddot{I}$. $R_{\ddot{I}}$ is neither monotone nor continuous.