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 $F_\rho$, where the parameter $\rho$ ranges over irrational numbers in $(0,1)$. Each $F_\rho$ is a Kwapisz-like diffeomorphism with a 2-dimensional non-polygonal rotation set $$\Lambda'_\rho = \operatorname{conv}\left(\left\{(\pm\frac{\lceil m\rho \rceil}{m+n+1}, \pm\frac{\lceil n\rho \rceil}{m+n+1}): m, n \in \mathbb{N} _0, \lceil m\rho\rceil - m\rho<\rho,\lceil n\rho\rceil - n\rho<\rho\right\}\right)$$ whose extreme point set contains exactly four (two-sided) accumulation points. We define the roundness of $\Lambda'_\rho$ as the ratio $R_\rho=\frac{\operatorname{Area}(\Lambda'_\rho)}{\pi\rho^2}$, and give its upper and lower bounds in terms of $\rho$. $R_\rho$ is neither monotone nor continuous.