On roundness of rotation sets

TL;DR

The paper analyzes how closely a non‑polygonal two‑dimensional rotation set $Λ'_ρ$ of a torus diffeomorphism can resemble a disk of radius $ρ$ by introducing a Kwapisz‑like family $F_ρ$ with rotation set $Λ'_ρ$ for irrational $ρ\in(0,1)$. It provides a precise combinatorial description of the underlying rotation set via index sets $I_ρ$, identifies a best diagonal point $D_ρ=(d_ρ,d_ρ)$, and determines when this point is an extreme point, revealing a nuanced extremality picture. The authors derive sharp lower and upper bounds for the roundness $R_ρ=\frac{\operatorname{Area}(Λ'_ρ)}{π ρ^2}$ in terms of $d_ρ$ and the slope parameter $γ_ρ=-ρ\lfloor 1/ρ\rfloor$, establishing asymptotic limits, and showing that $R_ρ$ is neither monotone nor continuous with respect to $ρ$ (including pronounced jumps near rational endpoints). An appendix generalizes Kwapisz’s construction to place rotation sets in all four quadrants, illustrating that the family of rotation sets is not simply a rescaling of a single template. Altogether, the work provides quantitative insights into how closely rotation sets can approximate a disk and clarifies the geometry of the extremal structure in this non‑polygonal setting.

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

Figures (8)

• Figure 1: Lower (red) and upper (blue) bounds for roundness $R_\rho$ of the rotation set $\Lambda'_\rho$ as a function of $\rho$
• Figure 2: An approximation of the non-polygonal rotation set $\Lambda'_\rho$ for $\rho=0.5^+$ (left) and $\rho=0.6^-$ (right)
• Figure 3: First 100 terms of sequence $(\alpha_{n}^\rho)_{n \in \mathbb{N}\setminus\left\{0\right\}}$ for $\rho=0.93+\pi.10^{-5}$. Red points are such that $n\notin I_\rho$ and blue ones are such that $n\in I_\rho$.
• Figure 4: First 100 terms of sequence $(\alpha_{n}^\rho)_{n \in \mathbb{N}\setminus\left\{0\right\}}$ for $\rho=0.31+\pi.10^{-5}$. Red points are such that $n\notin I_\rho$ and blue ones are such that $n\in I_\rho$.
• Figure 5: First 100 terms of sequence $(x_{n, n}^\rho)_{n \in \mathbb{N}\setminus\left\{0\right\}}$ for $\rho=0.93+\pi.10^{-5}$. Red points are such that $n\notin I_\rho$ and blue ones are such that $n\in I_\rho$.

Theorems & Definitions (73)

• Theorem 1.1: Kwapisz, 1994 Kwapisz2
• Theorem 1.2
• Theorem 1.3: Bounds on roundness of $\Lambda'_\rho$
• Definition 1
• Theorem 2.1: Misiurewicz-Ziemian, 1989
• Definition 2
• Definition 3
• Definition 4
• Definition 5
• Proposition 1