On roundness of rotation sets

2025/10/09 by Perrot, Boris, Boroński, Jan, Clark, Alex
#Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2510.08235

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= conv(\(±(\lceil mρ\rceil)/(m+n+1), ±(\lceil nρ\rceil)/(m+n+1)): m, n ∈ ℕ ₀, \lceil mρ\rceil - mρlt;ρ,\lceil nρ\rceil - nρlt;ρ\) whose extreme point set contains exactly four (two-sided) accumulation points. We define the roundness of Λ'_ρ as the ratio R_ρ=(Area(Λ'_ρ))/(πρ²), and give its upper and lower bounds in terms of ρ. R_ρ is neither monotone nor continuous.

Citations

• Topological horseshoes for surface homeomorphisms
• The Franks-Misiurewicz conjecture for extensions of irrational rotations
• Forcing theory for transverse trajectories of surface homeomorphisms
• New rotation sets in a family of torus homeomorphisms
• Denjoy constructions for fibred homeomorphisms of the torus
• Construction of curious minimal uniquely ergodic homeomorphisms on manifolds: the Denjoy-Rees technique
• A toral diffeomorphism with a nonpolygonal rotation set
• Every convex polygon with rational vertices is a rotation set
• Rotation Sets for Maps of Tori

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