Pré-Publication, Document De Travail Année : 2025

When Periodicity Fails to Guarantee the Existence of Rotation: A Counterexample on $\mathbb{T}^3$

Résumé

In this manuscript, we construct an explicit counterexample of a smooth \(C^{\infty}\), periodic dynamical system on the torus \(\mathbb{T}^3\) for which the rotation vector exists in a weak sense, but fails to exist in the strong sense of bounded deviation (also referred to as {\it {frequencies}} in parts of the physics and biology literature). The construction exploits Liouville-type arithmetic properties and demonstrates that smoothness and periodicity alone do not ensure bounded deviation, even within the class of integrable systems.

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Dates et versions

hal-05095775 , version 1 (03-06-2025)
hal-05095775 , version 2 (08-02-2026)
hal-05095775 , version 3 (21-04-2026)

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W Oukil. When Periodicity Fails to Guarantee the Existence of Rotation: A Counterexample on $\mathbb{T}^3$. 2026. ⟨hal-05095775v3⟩

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