A technical note on the arithmetic cone of smooth periodic vector fields

Abstract

The present note identifies the fundamental mechanism governing the extension of the contraction method. Although every periodic vector field defined by a finite trigonometric polynomial admits a bounded deviation from a linear drift, this property fails in general for smooth periodic vector fields. We show that the contraction argument underlying the original proof extends to any field satisfying a natural uniform summability condition, and that the counterexample violates this condition, thereby revealing the obstruction that prevents the method from extending beyond the finite-spectrum setting. For a smooth periodic vector field on the $n$-torus, the contraction method for establishing strong rotation vectors extends only to those asymptotic directions $\ddot{I}\in {\mathbb{R}}^n$ for which a certain spectral sum remains uniformly bounded along a sequence of rational approximations. We introduce the "arithmetic cone" $\mathfrak{C}(f)$, defined as the set of all $\ddot{I}$ admitting such an approximation. We establish its basic algebraic property: it is a cone. We prove that, under a uniform contraction condition, every element of $\mathfrak{C}(f)$ yields a strong rotation vector for the dynamics. The construction reveals a precise link between the Fourier asymptotics of $f$ and the arithmetic of admissible rotation directions. In the second part, we introduce the class of "spectrally admissible" fields ${\mathcal{A}}_{spec}$, for which the cone of the augmented field equals the whole space, and we show that it contains all finite trigonometric polynomials.

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