Birkhoff's conjecture and almost-periodic motions on torus \(T^2\) (Q1368235)

The paper deals with the Birkhoff conjecture for the existence of an analytical differential equation that has recurrent motion but has no almost-periodic motions [\textit{G. D. Birkhoff}, Bull. Soc. Math. Fr. 40, 305-323 (1912; JFM 43.0818.01)]. \textit{T. Ding} [Acta Math. Sin. 24, 64-68 (1981; Zbl 0473.34042)] verifies affirmatively the conjecture for a system on the torus \(T^2\) of the type \[ du/dt=1/F(u,v),\qquad dv/dt=\lambda/F(u,v).\tag{1} \] (the number \(\lambda\) is called then the Birkhoff number). On the other hand it is known that if the \(C^2\) system on \(T^2\) \[ du/dt=g(u,v),\qquad dv/dt=g(u,v)f(u,v).\tag{2} \] has neither critical points nor periodic motions, then every motion of \((2)\) is recurrent. This paper proofs results related to this matter, namely: \(1^0\). For almost all (in the sense of the Lebesgue measure) real numbers \(\lambda\), if \(f,g\in C^5\), \(g\neq 0\) and the rotation number \(\mu(f)\) of the system \( du/dt=1\), \(dv/dt=f(u,v) \) is \(\lambda\), then every motion of \((2)\) is Lyapunov stable and almost periodic. \(2^0\). A Birkhoff number must be a Liouville number (hence in particular not a real algebraic number), and the set of Birkhoff numbers is of Lebesgue measure zero.