On the existence of Siegel curves. (Sur l’existence des courbes de Siegel.) (French) Zbl 0943.37023

Hmissi, Mohamed; Sieveking, Malte

Grazer Math. Ber. 334, 139-146 (1997).

Summary: Let \(\Psi: \mathbb{R}\times T^2\mapsto T^2\) be a dynamical system without periodic trajectories, on the two-dimensional torus \(T^2\). In this work we suppose that \(\Psi\) is (only) continuous and we prove that there exists on \(T^2\) a simple closed curve which cuts every half-trajectory of the dynamical system \((T^2,\Psi)\). This result is a generalization of a theorem of C. L. Siegel who assumed that \(\Psi\) is defined by a vector field with continuous derivatives of the second order.

MSC:

37E35	Flows on surfaces
37C27	Periodic orbits of vector fields and flows

Keywords:

periodic trajectories; two-dimensional torus