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