Antonio Ureña

We present some results on the absence of a wide class of invariant measures for dynamical systems possessing attractors. We then consider a generalization of the classical nonholonomic Suslov problem which shows how previous investigations of existence of invariant measures for nonholonomic systems should necessarily be extended beyond the class of measures with strictly positive $C^1$ densities if one wishes to determine dynamical obstructions to the presence of attractors.

The Lambert problem consists in connecting two given points in a given lapse of time under the gravitational influence of a fixed center. While this problem is very classical, we are concerned here with situations where friction forces act alongside the Newtonian attraction. Under some boundedness assumptions on the friction, there exists exactly one rectilinear solution if the two points lie on the same ray, and at least two solutions traveling in opposite directions otherwise.

Poincaré and, later on, Carathéodory, showed that the Floquet multipliers of 1-dimensional periodic curves minimizing the Lagrangian action are real and positive. Even though Carathéodory himself observed that this result loses its validity in the general higherdimensional case, we shall show that it remains true for systems which are reversible in time. In this way, we also generalize a previous result by Offin on the hyperbolicity of nondegenerate symmetric minimizers. Our arguments rely on the higher-dimensional generalizations of the Sturm theory which were developed during the second half of the twentieth century by several authors, including Hartman, Morse or Arnol’d.