abstract: In this joint work with M. Gu\`ardia, V. Kaloshin and Pablo Rold\'an, we consider the restricted three-body problem, a good model for a Sun-Jupiter-asteroid system, in the neighborhood of a mean motion resonance (mean motions are the frequencies of evolution around the Sun, as opposed to the slower, secular frequencies). "Restricted" means that the asteroid, having a zero mass, is influenced by, but does not itself influences, the two primaries.
We study the elliptic problem (where the primaries describe a non-circular Keplerian ellipse) as a perturbation of the circular problem, thus taking the eccentricity of Jupiter as a small parameter, while the mass of Jupiter itself is fixed at a realistic value. The main result is that there are some solutions along which the eccentricity of the asteroid varies significantly (uniformly with respect to small eccentricities of the primaries), while the semi major axis of the asteroid is approximately blocked at a resonant value.
As the eccentricity of the asteroid changes, the asteroid might undergo some close encounter or collision with some other planet (as in the case of asteroids of the Asteroid Belt, with Mars), a point where the restricted three-body problem stops being relevant. This mechanism is a possible explanation for the absence of asteroids in the so-called Kirkwood gaps observed in the Asteroid Belt, and is backed up by some plausible, conjectural estimates of the time of diffusion.