TITLE
Consecutive quasi-collisions in the planar circular RTBP}

AUTHORS
Joaquim Font, Ana Nunes and Carles Sim\'o 

J.F. Departament de Matem\`atica Aplicada i An\`alisi
     Universitat de Barcelona
     Gran Via 585, 08007 Barcelona
     e-mail:quim@maia.ub.es
A.N. MAFUL/DFFCUL
     Universidade de Lisboa
     Av. Prof. Gama Pinto 2, 1619-003 Lisboa
     e-mail:anunes@lmc.fc.ul.pt
C.S. Departament de Matem\`atica Aplicada i An\`alisi
     Universitat de Barcelona
     Gran Via 585, 08007 Barcelona
     e-mail:carles@maia.ub.es 

ABSTRACT: 
In this paper we consider the planar
circular restricted three body problem and, in particular,
the existence of orbits which undergo consecutive close encounters with
the small primary. The number of revolutions of the small bodies around the 
larger one between successive encounter can be chosen to be two arbitrary 
sequences of natural number, with constraints depending on the Jacobi constant. 
We prove that such orbits exist as a consequence of
the fact that, when the mass parameter $\mu $ is small, the first return map 
defined on a region of phase space whose projection is a  circle around the 
small primary is a 'horseshoe' map. The proof is constructive, in the sense 
that it is based on the computation of an approximate expression for this 
return map. When $\mu $ is small, the approximate return map contains the 
essential information about the dynamics from the quantitative as well as 
from the qualitative point of view. Using this information, we have been 
able to carry out a numerical study of this problem for $\mu $ up to $10^{-3}$.
