TITLE: Invariant manifolds of $L_3$ and horseshoe motion in the restricted three-body problem AUTHORS: Esther Barrabés^1, Mercè Ollé^2 (1) Departament d'Informàtica i Matemàtica Aplicada Universitat de Girona Ed. P-IV, Campus Montilivi, Av. Lluís Santaló s/n 17071 Girona, Spain. (2) Departament de Matemàtica Aplicada I ETSEIB, Universitat Politècnica de Catalunya, Diagonal 647, 08028 Barcelona, Spain E-Mails: barrabes@ima.udg.es, merce.olle@upc.edu ABSTRACT: In this paper, we consider horseshoe motion in the planar restricted three-body problem. On one hand, we deal with the families of horseshoe periodic orbits (which surround three equilibrium points called $L_3$, $L_4 $ and $L_5$), when the mass parameter $\mu $ is positive and small; we describe the structure of such families from the two-body problem ($\mu =0$). On the other hand, the region of existence of horseshoe periodic orbits for any value of $\mu \in (0,1/2]$ implies the understanding of the behaviour of the invariant manifolds of $L_3$. So, a systematic analysis of such manifolds is carried out. As well the implications on the number of homoclinic connections to $L_3$, and on the {\sl simple} infinite and {\sl double} infinite period homoclinic phenomena are also analysed. Finally, the relationship between the horseshoe homoclinic orbits and the horseshoe periodic orbits are considered in detail.