Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2018, Director: 2018[en] This dissertation is divided into two parts. In the first part, we study the shift homeomorphism for a finite or denumerable alphabet set and we give sufficient conditions to conjugate an application in a Cantor subset of the domain with the shift. These conjugations have very important dynamic consequences. In the second part, we apply the results to Sitnikov’s problem. This is a restricted three body problem in which two bodies describe elliptic orbits while the center of mass is at rest. The third body is moving on a line perpendicular to the plane of motion of the first two bodies and going through the center of mass. The goal is to describe the motion of the third mass point which is periodically excited by the first two. This way we get the existence of a wide variety of oscillatory movements for this problem.