The simplest non trivial dynamical systems are the hyperbolic systems. The prototypical example is Smale’s Horseshoe, see Fig. 1.3 . A hyperbolic dynamical system is a differentiable system characterized by specific infinitesimal properties at each point. Namely, the tangent space at each point can be split into two parts, the stable and the unstable subspaces. The iterates of the derivatives along the unstable subspace expand exponentially fast while, the iterations of the derivatives along the stable subspace contract exponentially fast. This infinitesimal information of the map has global topological consequences. The starting point to describe the global topological structure of a hyperbolic system is the Stable Manifold Theorem.