 In this case, the symbolic representation is a powerful tool for investigating the properties of the dynamical system ("M","φ").A Markov partition is a finite cover of the invariant set of the manifold by a set of curvilinear rectangles formula_1 such that


Here, formula_10 and formula_11 are the unstable and stable manifolds of "x", respectively, and formula_12 simply denotes the interior of formula_13.These last two conditions can be understood as a statement of the Markov property for the symbolic dynamics; that is, the movement of a trajectory from one open cover to the next is determined only by the most recent cover, and not the history of the system.