The hyperbolic sets of Smale’s Axiom A systems have been defined in the topological setting by Ruelle. We will look at (co)homology and homotopy invariants for such systems both from a “classical” and noncommutative point of view. We will use symbolic dynamics to study a certain homology theory and its identification with the groupoid homology of equivariant sheaves. Finally we’ll exploit the triangulated structure of the KK-category to draw a connection with the operator K-theory of the C*-algebras associated to these dynamical systems.