On zeta functions and Anosov diffeomorphisms

This thesis considers some problems in Dynamical Systems concerned with zeta functions and with Anosov diffeomorphisms. In chapter 1 Bowen's method of expressing a basic set of an Axiom A diffeomorphism as a quotient of a subshift of finite type is used ,to calculate the numbers of periodic points of the diffeomorphism and show that its zeta function is ration31 which gives an affirmative answer to a question of Smale. The rest of the thesis is concerned with Anosov diffeomorphisms of nilmanifolds.Chapter 2 contains some facts about nilmanifolds describing them as twisted products of tori. Anilmanifold has a maximal torus factor. A hyperbolic nilmanifold automorphism projects onto an automorphism of this torus and we , say it has the toral automorphism as a factor. In chapter 3 we generalize this situation to show that many diffeomorphisms of other manifolds have toral automorphisms as factors and give some examples. In the last chapter we use a spectral sequence associated to another decomposition of a nilrnanifold into tori to calculate the Lefschetz number of any diffeomorphism of the nilmanifold. This enables us to prove a necessary condition on the map induced by an Anosov, diffeomorphism of a nilmanifold on its fundamental group. Then we consider the question of finding hyperbolic automorphisms of nilmanifolds from the decomposition into tori. Fin311y we calculate the zeta function of such an automorphism