ABSTRACT
This chapter investigates dynamical phenomena which are higher dimensional in nature. For the study of higher dimensional dynamical systems, the most important new ingredients are techniques from linear algebra. The chapter reviews some of the standard techniques and introduces a few more advanced topics. It provides a classical example due to Smale, the horseshoe map. This was the first example of a diffeomorphism which had infinitely many periodic points and yet was structurally stable. The chapter addresses a completely different class of dynamical system, the Anosov systems or hyperbolic toral automorphisms. It also provides a third type of dynamical phenomenon which is higher dimensional in nature, the attractor. Roughly speaking, an attractor is an invariant set to which all nearby orbits converge. Hence attractors are the sets that one "sees" when a dynamical system is iterated on a computer. The chapter explains two new and much more complicated attractors, the solenoid and the Plykin attractor.
