Abstract
This chapter reviews the notation, terminology, and some useful results of the theory of sets, topological spaces, metric spaces, and Banach spaces. It also presents a review of definitions and results on differentiability and differentiable manifolds in infinite dimension called Banach manifolds. The chapter discusses an important special class of dynamical systems and presents an analysis of the Axiom A systems. Usually, the discussion of Axiom A dynamical systems is restricted to diffeomorphisms and flows on compact manifolds. To pursue the study of Axiom A systems beyond the geometric treatment, it is necessary to use Markov partitions and symbolic dynamics. An Axiom A system closely resembles the geometric Lorenz attractor. The study of nonuniformly hyperbolic dynamical systems in general requires ergodic theory.
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