Axiom A

Summary

A diffeomorphism satisfies Axiom A if its non-wandering set is a hyperbolic set in which periodic points are dense. Smale introduced the condition to single out a large class of systems whose dynamics can be described completely. For such systems the non-wandering set splits into finitely many basic pieces by the spectral decomposition theorem. Each piece is coded by a subshift of finite type. Anosov diffeomorphisms, Morse–Smale systems and the Smale horseshoe satisfy Axiom A.