Let ϕ:M->M be a C^1 diffeomorphism on a compact Riemannian manifold M. Then ϕ satisfies Axiom A if the nonwandering set Ω(ϕ) of ϕ is hyperbolic and the periodic points of ϕ are dense in Ω(ϕ). Although it was conjectured that the first of these conditions implies the second, they were shown to be independent in or around 1977. Examples include the Anosov diffeomorphisms and Smale horseshoe map. In some cases, Axiom A can be replaced by the condition that the diffeomorphism is a hyperbolic diffeomorphism on a hyperbolic set (Bowen 1975, Parry and Pollicott 1990).
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