Axiom A Diffeomorphism

Let $\phi:M\to M$ be a $C^1$ Diffeomorphism on a compact Riemannian Manifold $M$. Then $\phi$ satisfies Axiom A if the Nonwandering set $\Omega(\phi)$ of $\phi$ is hyperbolic and the Periodic Points of $\phi$ are Dense in $\Omega(\phi)$. 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).

See also Anosov Diffeomorphism, Axiom A Flow, Diffeomorphism, Dynamical System, Riemannian Manifold, Smale Horseshoe Map

References

Bowen, R. Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms. New York: Springer-Verlag, 1975.

Ott, E. Chaos in Dynamical Systems. New York: Cambridge University Press, p. 143, 1993.

Parry, W. and Pollicott, M. ``Zeta Functions and the Periodic Orbit Structure of Hyperbolic Dynamics.'' Astérisque No. 187-188, 1990.

Smale, S. ``Differentiable Dynamical Systems.'' Bull. Amer. Math. Soc. 73, 747-817, 1967.