Axiom A

Definition

Let M be a smooth manifold with a diffeomorphism f: M→M. Then f is an axiom A diffeomorphism if the following two conditions hold:

1. The nonwandering set of f, Ω(f), is a hyperbolic set and compact.
2. The set of periodic points of f is dense in Ω(f).

For surfaces, hyperbolicity of the nonwandering set implies the density of periodic points, but this is no longer true in higher dimensions. Nonetheless, axiom A diffeomorphisms are sometimes called hyperbolic diffeomorphisms, because the portion of M where the interesting dynamics occurs, namely, Ω(f), exhibits hyperbolic behavior.

Axiom A diffeomorphisms generalize Morse–Smale systems, which satisfy further restrictions (finitely many periodic points and transversality of stable and unstable submanifolds). Smale horseshoe map is an axiom A diffeomorphism with infinitely many periodic points and positive topological entropy.

Properties

Any Anosov diffeomorphism satisfies axiom A. In this case, the whole manifold M is hyperbolic (although it is an open question whether the non-wandering set Ω(f) constitutes the whole M).

Rufus Bowen showed that the non-wandering set Ω(f) of any axiom A diffeomorphism supports a Markov partition. Thus the restriction of f to a certain generic subset of Ω(f) is conjugated to a shift of finite type.

The density of the periodic points in the non-wandering set implies its local maximality: there exists an open neighborhood U of Ω(f) such that

: \cap_{n\in \mathbb Z} f^{n} (U)=\Omega(f).

Omega stability

An important property of Axiom A systems is their structural stability against small perturbations. That is, trajectories of the perturbed system remain in 1-1 topological correspondence with the unperturbed system. This property is important, in that it shows that Axiom A systems are not exceptional, but are in a sense 'robust'.

More precisely, for every C1-perturbation f*ε* of f, its non-wandering set is formed by two compact, f*ε*-invariant subsets *Ω*1 and Ω2. The first subset is homeomorphic to Ω(f) via a homeomorphism h which conjugates the restriction of f to Ω(f) with the restriction of fε to *Ω*1:

: f_\epsilon\circ h(x)=h\circ f(x), \quad \forall x\in \Omega(f).

If Ω2 is empty then h is onto Ω(fε). If this is the case for every perturbation f*ε* then f is called omega stable. A diffeomorphism f is omega stable if and only if it satisfies axiom A and the no-cycle condition (that an orbit, once having left a transitive subset of Ω(f), does not return).

References

References

1. Smale, S.. (1967). "Differentiable Dynamical Systems". Bull. Amer. Math. Soc..
2. Ruelle (1978) p.149
3. See [http://www.scholarpedia.org/article/Chaotic_hypothesis Scholarpedia, Chaotic hypothesis]
4. Bowen, R.. (1970). "Markov partitions for axiom A diffeomorphisms". Am. J. Math..
5. Abraham and Marsden, ''Foundations of Mechanics'' (1978) Benjamin/Cummings Publishing, ''see Section 7.5''