Axiom A

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.

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.{{citation | last=Bowen | first=R. | authorlink=Rufus Bowen | title=Markov partitions for axiom A diffeomorphisms | journal=Am. J. Math. | volume=92 | pages=725–747 | year=1970 | issue=3 | zbl=0208.25901 | doi=10.2307/2373370| jstor=2373370 }} 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

: $\backslash cap\_\{n\backslash in\; \backslash mathbb\; Z\}\; f^\{n\}\; (U)=\backslash Omega(f).$

An important property of Axiom A systems is their structural stability against small perturbations.Abraham and Marsden, Foundations of Mechanics (1978) Benjamin/Cummings Publishing, see Section 7.5 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 Ω₁ 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 f_ε to Ω₁:

: $f\_\backslash epsilon\backslash circ\; h(x)=h\backslash circ\; f(x),\; \backslash quad\; \backslash forall\; x\backslash in\; \backslash Omega(f).$

If Ω₂ is empty then h is onto Ω(f_ε). 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).

• Ergodic flow

{{reflist}}

• {{cite book | last=Ruelle | first=David | authorlink=David Ruelle | title=Thermodynamic formalism. The mathematical structures of classical equilibrium | series=Encyclopedia of Mathematics and its Applications | volume=5 | location=Reading, Massachusetts | publisher=Addison-Wesley | year=1978 | isbn=0-201-13504-3 | zbl=0401.28016 }}
• {{cite book | last=Ruelle | first=David | authorlink=David Ruelle | title=Chaotic evolution and strange attractors. The statistical analysis of time series for deterministic nonlinear systems | url=https://archive.org/details/chaoticevolution0000ruel | url-access=registration | others=Notes prepared by Stefano Isola | series=Lezioni Lincee | publisher=Cambridge University Press | year=1989 | isbn=0-521-36830-8 | zbl=0683.58001 }}

Category:Ergodic theory

Category:Diffeomorphisms