In mathematics, Smale's axiom A defines a class of dynamical systems which have been extensively studied and whose dynamics is relatively well understood. A prominent example is the Smale horseshoe map. The term "axiom A" originates with Stephen Smale.[1] The importance of such systems is demonstrated by the chaotic hypothesis, which states that, 'for all practical purposes', a many-body thermostatted system is approximated by an Anosov system.[2]
Let M be a smooth manifold with a diffeomorphism f: M→M. Then f is an axiom A diffeomorphism ifthe following two conditions hold:
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.[1] 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
\capn\infn(U)=\Omega(f).
An important property of Axiom A systems is their structural stability against small perturbations.[3] 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 fε of f, its non-wandering set is formed by two compact, fε-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 fε to Ω1:
f\epsilon\circh(x)=h\circf(x), \forallx\in\Omega(f).
If Ω2 is empty then h is onto Ω(fε). If this is the case for every perturbation fε 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 an invariant subset, does not return).
. David Ruelle . Thermodynamic formalism. The mathematical structures of classical equilibrium . Encyclopedia of Mathematics and its Applications . 5 . Reading, Massachusetts . Addison-Wesley . 1978 . 0-201-13504-3 . 0401.28016 .
. David Ruelle . Chaotic evolution and strange attractors. The statistical analysis of time series for deterministic nonlinear systems . registration . Notes prepared by Stefano Isola . Lezioni Lincee . . 1989 . 0-521-36830-8 . 0683.58001 .