Axiom A

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][2] 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.^[3]

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.^[2][4] 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.^[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 'generic'.

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:

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

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 an invariant subset, does not return).

References

1. Smale, S. (1967), "Differentiable Dynamical Systems", Bull. Amer. Math. Soc. 73: 747–817, doi:10.1090/s0002-9904-1967-11798-1, Zbl 0202.55202
2. Ruelle (1978) p.149
3. See Scholarpedia, Chaotic hypothesis
4. Bowen, R. (1970), "Markov partitions for axiom A diffeomorphisms", Am. J. Math. 92: 725–747, doi:10.2307/2373370, Zbl 0208.25901
5. Abraham and Marsden, Foundations of Mechanics (1978) Benjamin/Cummings Publishing, see Section 7.5

• Ruelle, David (1978). Thermodynamic formalism. The mathematical structures of classical equilibrium. Encyclopedia of Mathematics and its Applications 5. Reading, Massachusetts: Addison-Wesley. ISBN 0-201-13504-3. Zbl 0401.28016.
• Ruelle, David (1989). Chaotic evolution and strange attractors. The statistical analysis of time series for deterministic nonlinear systems. Lezioni Lincee. Notes prepared by Stefano Isola. Cambridge University Press. ISBN 0-521-36830-8. Zbl 0683.58001.