# 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:

- The nonwandering set of
*f*,*Ω*(*f*), is a hyperbolic set and compact. - 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

## 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 *C*^{1}-perturbation *f*_{ε} of *f*, its non-wandering set is formed by two compact, *f*_{ε}-invariant subsets *Ω*_{1} and *Ω*_{2}. 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 *Ω*_{1}:

If *Ω*_{2} 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

- ↑ Smale, S. (1967), "Differentiable Dynamical Systems",
*Bull. Amer. Math. Soc.*,**73**: 747–817, doi:10.1090/s0002-9904-1967-11798-1, Zbl 0202.55202 - 1 2 Ruelle (1978) p.149
- ↑ See Scholarpedia, Chaotic hypothesis
- ↑ Bowen, R. (1970), "Markov partitions for axiom A diffeomorphisms",
*Am. J. Math.*,**92**: 725–747, doi:10.2307/2373370, Zbl 0208.25901 - ↑ 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.