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| {{About|a property of dynamical systems|the property of arithmetical semigroups|Abstract analytic number theory}}
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| In [[mathematics]], '''Smale's axiom A''' defines a class of [[dynamical system]]s 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]].<ref>{{citation | first=S. | last=Smale | url=http://www.ams.org/bull/1967-73-06/S0002-9904-1967-11798-1/home.html | title=Differentiable Dynamical Systems | journal=Bull. Amer. Math. Soc. | volume=73 | year=1967 | pages=747–817 | zbl=0202.55202 }}</ref><ref name=R78149>Ruelle (1978) p.149</ref> The importance of such systems is demonstrated by the [[chaotic hypothesis]], which states that, 'for all practical purposes', that a many-body [[thermostatted system]] is approximated by an [[Anosov system]].<ref>See [http://www.scholarpedia.org/article/Chaotic_hypothesis Scholarpedia, Chaotic hypothesis]</ref>
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| == Definition ==
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| Let ''M'' be a [[smooth manifold]] with a [[diffeomorphism]] ''f'': ''M''→''M''. Then ''f'' is an '''axiom A diffeomorphism''' if
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| the following two conditions hold:
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| #The [[nonwandering set]] of ''f'', ''Ω''(''f''), is a [[hyperbolic set]] and [[Compact space|compact]].
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| #The set of [[periodic point]]s of ''f'' is [[dense subset|dense]] in ''Ω''(''f'').
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| 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.
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| Axiom A diffeomorphisms generalize [[Morse–Smale system]]s, 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]].
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| == Properties ==
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| 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'').
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| [[Rufus Bowen]] showed that the non-wandering set ''Ω''(''f'') of any axiom A diffeomorphism supports a [[Markov partition]].<ref name=R78149/><ref>{{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 | zbl=0208.25901 }}</ref> Thus the restriction of ''f'' to a certain generic subset of ''Ω''(''f'') is conjugated to a [[shift of finite type]].
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| 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
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| : <math>\cap_{n\in \mathbb Z} f^{n} (U)=\Omega(f).</math>
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| == Omega stability ==
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| An important property of Axiom A systems is their structural stability against small perturbations.<ref>Abraham and Marsden, ''Foundations of Mechanics'' (1978) Benjamin/Cummings Publishing, ''see Section 7.5''</ref> 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'.
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| More precisely, for every ''C''<sup>1</sup>-[[perturbation theory|perturbation]] ''f''<sub>''ε''</sub> of ''f'', its non-wandering set is formed by two compact, ''f''<sub>''ε''</sub>-invariant subsets ''Ω''<sub>1</sub> and ''Ω''<sub>2</sub>. 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''<sub>''ε''</sub> to ''Ω''<sub>1</sub>:
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| : <math>f_\epsilon\circ h(x)=h\circ f(x), \quad \forall x\in \Omega(f).</math>
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| If ''Ω''<sub>2</sub> is empty then ''h'' is onto ''Ω''(''f''<sub>''ε''</sub>). If this is the case for every perturbation ''f''<sub>''ε''</sub> 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).
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| == References ==
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| {{reflist}}
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| * {{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, MA | publisher=Addison-Wesley | year=1978 | isbn=0-201-13504-3 | zbl=0401.28016 }}
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| * {{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 | others=Notes prepared by Stefano Isola | series=Lezioni Lincee | publisher=[[Cambridge University Press]] | year=1989 | isbn=0-521-36830-8 | zbl=0683.58001 }}
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| [[Category:Ergodic theory]]
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| [[Category:Diffeomorphisms]]
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