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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.〔Ruelle (1978) p.149〕 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.〔See (Scholarpedia, Chaotic hypothesis )〕 == 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Axiom A」の詳細全文を読む スポンサード リンク
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