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In mathematics, in the study of iterated functions and dynamical systems, a periodic point of a function is a point which the system returns to after a certain number of function iterations or a certain amount of time. == Iterated functions == Given an endomorphism ''f'' on a set ''X'' : a point ''x'' in ''X'' is called periodic point if there exists an ''n'' so that : where is the ''n''th iterate of ''f''. The smallest positive integer ''n'' satisfying the above is called the ''prime period'' or ''least period'' of the point ''x''. If every point in ''X'' is a periodic point with the same period ''n'', then ''f'' is called ''periodic'' with period ''n''. If there exists distinct ''n'' and ''m'' such that : then ''x'' is called a preperiodic point. All periodic points are preperiodic. If ''f'' is a diffeomorphism of a differentiable manifold, so that the derivative is defined, then one says that a periodic point is ''hyperbolic'' if : that it is ''attractive'' if : and it is ''repelling'' if : If the dimension of the stable manifold of a periodic point or fixed point is zero, the point is called a ''source''; if the dimension of its unstable manifold is zero, it is called a ''sink''; and if both the stable and unstable manifold have nonzero dimension, it is called a ''saddle'' or saddle point. 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「Periodic point」の詳細全文を読む スポンサード リンク
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