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In [[mathematics]], in the study of [[iterated function]]s and [[dynamical system]]s, a '''periodic point''' of a [[function (mathematics)|function]] is a point which the system returns to after a certain number of function iterations or a certain amount of time.
In [[mathematics]], in the study of [[iterated function]]s and [[dynamical system]]s, a '''periodic point''' of a [[function (mathematics)|function]] is a point which the system returns to after a certain number of function iterations or a certain amount of time.
在数学中，特别是在迭代函数和[[动力系统]]的研究领域中，函数的周期点是系统在一定次数的函数迭代或一定时间后返回的点。这里的迭代次数叫做周期。周期为1的周期点被称为不动点。
== 迭代函数 ==
== Iterated functions ==
Given a [[mapping (mathematics)|mapping]] ''f'' from a [[set (mathematics)|set]] ''X'' into itself,
Given a [[mapping (mathematics)|mapping]] ''f'' from a [[set (mathematics)|set]] ''X'' into itself,
给定一个从集合<math>X</math>到自身的映射<math>f</math>,
<math>f: X \to X,</math>
a point ''x'' in ''X'' is called periodic point if there exists an ''n'' so that
a point ''x'' in ''X'' is called periodic point if there exists an ''n'' so that
<math>X</math>中的点<math>x</math>称为周期点，如果存在一个<math>n</math>使
<math>\ f_n(x) = x</math>
where <math>f_n</math> is the ''n''th [[iterated function|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'' (this is not to be confused with the notion of a [[periodic function]]).
where <math>f_n</math> is the ''n''th [[iterated function|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'' (this is not to be confused with the notion of a [[periodic function]]).
其中<math>f_n</math>为<math>f</math>的第<math>n</math>次迭代。满足上述条件的最小正整数<math>n</math>称为点<math>x</math>的素数周期或最小周期。如果X中的每一个点都是周期为n的周期点，那么 f被称为周期点，周期为n(这不能和周期函数的概念混淆)。