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:<math>f: X \to X,</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>\ f_n(x) = x</math>
:<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]]).
If there exist distinct ''n'' and ''m'' such that
If there exist distinct ''n'' and ''m'' such that
如果存在不同的n和m使
如果存在不同的n和m使
:<math>f_n(x) = f_m(x)</math>
:<math>f_n(x) = f_m(x)</math>
then ''x'' is called a '''preperiodic point'''. All periodic points are preperiodic.
then ''x'' is called a '''preperiodic point'''. All periodic points are preperiodic.
* [[Limit cycle]]
* [[Limit cycle]]
* [[限制周期]]
* [[Limit set]]
* [[Limit set]]
* [[限量套]]
* [[Stable manifold|Stable set]]
* [[Stable manifold|Stable set]]
* [[稳定歧管|稳定集]]
* [[Sharkovsky's theorem]]
* [[Sharkovsky's theorem]]
* [[Sharkovsky定理]]
* [[Stationary point]]
* [[Stationary point]]
* [[固定点]]
* [[Periodic points of complex quadratic mappings]]
* [[Periodic points of complex quadratic mappings]]
* [[复杂二次映射的周期点]]
* [[复杂二次映射的周期点]]
{{PlanetMath attribution|id=4516|title=hyperbolic fixed point}}
{{PlanetMath attribution|id=4516|title=hyperbolic fixed point}}