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