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iterated_function [2013/09/13 20:22] nikolaj |
iterated_function [2014/03/21 11:11] (current) |
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| ===== Iterated function ===== | ===== Iterated function ===== | ||
| - | ==== Definition ==== | + | ==== Set ==== |
| - | | @#88DDEE: $ f:X\to X $ | | + | | @#55CCEE: context | @#55CCEE: $ f:X\to X $ | |
| - | | @#88DDEE: $ n\in \mathbb N, n\neq 0 $ | | + | | @#55CCEE: context | @#55CCEE: $ n\in \mathbb N, n\neq 0 $ | |
| - | | @#FFBB00: $ f^n $ | | + | | @#FFBB00: definiendum | @#FFBB00: $ f^n $ | |
| Iteratively defined as follows: | Iteratively defined as follows: | ||
| - | | @#FFBB00: $ f^1:=f $ | | + | | @#FFBB00: definiendum | @#FFBB00: $ f^1:=f $ | |
| - | | @#FFBB00: $ f^{n}:=f\circ f^{n-1} $ | | + | | @#FFBB00: definiendum | @#FFBB00: $ f^{n}:=f\circ f^{n-1} $ | |
| ==== Discussion ==== | ==== Discussion ==== | ||
| - | We call a function //idempotent// if $f^2=f$. We say it's an //involution// if $f^2=\text{id}$. | ||
| - | |||
| - | | @#EEEE55: $f$ ... idempotent $\equiv f^2=f$ | | ||
| - | | @#EEEE55: $f$ ... involution $\equiv f^2=\text{id}$ | | ||
| - | |||
| ==== Parents ==== | ==== Parents ==== | ||
| === Subset of === | === Subset of === | ||
| [[Unary operation]] | [[Unary operation]] | ||
| - | === Requirements === | + | === Context === |
| [[Relation concatenation]], | [[Relation concatenation]], | ||
| [[Natural number]] | [[Natural number]] | ||
