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iterated_function [2013/12/21 19:57] 127.0.0.1 external edit |
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| - | ===== Iterated function ===== | ||
| - | ==== Set ==== | ||
| - | | @#88DDEE: $ f:X\to X $ | | ||
| - | | @#88DDEE: $ n\in \mathbb N, n\neq 0 $ | | ||
| - | | @#FFBB00: $ f^n $ | | ||
| - | |||
| - | Iteratively defined as follows: | ||
| - | |||
| - | | @#FFBB00: $ f^1:=f $ | | ||
| - | | @#FFBB00: $ f^{n}:=f\circ f^{n-1} $ | | ||
| - | |||
| - | ==== 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 ==== | ||
| - | === Subset of === | ||
| - | [[Unary operation]] | ||
| - | === Requirements === | ||
| - | [[Relation concatenation]], | ||
| - | [[Natural number]] | ||
