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iterated_function [2013/09/09 15:52] nikolaj |
iterated_function [2013/12/21 19:57] 127.0.0.1 external edit |
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| ===== Iterated function ===== | ===== Iterated function ===== | ||
| - | ==== Definition ==== | + | ==== Set ==== |
| - | | $ f:X\to X $ | | + | | @#88DDEE: $ f:X\to X $ | |
| - | | $ n\in \mathbb N, n\neq 0 $ | | + | | @#88DDEE: $ n\in \mathbb N, n\neq 0 $ | |
| - | ^ $ f^n $ ^ | + | | @#FFBB00: $ f^n $ | |
| - | ^ $ f^1=f $ ^ | + | Iteratively defined as follows: |
| - | ^ $ f^{n}=f\circ f^{n-1} $ ^ | + | |
| - | ==== Ramifications ==== | + | | @#FFBB00: $ f^1:=f $ | |
| - | === Predicates === | + | | @#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 $f^2=\text{id}$. | 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 === | ||
