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 Functional super-iteration and hierarchy of functional hyper-iterations Base-Acid Tetration Fellow Posts: 94 Threads: 15 Joined: Apr 2009 05/02/2009, 02:23 AM (This post was last modified: 05/12/2009, 02:12 AM by Base-Acid Tetration.) We all know the functional "exponentiation" or "power" for functional iteration, $f^{\circ n} (x).$ (I use F^circle n, f^n (x), or f^circle n (x) if clarity is needed, for iteration) That means we can do this for some function: $f^{g(x)}(x),$ where the function f is iterated g(x) times at x. We could investigate the properties and results of functional "tetration", or super-iteration. The "upper-left exponent" notation can be abuse, or use the circle notation like ${}^{\circ n} f(x)$ Initial Definition: (might have gotten my variables wrong) Given a $f : S \rightarrow S$ and a $n \in \mathbb{N},$ a functional super-iteration is defined by the recurrence relation: $(\forall n \, \forall x) \mbox{ so that the result of the iteration is defined, } {}^{\circ n+1} f(x) =[ f^ {{}^{\circ{n}} f(x)}(x) ] ,$ ${}^{\circ 1} f(x) = f(x).$ Already a hierarchy of operations on functions that is much like the arithmetic hyper-operations is evident: Function composition: $g \circ f (x) = g(f(x)), \, g(x) = c \rightarrow g(f(x)) = c.$ Function iteration: $f^{n+1}(x) = f (f^n (x)), \, f^0(x) = x.$ A function can be iterated a constant times, or by g(x) times. Functional superiteration: ${}^{\circ n+1} f(x) = \left [ {}^{\circ n} f^{\circ f(x)} \right ] (x)}, {}^{\circ 1} f(x) = f(x).$ « Next Oldest | Next Newest »

 Messages In This Thread Functional super-iteration and hierarchy of functional hyper-iterations - by Base-Acid Tetration - 05/02/2009, 02:23 AM RE: Functional super-iteration and hierarchy of functional composition-based operations - by bo198214 - 05/02/2009, 07:48 AM RE: Functional super-iteration and hierarchy of functional composition-based operations - by andydude - 05/02/2009, 08:32 AM RE: Functional super-iteration and hierarchy of functional composition-based operations - by andydude - 05/02/2009, 09:23 AM RE: Functional super-iteration and hierarchy of functional composition-based operations - by tommy1729 - 05/02/2009, 12:22 PM RE: Functional super-iteration and hierarchy of functional composition-based operations - by bo198214 - 05/02/2009, 01:14 PM RE: Functional super-iteration and hierarchy of functional composition-based operations - by andydude - 05/02/2009, 06:38 PM RE: Functional super-iteration and hierarchy of functional composition-based operations - by Base-Acid Tetration - 05/02/2009, 06:14 PM RE: Functional super-iteration and hierarchy of functional composition-based operations - by bo198214 - 05/02/2009, 07:27 PM RE: Functional super-iteration and hierarchy of functional composition-based operations - by Base-Acid Tetration - 05/03/2009, 01:08 PM RE: Functional super-iteration and hierarchy of functional composition-based operations - by bo198214 - 05/03/2009, 03:25 PM RE: Functional super-iteration and hierarchy of functional composition-based operations - by Base-Acid Tetration - 05/03/2009, 07:21 PM RE: Functional super-iteration and hierarchy of functional composition-based operations - by bo198214 - 05/03/2009, 08:13 PM RE: Functional super-iteration and hierarchy of functional composition-based operations - by Base-Acid Tetration - 05/03/2009, 09:45 PM RE: Functional super-iteration and hierarchy of functional composition-based operations - by bo198214 - 05/03/2009, 10:27 PM RE: Functional super-iteration and hierarchy of functional composition-based operations - by Base-Acid Tetration - 05/04/2009, 01:06 AM RE: Functional super-iteration and hierarchy of functional composition-based operations - by bo198214 - 05/04/2009, 07:56 AM RE: Functional super-iteration and hierarchy of functional composition-based operations - by Base-Acid Tetration - 05/04/2009, 08:04 PM RE: Functional super-iteration and hierarchy of functional composition-based operations - by bo198214 - 05/04/2009, 08:50 PM RE: Functional super-iteration and hierarchy of functional hyper-iterations - by Base-Acid Tetration - 05/04/2009, 08:57 PM RE: Functional super-iteration and hierarchy of functional hyper-iterations - by bo198214 - 05/04/2009, 09:01 PM RE: Functional super-iteration and hierarchy of functional hyper-iterations - by Base-Acid Tetration - 05/04/2009, 09:06 PM RE: Functional super-iteration and hierarchy of functional hyper-iterations - by bo198214 - 05/04/2009, 09:12 PM RE: Functional super-iteration and hierarchy of functional hyper-iterations - by Base-Acid Tetration - 05/12/2009, 02:29 AM RE: Functional super-iteration and hierarchy of functional hyper-iterations - by bo198214 - 05/12/2009, 07:11 AM

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