Iteration basics
#21
bo198214 Wrote:The Abel function counts the iterations of \( f \):
\( \phi(f(f(x)))=\phi(f(x))+1=f(x)+2 \)
\( \phi(f(f(f(x))))=\phi(f(f(x)))+1=f(x)+2+1=f(x)+3 \)
\( \phi(f^{\circ n}(x))=\phi(x)+n \).

The fact that Abel function extends COUNTING to non-integer , negative(?) and complex values of COUNTING is very interesting concept besides the fact it is useful in computations and analysis of iterates.

\( \phi(f^{\circ t}(x))=\phi(x)+t \).

e.g.

\( \phi(f^{\circ I}(x))=\phi(x)+I \).

If we turn it upside down, we can may be use the result of t iterations of a function to define what COUNTING with t (e.g. t=I) means.

We may get functions which are only COUNTABLE in certain ways, so that Abel equation is not possible to solve for all t.

Would that make sense?

Ivars


Messages In This Thread
Iteration basics - by Ivars - 03/20/2008, 10:34 AM
RE: Iteration basics - by Ivars - 03/20/2008, 05:24 PM
RE: Iteration basics - by Gottfried - 03/21/2008, 08:37 AM
RE: Iteration basics - by Ivars - 04/03/2008, 08:05 AM
RE: Iteration basics - by Gottfried - 04/03/2008, 08:19 AM
RE: Iteration basics - by Ivars - 04/03/2008, 10:13 AM
RE: Iteration basics - by andydude - 04/05/2008, 11:35 PM
RE: Iteration basics - by Gottfried - 04/06/2008, 07:16 AM
RE: Iteration basics - by Ivars - 04/06/2008, 08:55 AM
RE: Iteration basics - by andydude - 04/07/2008, 12:01 AM
RE: Iteration basics - by Ivars - 04/20/2008, 10:15 PM
RE: Iteration basics - by bo198214 - 04/21/2008, 08:21 PM
RE: Iteration basics - by andydude - 04/22/2008, 05:30 AM
RE: Iteration basics - by Ivars - 04/22/2008, 07:02 AM
RE: Iteration basics - by Ivars - 04/09/2008, 07:56 PM
RE: Iteration basics - by Ivars - 05/09/2008, 09:45 AM
RE: Iteration basics - by bo198214 - 05/09/2008, 02:49 PM
RE: Iteration basics - by Ivars - 05/27/2008, 10:33 AM
RE: Iteration basics - by Gottfried - 05/27/2008, 07:35 PM
RE: Iteration basics - by Ivars - 05/30/2008, 06:08 AM
RE: Iteration basics - by Xorter - 01/02/2017, 05:21 PM
RE: Imaginary iterates of exponentiation - by Ivars - 05/24/2008, 08:35 AM

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