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i was thinking about tetration and came up with important questions.

for the 3rd time !!

are there meromorphic functions f(x) so that they commute with exp(x) ?

thus :

exp(f(x)) = f(exp(x))

related 2nd question :

t ( t(x) ) = exp(x) ( any real analytic t(x) )

g(x) a meromorphic function

t(x) = inverse_g( 1-fixpoint-regular-half-iterate[ exp(g(x)) ] )

related 3rd question :

q(x) a meromorphic function

t ( t(x) ) = exp(x) ( any real analytic t(x) )

1-fixpoint-regular-half-iterate[ exp(q(x)) ] = t( q(x) )

thanks in advance

regards

tommy1729

Posts: 510

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05/13/2009, 09:51 PM
(This post was last modified: 05/13/2009, 09:55 PM by andydude.)
(05/13/2009, 04:52 PM)tommy1729 Wrote: are there meromorphic functions f(x) so that they commute with exp(x) ?

I'm not sure if this is important, but I think what is important is whether or not:

"For all f(x) that satisfy \( f(\exp(x)) = \exp(f(x)) \), there exists a unique real number t such that \( f(x) = \exp^t(x) \)."

I'm not convinced that this is always true for holomorphic/meromorphic functions. I'm sure its false for for arbitrary (or piecewise-defined) functions. I also think this would be useful in characterizing fractional iterates.

Andrew Robbins

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andydude wrote :

(05/13/2009, 09:51 PM)andydude Wrote: (05/13/2009, 04:52 PM)tommy1729 Wrote: are there meromorphic functions f(x) so that they commute with exp(x) ?

I'm not sure if this is important, but I think what is important is whether or not:

"For all f(x) that satisfy \( f(\exp(x)) = \exp(f(x)) \), there exists a unique real number t such that \( f(x) = \exp^t(x) \)."

I'm not convinced that this is always true for holomorphic/meromorphic functions. I'm sure its false for for arbitrary (or piecewise-defined) functions. I also think this would be useful in characterizing fractional iterates.

Andrew Robbins

i kinda asked this question before - more or less -

see thread :

http://math.eretrandre.org/tetrationforu...hp?tid=270
yes , this is useful in characterizing fractional iterates.

and i believe it is important for some half-iterate questions , though i cant prove it to be for tetration , but im working on it.

regards

tommy1729