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Iterating at fixed points of b^x
#3
Now let us take it a bit further for a general real base .

Proposition. The only real base for which the complex function has a fixed point with is and in this case the only such fixed point is .

Simalarly to the previous post we assume a fixed point being given by
then for parabolicity we have to prove



and we have the equation system:

(1) and (2)

Both equation squared and added yields
and hence



For the case : and and so
.

Now we put this into equation (1) with :



We substitute ,
. But the functions on both sides are well known. The right side is a circle with radius 1 and the left side is always above or below the right side in the region . So equality happens exactly at .

This implies however only satisfies equation (1). Then is the only real base for which has a parabolic fixed point and this fixed point is given by and and is hence the real number .
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Messages In This Thread
Iterating at fixed points of b^x - by bo198214 - 09/08/2007, 10:02 AM
The fixed points of e^x - by bo198214 - 09/08/2007, 10:34 AM
The fixed points of b^x - by bo198214 - 09/08/2007, 11:36 AM
RE: Iterating at fixed points of b^x - by jaydfox - 09/12/2007, 06:23 AM
RE: Iterating at fixed points of b^x - by GFR - 10/03/2007, 11:03 PM
RE: Iterating at fixed points of b^x - by GFR - 01/31/2008, 03:07 PM

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