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Iterating at fixed points of b^x
#2
Proposition. All fixed points of are hyperbolic, i.e. for each complex with .

Clearly . So we want to show that .
We can exclude the case as this implies and we know that 0 is not a fixed point of .

For the other case we set and get
and hence the equation system:
(1) and (2).
We square both equation and add them:

.
But beware, this is only a necessary condition on the fixed points. The fixed points lie discretely on the complex plane. Not every point satisfying this equation is a fixed point.

But from this condition we can look what happens for . For we get and we see that both values of do not satisfy equation (2). So there is no fixed point with .
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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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