07/28/2022, 12:39 AM
(07/26/2022, 10:50 AM)Daniel Wrote: Ask and you shall receive. Thanks Bo, this is very helpful!
I'm impressed with what can be done with two fixed points. In trying to make a contribution to real tetration I considered what could be done with the entire set of fixed points. Couldn't it be possible to construct real tetration using a countable infinity of fixed points to fix the interval between \(^nb \textrm{ and } ^{n+1}b\) at a countable infinity of points on the real line and thus fixing all points on the real line.
As far as regular iteration from a single single fixed point not being holomorphic, I did a test which indicated that the regular iteration of the exponential map provides not only the correct multiplier at the fixed point, it predicted the location of the next fixed point and it's multiplier. Of course I would need to recreate the experiment if folks were to believe me.
My goal in generating fractals is to make the underlying mathematics transparent. My professional background is as a programmer, so I expect I can write terse clean code that is easy to read and understand. I trying doing some stuff with Julia but most of the packages were broken. On the other hand with Pari-GP I found the code for computing Bell polynomials which I can make good use of.
Regular iteration at a single fixed point is absolutely holomorphic..? I'm confused. What was the problem, is that if we perform regular iteration about \(z \approx L\) the fixed point, then it won't be real valued on the real line, but it's still holomorphic. Also, if you perform the regular iteration on the real line, as Kouznetsov refers to it, you get the crescent iteration, which is again holomorphic. The absolute power of regular iteration is that it's always holomorphic.
Perhaps you got confused by what I wrote, using \(\theta\) mappings you can perturb the regular iteration, but then it is no longer holomorphic in a neighborhood hood of the fixed point, is that what you meant to say?
