09/16/2021, 10:54 PM
(09/16/2021, 01:41 AM)JmsNxn Wrote: ..., the orbits \( \exp^{\circ n}(z) \) get arbitrarily close to the orbit \( \exp^{\circ n}(0) \). The missing points are the periodic points and cycles. And also that the julia set of the exponential is the entire complex plane.Just commenting on this small thing.
You bring it up often.
And I have mixed feeling about it :^)
The thing is , it is correct and relevant to many tetration related ideas. It has its upsides and downsides.
However notice for MOST \( z \) , the orbits \( \exp^{\circ R}(z) \) USUALLY DO NOT get arbitrarily close to the orbit \( \exp^{\circ R}(0) \) where \( R \) is a positive real that is not an integer.
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IMO the " beta method " or whatever we wanna call it , is by design made to have no singularities in certain locations.
Those locations depends on the (analytic) " helping functions " we picked ( the functions that go fast to 1 so our base gets close to e ).
This implies that different " helping functions " give different (connected) locations.
Those different locations give different (connected) boundaries ; and those boundaries can be functionally inverted , which implies different slog's.
So I conclude for instance that my gaussian method is distinct from earlier type solutions.
( I planned future similar solutions which might further strengthen or clarify that idea ... more later )
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I have to think about the remaining things everyone said. I just wanted to comment this.
regards
tommy1729