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Is successor function analytic?
#1
Consider the Ackermann successor function $$\varphi(u)$$ such that $$\varphi(a \uparrow ^m b)=a \uparrow ^{m+1} b$$. Is $$\varphi(u)$$ analytic? If so then the complex dynamics of $$\varphi^n(u)$$ can be studied where there is a finite real fixed at  point $$a\uparrow^\infty\infty$$. For example where $$1 \leq a < e^\frac{1}{e}$$.
Daniel
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#2
(09/22/2022, 12:19 PM)Daniel Wrote: Consider the Ackermann successor function $$\varphi(u)$$ such that $$\varphi(a \uparrow ^m b)=a \uparrow ^{m+1} b$$. 

Is this well-defined? I mean to define it that way you would need something like:  if \(a_1\uparrow^m b_1=a_2\uparrow^m b_2\) then \(a_1=a_2\) and \(b_1=b_2\) ?
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