Tetration fixed points

[MphLee]

Yes, forum members have started the study of this matter. I'd name first Rubtosv and Romerio's 2006 report

(2006) Notes on Hyper-operations
Progress Report NKS Forum III

I have not time to brush of the rust, but If I recall correctly everything starts with this heuristics. Let \(H_n(x)=b[n]x\) be n-th rank hyperexponentiation in base \(b\)
then we have \(V_n(b)=\lim_{x\to \infty} b[n]x\) then \[H_n(V_{n+1}(b))=V_{n+1}(b)\] thus if \({\rm hrt}_n(d,x)\) is the \(d\)-th degree n-hyperroot function the equation \(b[n]p=p\) is solved by \(p=V_{n+1}(b)\) or by the inverse of \(b={\rm hrt}_n(p,p)\): call that \(W_n(b)=p\). So we use the next rank... and roots.

All of this obviously depends on attracting or repelling fixedpoints. Separate analysis needs to be carried for odd rank (exp, pentation, heptation,...) and even ranks (tetration, hexation, octation,...).

I suggest you these threads:

(2007) Generalized recursive operators

(2007) Exploring Pentation - Base e

(2015) Mizugadro, pentation, Book

(2017) pentation and hexation

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Note that \(V_{n+1}(b)\) and \(W_n(b)\) must be related somehow. Also we could go also make the distinction \(V_n^-(b)=\lim_{x\to -\infty} b[n] x\). It is important to highlight that \(W_3\) is not Lambert W function, so I apologize for choosing the letter W for it.

• Lambert W is the inverse of \(b\mapsto be^b\) because \(W(b)e^{W(b)}=b\);
  
• while \(W_3\) inverts \(b\mapsto b^{1/b}\) because \(b^{W_3 (b)}=W_3(b)\);
  
• the function \(V^+_4(b)={}^{\infty}b\) is the infinite tower, so it somehow inverts \(b\mapsto {\rm srt}(\infty,b)\);
  

From this heuristics Rubtsov and Romerio derive the concept that self hyperrots are deeply related to infinite degree hyperroots of the next ranks. Because \[\boxed{{\rm hrt}_{n+1}(\infty,b)\simeq {\rm hrt}_{n}(b,b)}\]