Semi-group iso , tommy's limit fix method and alternative limit for 2sinh method
I have been talking alot about the semi-group iso lately.

And the claim that the 2sinh method has it occured often.

I also mentioned the idea of other paths of tetration aka non-real tetration.

And I talked about the primary fixpoint method used in knesers solution.

All of that is related in a way I have not mentioned before.


Consider the primary fixpoint expansion.

Now we construct the sexp(z) from that.

We now have a sexp(z) that has the points 1,e,e^e etc.

But it is not a real tetration.

it does satisfy the semi-group iso though just like 2sinh method.

So how do they relate ?

tommy's limit fix method

Well , take the unit length interval [x,x+1] mapping to sexp[x,x+1].

then consider taking the exp repeatedly.

We know exp iterations are chaotic but also know the positive real line is infinitely often an attractor.

so for a set of large integers n 

sexp[x+n,x+n+1] We will get arbitrary close to a real range output of [0,1] and later [1,e].


So this limit results in the real tetration satisfying the semi-group iso which is UNIQUE.

notice that the neighbourhood of sexp[x+n,x+n+1] is chaotic for nonreal z in domain [x+n+z,x+n+1+z] but the limit only is considered for the real range.

Since the limit is only considered for the real output we do not know if the function is analytic and the chaotic nature of exp iterates makes us doubt.

This also happens with the 2sinh but notice we also did NOT CLAIM that they are equal away from the real line , since that limit definition did not imply that.

The limit is probably not a uniform limit but nevertheless has to work.

Unfortunately not uniform limits and exp iterations are hard to work with.

But in theory it works.

So tommy's limit fix method and tommy's 2sinh method are equivalent.

2 sides of the same coin.

Btw , ofcourse this method can be used for other real bases and other functions with similar chaotic nature.

Usually having 2 ways to compute something makes it easier but this is ofcourse not trivial here.

This is the short version of the idea , there are many implications and further ideas but those are not easy to explain.


ok it seems this does not work out.

iterations of exp is too chaotic to make this work.

the points 0,1,e,e^e etc are attractive but the real interval [0,1] is not without any iteration intersections.

it gets only worse everytime we get close to 0 , like a hairy ball or so.

So this method goes to the garbage can.



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