cyclic points
i want to adress attention to cyclic points.

for instance f(z)^[a/2] must have the same fixpoint as f(z)^[a].

but if f(w) = q and f(q) = s and f(s) = w we are " in trouble ".

certain function are thus " in trouble " at certain points.

the super of f(z) or f(f(z)) should be similar.

is exp(z) " in trouble " ?

to avoid " trouble " we analyse f(z)^+real = z

or not ?

Well, the half-iterate of sin(x) isn't "in trouble", and it's the prime cyclic function. I don't see how exp(z) could be any worse.

maybe that clarifies it since i think there might be a misunderstanding.

therefore i am intrested in the equation(s) f(z)^^[+real x] = z in particular for f(z) = exp(z) since they lead to the " loops " or " cyclic points " in Gottfried's pics / the continu iterations of exp(z)/ sexp(slog(z_0) + (+real x) ).

those solutions of f(z)^^[+real x] = z relate to the branch structure of the super and inversesuper of f(z).

since the branch structures of sexp and slog are complicated and quite unexplored , i consider the " helping " equation(s) f(z)^^[+real x] = z for f(z) = exp(z).

since f(z)^^+oo is a fractal , fractals are related and strongly connected.

loop or cycle detection is also intresting and related.

ofcourse the kind of sexp / slog we choose matters as well.
( i wont go into details about that now )

tetration is more than constructing a coo sexp(z) , we need to understand the branch structure.

also of intrest is that if we can show ( prove / compute ) two distinct branch structure this implies two distinct superfunctions !!

a lot of research needs to be done , and thread 499 is epic but i thought id start a new thread about the equation f(z)^^[+real x] = z.

in my nightmares we end up with undecidable halting problems and self-reference , but i think this will not be the case.

not sure what you meant by your comment about sine , if it is still relevant plz inform me.

are you saying sin or its super doesnt loop ?


no you're right, miscommunication. I didn't realize you were talking about complex spirals, I was thinking along the real line.

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