+- Tetration Forum (https://tetrationforum.org)
+-- Forum: Tetration and Related Topics (https://tetrationforum.org/forumdisplay.php?fid=1)
+--- Forum: Mathematical and General Discussion (https://tetrationforum.org/forumdisplay.php?fid=3)
+--- Thread: cyclic points (/showthread.php?tid=616)
cyclic points - tommy1729 - 04/01/2011
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 ?
RE: cyclic points - JmsNxn - 04/04/2011
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.
RE: cyclic points - tommy1729 - 04/06/2011
http://math.eretrandre.org/tetrationforum/showthread.php?tid=499
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 ?
regards
tommy1729
RE: cyclic points - JmsNxn - 04/07/2011
no you're right, miscommunication. I didn't realize you were talking about complex spirals, I was thinking along the real line.