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 cyclic points tommy1729 Ultimate Fellow Posts: 1,370 Threads: 335 Joined: Feb 2009 04/01/2011, 12:27 PM 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 ? JmsNxn Long Time Fellow Posts: 291 Threads: 67 Joined: Dec 2010 04/04/2011, 12:55 AM 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. tommy1729 Ultimate Fellow Posts: 1,370 Threads: 335 Joined: Feb 2009 04/06/2011, 03:50 PM http://math.eretrandre.org/tetrationforu...hp?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 JmsNxn Long Time Fellow Posts: 291 Threads: 67 Joined: Dec 2010 04/07/2011, 07:57 PM no you're right, miscommunication. I didn't realize you were talking about complex spirals, I was thinking along the real line. « Next Oldest | Next Newest »

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