Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
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.

Possibly Related Threads...
Thread Author Replies Views Last Post
  Are tetrations fixed points analytic? JmsNxn 2 3,452 12/14/2016, 08:50 PM
Last Post: JmsNxn
  Polygon cyclic fixpoint conjecture tommy1729 1 2,534 05/18/2016, 12:26 PM
Last Post: tommy1729
  Removing the branch points in the base: a uniqueness condition? fivexthethird 0 1,770 03/19/2016, 10:44 AM
Last Post: fivexthethird
  Natural cyclic superfunction tommy1729 3 3,517 12/08/2015, 12:09 AM
Last Post: tommy1729
  Cyclic dynamics f(-x) = T (f(x)) tommy1729 2 2,963 08/25/2015, 08:23 AM
Last Post: tommy1729
  Branch points of superlog mike3 0 2,486 02/03/2010, 11:00 PM
Last Post: mike3
  Complex fixed points of base-e tetration/tetralogarithm -> base-e pentation Base-Acid Tetration 19 33,095 10/24/2009, 04:12 AM
Last Post: andydude
  Iterating at fixed points of b^x bo198214 28 27,626 05/28/2008, 07:37 AM
Last Post: Kouznetsov
  Cyclic complex functions and uniqueness jaydfox 12 13,856 04/24/2008, 08:50 PM
Last Post: bo198214
  Migration of inflection points in y = b # x, for e^(1/e) < b < +oo GFR 9 9,424 01/25/2008, 11:48 PM
Last Post: GFR

Users browsing this thread: 1 Guest(s)