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 how many superfunctions? [was superfunctions of eta converge towards each other] bo198214 Administrator Posts: 1,389 Threads: 90 Joined: Aug 2007 05/27/2011, 09:33 AM (This post was last modified: 05/27/2011, 09:35 AM by bo198214.) (05/26/2011, 10:09 PM)tommy1729 Wrote: another question is : how many superfunctions can a function have ? There are different answers. If you just ask about the number of superfunctions, then there are infinitely many. We discussed that already, when ever you have a superfunction F, F(x+1)=f(F(x)) then also the function $G(x)=G(x+\theta(x))$ is a superfunction, for $\theta$ 1-periodic, this should not be new for you. If you however ask, how many *regular* super-functions you have at a given fixpoint, i.e. superfunction from regular fractional iterations, i.e. which have an asymptotic powerseries development at the fixpoint, which is equal to the formal fractional iteration powerseries, then there is a clear answer: You look at the powerseries development of the corresponding function, for simplicity we assume fixpoint at 0. $f(x)=f_1 x + f_2 x^2 + \dots$, assume $f_1\neq 0$ Hyperbolic: $|f_1|\ne 1$: there is exactly one regular superfunction Parabolic: $f(x)=x + f_m x^m + f_{m+1}x^{m+1} + \dots$: There are exactly 2(m-1) regular superfunctions. For example $e^x-1=x+x^2/2+\dots$, that's why we have 2*(2-1)=2 regular superfunctions. One from left and one from right. Generally there are $2(m-1)$ petals around the fixpoint, which are alternatingly attractive and repellant (in our example coming from left is attractive and coming from right repellant), on each petal there is defined a different regular Abel function (which is the inverse of a superfunction). The whole thing is called the Leau-Fatou-flower and is kinda standard in holomorphic dynamics (see for example the book of Milnor mentioned on the forum). « Next Oldest | Next Newest »

 Messages In This Thread how many superfunctions? [was superfunctions of eta converge towards each other] - by tommy1729 - 05/26/2011, 10:09 PM RE: how many superfunctions? [was superfunctions of eta converge towards each other] - by bo198214 - 05/27/2011, 09:33 AM RE: how many superfunctions? [was superfunctions of eta converge towards each other] - by tommy1729 - 05/27/2011, 10:48 PM RE: how many superfunctions? [was superfunctions of eta converge towards each other] - by bo198214 - 05/28/2011, 09:18 AM RE: how many superfunctions? [was superfunctions of eta converge towards each other] - by sheldonison - 05/31/2011, 07:38 PM RE: how many superfunctions? [was superfunctions of eta converge towards each other] - by tommy1729 - 05/28/2011, 12:12 PM RE: how many superfunctions? [was superfunctions of eta converge towards each other] - by bo198214 - 05/28/2011, 12:25 PM RE: how many superfunctions? [was superfunctions of eta converge towards each other] - by tommy1729 - 05/29/2011, 05:27 PM RE: how many superfunctions? [was superfunctions of eta converge towards each other] - by bo198214 - 05/29/2011, 09:20 PM

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