Wild conjecture about 2 fixpoints. tommy1729 Ultimate Fellow     Posts: 1,906 Threads: 409 Joined: Feb 2009 05/03/2014, 10:56 PM Let f(z) be a real entire function that is strictly increasing on the real line and has exactly 2 fixpoints A,B both being positive reals. Also f(0)=1 , f ' (A) =/= 0,1 , f ' (B) =/= 0,1. This f(z) has 2 regular superfunctions ; superf_A(z) and superf_B(z) based on what fixpoint was used. Conjecture : the functional equations that hold on the other branches of superf_A(z) are identical to the functional equations that hold on the other branches of superf_B(z). ------------------------------------------------------------------------------ Note : We assumed the existance of superfunctions , some functions do not even have that ! ( such as polynomials of degree 2 ) ------------------------------------------------------------------------------ Remark : We can define a half-iterate that agrees on both fixpoints withing the interval [A,B] but it has singularies at A,B. A similar conjecture can be made about those singularities. Or about the singularities of its superfunction. Example of Remark : f(z) = (5/4)^z. For A < z < B define : g(z) = SUM_k cos(2pi k) f^[k](z) where the sum runs over all the integers k. Now g(z) = g((5/4)^z) So if g(w) = 0 with z

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