Wild conjecture about 2 fixpoints. - Printable Version +- Tetration Forum ( https://math.eretrandre.org/tetrationforum)+-- Forum: Tetration and Related Topics ( https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1)+--- Forum: Mathematical and General Discussion ( https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3)+--- Thread: Wild conjecture about 2 fixpoints. ( /showthread.php?tid=857) |

Wild conjecture about 2 fixpoints. - tommy1729 - 05/03/2014
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<w<(5/4)^z then w = f^[1/2](z). Now by assuming everything to be analytic ( is g(z) analytic ? is f^[1/2](z) analytic ? ) ... ------------------------------------------------------------------------- this example unfortunately holds a new conjecture about the analyticity of these functions ... or has this been (dis)proved before ? ------------------------------------------------------------------------- ...we can consider these singularies at A,B for the functions in the "same way" (i mean the first conjecture made here ) ... or construct the super and consider those singularities in the same way (ditto). regards tommy1729 |