Consider a function f that is strictly increasing , u-shaped and real-analytic for real > 0.
With fixpoints at 0 with derivative 0 and fixpoint at 1 with derivative 2.
No other fixpoints.
Then the superfunction starting at \( x_o >0 \) is given by :
Lim n to oo
\( f^{[-n]} ( x_o ^{2^{(n+z + \theta(z) )}} ). \).
Where \( \theta \) is a 1 periodic real function.
How about other cases ?
Like also 2 fixpoints but at other positions with other derivatives.
This solution was based on x^2 because that has a closed form.
For instance \( sinh ( m. arcsinh(x) ) ^{[z] } \) also has a closed form for positive integer m, just like x^2.
A second question is how to get An analytic half-iterate out of this.
1) we need such a closed form
2) the super needs to be analytic
3) we need analytic continuation from x_a to x_b , where
Small fixpoint < x_a < Large fixpoint < x_b.
Leading to question 3,5 : the Roc of the Taylor series at expansion points and the agreement of 2 taylors when expanded at different points.
In the Bummer thread this has been adressed a Tiny bit , but that was for Natural iteration ( theta = 0 , koenigs etc )
Here the attention Goes to the right theta , the correct helping function with a closed form like x^2 , and the questions above.
Also Sheldon has investigated the natural method with An added theta_1 and theta_2 for resp. Each fixpoint , then setting them equal with the equation super_1 = super_2 and hoping to find the correct thetas like that so that the equality is fullfilled and analytic in An interval containing both fixpoints.
But this is different thus.
Also Bo asked questions about supers with closed forms.
Sorry for the late edit.
Regards
tommy1729
With fixpoints at 0 with derivative 0 and fixpoint at 1 with derivative 2.
No other fixpoints.
Then the superfunction starting at \( x_o >0 \) is given by :
Lim n to oo
\( f^{[-n]} ( x_o ^{2^{(n+z + \theta(z) )}} ). \).
Where \( \theta \) is a 1 periodic real function.
How about other cases ?
Like also 2 fixpoints but at other positions with other derivatives.
This solution was based on x^2 because that has a closed form.
For instance \( sinh ( m. arcsinh(x) ) ^{[z] } \) also has a closed form for positive integer m, just like x^2.
A second question is how to get An analytic half-iterate out of this.
1) we need such a closed form
2) the super needs to be analytic
3) we need analytic continuation from x_a to x_b , where
Small fixpoint < x_a < Large fixpoint < x_b.
Leading to question 3,5 : the Roc of the Taylor series at expansion points and the agreement of 2 taylors when expanded at different points.
In the Bummer thread this has been adressed a Tiny bit , but that was for Natural iteration ( theta = 0 , koenigs etc )
Here the attention Goes to the right theta , the correct helping function with a closed form like x^2 , and the questions above.
Also Sheldon has investigated the natural method with An added theta_1 and theta_2 for resp. Each fixpoint , then setting them equal with the equation super_1 = super_2 and hoping to find the correct thetas like that so that the equality is fullfilled and analytic in An interval containing both fixpoints.
But this is different thus.
Also Bo asked questions about supers with closed forms.
Sorry for the late edit.
Regards
tommy1729