03/24/2014, 12:58 AM
Hi mike3.
Thanks for your answer.
Its a bit helpful but I feel there are still issues.
The main one being F(x)=2 having multiple solutions.
Maybe its my lack of understanding but lets compare to sexp.
The sexp has all its fixpoints at +/- oo i.
If sexp had one of its fixpoints at say a + bi for 0 <a,b< oo then that would create a problem since that would require we have a straith line.
And hence sexp would then not be analytic near c + bi for any real c.
So what does that mean ?
Is not 2 a fixpoints of the half-iterate of f ?
Is there no line in this case ?
But whenever we get F(x)=2 , because of f(2)=2 we know that we get a local periodic behaviour. (period =< 1)
Guess similar question arise for most superfunctions of polynomials of degree 2 since they are all of double exponential nature.
(related : chebyshev polynomial , sinh(2^x) , mandelbrot , ... )
My confusion is probably because of the imho weird positions of solutions to F(x) = 2. ( because of ln branches ). In combination with the idea above ofcourse.
In other cases of superfunctions I can imagine the fixpoints (of the original function , not the superfunction) to be on other branches. Here that is problematic. Hence my special intrest.
regards
tommy1729
Thanks for your answer.
Its a bit helpful but I feel there are still issues.
The main one being F(x)=2 having multiple solutions.
Maybe its my lack of understanding but lets compare to sexp.
The sexp has all its fixpoints at +/- oo i.
If sexp had one of its fixpoints at say a + bi for 0 <a,b< oo then that would create a problem since that would require we have a straith line.
And hence sexp would then not be analytic near c + bi for any real c.
So what does that mean ?
Is not 2 a fixpoints of the half-iterate of f ?
Is there no line in this case ?
But whenever we get F(x)=2 , because of f(2)=2 we know that we get a local periodic behaviour. (period =< 1)
Guess similar question arise for most superfunctions of polynomials of degree 2 since they are all of double exponential nature.
(related : chebyshev polynomial , sinh(2^x) , mandelbrot , ... )
My confusion is probably because of the imho weird positions of solutions to F(x) = 2. ( because of ln branches ). In combination with the idea above ofcourse.
In other cases of superfunctions I can imagine the fixpoints (of the original function , not the superfunction) to be on other branches. Here that is problematic. Hence my special intrest.
regards
tommy1729