08/02/2022, 07:21 PM
I just was investigating the subject with linear fractional functions, and I came up with this example:
\(f(z)=\frac{2z}{1+z}\)
This function has exactly two fixed points, one at 0 and one at 1.
With a bit of calculating one can come up with an explicit formula for the iteration of this function:
\(f^{\circ t}(z)=\frac{2^tz}{1+(2^t-1)z}\)
As all the iterates of the function are analytic at both fixed points, it must be the regular iteration at both fixed points.
And hence we have an example of a function that can be analytically iterated at 2 fixed points at the same time.
Of course one can say linear fractional functions are somehow the trivial regular iterations (though I am not sure whether I could really give an explicit formula for the general iteration of \(f(z)=\frac{a+bz}{c+dz}\) ). But then for which other functions is that also possible?!
\(f(z)=\frac{2z}{1+z}\)
This function has exactly two fixed points, one at 0 and one at 1.
With a bit of calculating one can come up with an explicit formula for the iteration of this function:
\(f^{\circ t}(z)=\frac{2^tz}{1+(2^t-1)z}\)
As all the iterates of the function are analytic at both fixed points, it must be the regular iteration at both fixed points.
And hence we have an example of a function that can be analytically iterated at 2 fixed points at the same time.
Of course one can say linear fractional functions are somehow the trivial regular iterations (though I am not sure whether I could really give an explicit formula for the general iteration of \(f(z)=\frac{a+bz}{c+dz}\) ). But then for which other functions is that also possible?!
