And now it gets really cool!
We can connect back to the question which functions do have all analytic iterates at the fixed point and which not.
The previous functions in this thread are all polynomial or are entire, hence the logit will not converge (as well as most of the corresponding parabolic iterations).
But in my post here, I constructed a parabolic function where all iterates are analytic at the fixed point.
This function was \(\arctan(1+\tan(x)) + \pi\left\lfloor \frac{x+\frac{\pi}{2}}{\pi}\right\rfloor\)
with the fixed point \(\frac{\pi}{2}\). When we conjugate the fixed point to 0, the function can be written as
\[f(x)=-{\rm arccot}(1-\cot(x))\]
As we are only interested in a small vicinity around 0, I omit the branch compensation. So we are looking for a powerseries expansion of this function. Actually Sage has difficulties calculating the powerseries because division of 0, so I took the detour differentiating the function \(f'(x) = \frac{\cot(x)^2 + 1}{(1-\cot(x))^2 + 1}\), then calculating the formal powerseries and integrating it, so these are the coefficients of f
0, 1, 1, 1, 2/3, 0, -43/45, -29/15, -778/315, -374/189, 122/14175, ...
From the example before one could conclude that the coefficient growth would be \(\frac{(k-3)!}{(2\pi)^k}\). But this is totally not the case, the logit is converging! The logit j has the coefficients:
0, 0, 1, 0, -1/3, 0, 2/45, 0, -1/315, 0, 2/14175, 0, -2/467775, 0, ...
the repeating 0 are interesting, because you can not produce these e.g. with polynomials. So here we have a completely different behaviour:
And now that I am experimenting with I even can give an explicit formula for the coefficients of the logit of \(-{\rm arccot}(1-\cot(x))\)!
\[j_k = \frac{-\cos(\frac{\pi}{2}k)}{2}\frac{2^k}{k!}, \quad k\ge 1\]
So all the global behaviour of a holomorphic function is concentrated in the powerseries development in one point (the whole function can be reconstructed by analytic continuation). And for the logit to converge (equivalent to all regular iterates are analytic at the fixed point) seems quite to depend on the global behaviour of the function - meromorphic function with countable isolated singularities can not have a convergent logit (only talking about parabolic fixed points here). While some multivalued functions (i.e. they have branch points) can have a convergent logit - as I just showed. But we can not read all these properties from the coefficients, so the convergence of the logit will remain a mystery!
We can connect back to the question which functions do have all analytic iterates at the fixed point and which not.
The previous functions in this thread are all polynomial or are entire, hence the logit will not converge (as well as most of the corresponding parabolic iterations).
But in my post here, I constructed a parabolic function where all iterates are analytic at the fixed point.
This function was \(\arctan(1+\tan(x)) + \pi\left\lfloor \frac{x+\frac{\pi}{2}}{\pi}\right\rfloor\)
with the fixed point \(\frac{\pi}{2}\). When we conjugate the fixed point to 0, the function can be written as
\[f(x)=-{\rm arccot}(1-\cot(x))\]
As we are only interested in a small vicinity around 0, I omit the branch compensation. So we are looking for a powerseries expansion of this function. Actually Sage has difficulties calculating the powerseries because division of 0, so I took the detour differentiating the function \(f'(x) = \frac{\cot(x)^2 + 1}{(1-\cot(x))^2 + 1}\), then calculating the formal powerseries and integrating it, so these are the coefficients of f
0, 1, 1, 1, 2/3, 0, -43/45, -29/15, -778/315, -374/189, 122/14175, ...
From the example before one could conclude that the coefficient growth would be \(\frac{(k-3)!}{(2\pi)^k}\). But this is totally not the case, the logit is converging! The logit j has the coefficients:
0, 0, 1, 0, -1/3, 0, 2/45, 0, -1/315, 0, 2/14175, 0, -2/467775, 0, ...
the repeating 0 are interesting, because you can not produce these e.g. with polynomials. So here we have a completely different behaviour:
And now that I am experimenting with I even can give an explicit formula for the coefficients of the logit of \(-{\rm arccot}(1-\cot(x))\)!
\[j_k = \frac{-\cos(\frac{\pi}{2}k)}{2}\frac{2^k}{k!}, \quad k\ge 1\]
So all the global behaviour of a holomorphic function is concentrated in the powerseries development in one point (the whole function can be reconstructed by analytic continuation). And for the logit to converge (equivalent to all regular iterates are analytic at the fixed point) seems quite to depend on the global behaviour of the function - meromorphic function with countable isolated singularities can not have a convergent logit (only talking about parabolic fixed points here). While some multivalued functions (i.e. they have branch points) can have a convergent logit - as I just showed. But we can not read all these properties from the coefficients, so the convergence of the logit will remain a mystery!
