08/18/2022, 07:05 PM
Interestingly the iterative logarithm of \(e^x-1\) follows a similar pattern.
The iterative logarithm is an indicator whether all iterates at the fixed point are analytic, the defining equation is \(j(f(x))=f'(x)j(x)\).
It also satisfies \(j(x)=\frac{\partial f^{\circ t}(x)}{\partial t}\big|_{t=0}\).
It is called iterative logarithm because if you consider it a functional mapping f to j, i.e. \(j=\text{logit}[f]\) then you have \(\text{logit}[f^{\circ t}] = t\;\text{logit}[f]\).
Anyways this is the pattern:
with the advantage that it doesn't depend on t, which we took as t=1/2.
The iterative logarithm is an indicator whether all iterates at the fixed point are analytic, the defining equation is \(j(f(x))=f'(x)j(x)\).
It also satisfies \(j(x)=\frac{\partial f^{\circ t}(x)}{\partial t}\big|_{t=0}\).
It is called iterative logarithm because if you consider it a functional mapping f to j, i.e. \(j=\text{logit}[f]\) then you have \(\text{logit}[f^{\circ t}] = t\;\text{logit}[f]\).
Anyways this is the pattern:
with the advantage that it doesn't depend on t, which we took as t=1/2.
