03/25/2023, 12:57 PM
but f'(x) = f(f(x)) is a problematic equation for x near a fixpoint.
Say the fixpoint is zero.
take the truncated taylor series for an infinitesimal h that vanishes at h^n = 0.
then
f(h) = 0 + a h + b h^2 + ... + 0.
f'(h) =/= f(f(h))
not even close.
So we already have issues with integer iterations and integer derivatives.
Similarly the carleman matrix A(f) does not satisfy
D^n A(f) = A(f)^n
Or take an example without fixpoints :
t(x) = x + 1.
the derivatives are 0 eventually.
while the iterations are not.
regards
tommy1729
Say the fixpoint is zero.
take the truncated taylor series for an infinitesimal h that vanishes at h^n = 0.
then
f(h) = 0 + a h + b h^2 + ... + 0.
f'(h) =/= f(f(h))
not even close.
So we already have issues with integer iterations and integer derivatives.
Similarly the carleman matrix A(f) does not satisfy
D^n A(f) = A(f)^n
Or take an example without fixpoints :
t(x) = x + 1.
the derivatives are 0 eventually.
while the iterations are not.
regards
tommy1729