Daniel Wrote:Two techniques were published in the mid nineties, one using Bell matrices and the other using another matrix technique, the Carleman linearization technique. So there are three different techniques for continuously iterating functions from the nineties.
What do you mean by techniques?
Uniqueness and existence for solutions in the hyperbolic case are quite old: publications reach back to the beginning of the 20th century while the most known results including the parabolic case were found in the 60s. This means for most types of differentiable functions with parabolic or hyperbolic fixed point there is "the" (i.e. the regular) solution of continuous iteration.
The equivalence of powerseries composition with bell matrix multiplication and hence the derivation of continuous iteration via matrix exponentiation was already known to Jabotinsky 1961 (especially for the parabolic case, where you have a non-limit formula for the coefficients as opposed to the hyperbolic case).
Quote:Their main limitation is that they require a hyperbolic fixed point.
For me the limitation is that they require a fixed point at all.
Thatswhy it is so interesting to consider the real iteration of \( e^x \) or tetration for bases greater than \( e^{1/e} \) because it has no fixed point, and we know from Kneser (1950) that the development at a complex fixed point yields complex values for real arguments. Is there anyway a proof that developing at different complex fixed points yields always the same continuous iteration?
