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Parabolic Iteration
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bo198214 Wrote:A famous example for this second case is , see [2]. It has a fixed point at 0, and is parabolic: . But the parabolic iterates converge only for integer s.

However this looks more terrible than it is, because there are so called asymptotic expansions. That means that in the development point the series does not converge, but the function approximates in a certain way the (formal) power series in the point of development, see [5]. For this case Ecalle [3] showed, that there is a unique continuous iteration that has the formal continuous iteration as its asymptotic expansion (for series with and ). The first however who treated this case was Szekeres in [4].

Sweet, I had concluded the same thing and was trying to turn my conceptual proof into a formal proof. But if the work's been done already, then I can breathe easy.

Essentially, if k is the number of terms at which we truncate the series expansion, then there is a non-zero radius for which the series is initially convergent (i.e., the root-test for terms 1 through k would all be less than 1).

As k is increased the radius of initial convergence decreases towards zero, but the evalutation of the series well inside that radius (e.g., within 1/2 that radius) converges asymptotically, and we can define an alternating Cauchy sequence (2 terms for each k, a least upper bound and a greatest lower bound) that has a definite limit. Within the 1/2 radius, for example, we can exponentially constrain the change in each successive term of the sequence (up to k), allowing us to define a least upper bound and a greatest lower bound for the asymptote, with the distance between these bounds decreasing with increasing k.

Sounds nice, but showing it formally is proving difficult with my lack of rigorous formal training in mathematics.

Regardless, if the proof has already been shown, then combined with my change of base formula, we now have a unique solution to tetration of bases greater than eta.

By the way, for the reference to Ecalle, where can I get a copy, and more importantly, is there an English translation available?
~ Jay Daniel Fox
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Parabolic Iteration - by andydude - 08/10/2007, 08:39 PM

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