06/22/2011, 03:23 PM
(This post was last modified: 06/23/2011, 01:50 PM by sheldonison.)

(06/21/2011, 12:13 PM)Gottfried Wrote: ....I made a picture, how the fixpoint, the log of the fixpoint, and the base according to the Shell-Thron-description are connected. (I've also put it in the hyperop-wiki). In my notation I always used u (for the log of the fixpoint), t =exp(u) for the fixpoint and b=exp(u/t) for the base....My conjecture is that as long as the period is an irrational real number, then I suspect the superfunction is analytic. As we have seen, for the base with a real period=3, starting with a point near L, and iterating the function x=b^x three times, doesn't get you back to the initial starting point, which seems to be a big problem, which prevents the definition of the superfunction as being real periodic, with a period=3.

The function f(z)=b^(L+z) for a base on the Shell-Thron region, has a Taylor series developed in the neighborhood of the fixed point L with the following form:

, which is also number on the unit circle, since abs(log(L))=1. In Gottfried's notation, a1=u.

, where the period is a real number

The goal is to show that if the period is irrational, then for good rational approximations of the period (developed from the continued fraction), where:

and that as the approximation for the period becomes more and more exact, then iterating b^(L+z) m times also becomes more and more exact, so that in the limit, for a small enough z (to avoid fractal chaos), then

, where m is the numerator of a continued fraction approximation of the period. In that case, Then the period "works". For all integer values of m, we can define the superfunction for x=(m mod period), starting from initial value L+z:

where . And for any real x, a sequence of more and more accurate approximations can be used, (edit not sure how to define this sequence).

, and then there is justification for developing a superfunction of b^z, which has a real period, and decays to the constant L as .

Going back to the series for

I conjecture that there aren't restrictions on a2..an, for b^(L+z), so that real period works, so long as the period is irrational, and the superfuntion is developed with z small enough to avoid fractal chaos. But if the period is a rational number, there are more strict restrictions on a2..an, since for a small integer number of iterations, then iterating must be exact if period=m/n. For example, a2..an=0 always works, and many other examples also work for a rational period, but in the general case for arbitrary a2..an, the rational period doesn't work. That means that in the general case, the superfunction on the Shell Thron boundary with a rational period doesn't work, and we can't develop the superfunction with that rational period. Whereas, the conjecture is that we can develop the superfunction whenever the period is an irrational number on the Shell-Thron boundary. It will be interesting to see if the conjecture turns out to be true, and how it can be proven (or disproven).

- Sheldon