I've a small contribution to this thread, however while I initially thought I could provide a nice solution to give an eich-value for synchronizing the fixpoint-specific results and a functional relation between the values of the two fixpoint-specific versions, this idea finally failed, but maybe there is still some extentable idea in it.

We look at the curve for b^x , where b=sqrt(2), and find the three segments

s0 = -inf..(2

s1 = 2)..(4

s2 = 4).. inf

corresponding to the fixpoints.

We also look at the display of the same thing in terms of the function, depending on h, so

f(h) = exp_b°h(x)

where we assign to x some constant value and write f_2(h) for the function developed at fixpoint 2 and f_4(h) for that at fixpoint 4.

In the graphs for f(h) in the beginning of this thread

(see Dmitri's graph) we have a natural norm for the horizontal centering of the curve in the segment s0; we simply have, that at height h=0 the value of f(h) is x, or if as usual x=1, then f(0) = 1 - and this gives the required norm, or "natural center", for the curve.

For the curves in the other segments s1 and s2 we don't have such a natural center, it was just arbitrarily taken as h=0 where x=3 and f(0)=x for the segment s1 and x=5 f(0)=x for the segment s2.

Now we consider only the curve in the segment s1.

On the other hand we look at the differences of f(h) when developed at the different fixpoints t0=2 and t1=4 in the segment s1. A rough curve for f_2(h)-f_4(h) is also given in the plot of Dmitri; it is nearly sinusoidal with the height-parameter h and periodic with h (mod 1).

However, in some more detailed versions of our graphs we can see, that the curve of the differences not only has decreasing amplitude when h approaches the infinity, but also, in general is not very well symmetric wrt x-axis. ( see some curves in the plot some curves in the posting in the thread)

Here comes now my observation/idea.

The asymmetry depends on the initial choice of x; if we select a "good" x, the curve of differences is nicely symmetric wrt to the x-axis (or differently stated: to the occuring y-values). That also means, that the fixpoint-2 curve and fixpoint4-curve are sometimes in phase and sometimes not, depending on the "eich-value" x for h=0.

I was searching for some special property of such x. and the striking finding was, that if I used the imaginary height of Pi*I, beginning at x=1, then the curve of differences is nicely symmetric and "in phase".

Practically it is simply to negate the value of the schröder-function at x=1.

Example:

If we use the fixpoint-2-function, compute

we get some (real!) value x1 in the segment s1, which taken as norm for that segment x=x1, f(0)=x1 gives a very accurate symmetric image for the curve of differences.

However - I first thought here I got the perfect match, but this was only a near-match.

Instead I needed a binary search to find an even better norm-value, where symmetry was perfect; this could be seen, when not only the integer iterates but also the half-iterates f_2(0.5) and f_4(0.5) in that segment were equal. Then also the quarter-iterates f_2(0.25) - f_4(0.25) give the maximal difference dmax and using that value for the definition of the amplitude of a sinus-function amp=dmax one can relate the two functions by

This is a very nice functional relation.

...

This *were* a very nice functional relation...

However - again this is not the end. There is again a (smaller though) difference of the curves, and again not perfectly in phase.

May be one can continue this using fourier-analysis to some reasonable extend; unfortunately I've no clue about this matter.

So - my idea was to propose one eich-value x for the segments; I didn't actually succeed but the consideration may not be without worth. I think, the idea to use the value at the height I*Pi iterated from 1 has some charme, maybe someone finds an idea how to realize that.

Gottfried

We look at the curve for b^x , where b=sqrt(2), and find the three segments

s0 = -inf..(2

s1 = 2)..(4

s2 = 4).. inf

corresponding to the fixpoints.

We also look at the display of the same thing in terms of the function, depending on h, so

f(h) = exp_b°h(x)

where we assign to x some constant value and write f_2(h) for the function developed at fixpoint 2 and f_4(h) for that at fixpoint 4.

In the graphs for f(h) in the beginning of this thread

(see Dmitri's graph) we have a natural norm for the horizontal centering of the curve in the segment s0; we simply have, that at height h=0 the value of f(h) is x, or if as usual x=1, then f(0) = 1 - and this gives the required norm, or "natural center", for the curve.

For the curves in the other segments s1 and s2 we don't have such a natural center, it was just arbitrarily taken as h=0 where x=3 and f(0)=x for the segment s1 and x=5 f(0)=x for the segment s2.

Now we consider only the curve in the segment s1.

On the other hand we look at the differences of f(h) when developed at the different fixpoints t0=2 and t1=4 in the segment s1. A rough curve for f_2(h)-f_4(h) is also given in the plot of Dmitri; it is nearly sinusoidal with the height-parameter h and periodic with h (mod 1).

However, in some more detailed versions of our graphs we can see, that the curve of the differences not only has decreasing amplitude when h approaches the infinity, but also, in general is not very well symmetric wrt x-axis. ( see some curves in the plot some curves in the posting in the thread)

Here comes now my observation/idea.

The asymmetry depends on the initial choice of x; if we select a "good" x, the curve of differences is nicely symmetric wrt to the x-axis (or differently stated: to the occuring y-values). That also means, that the fixpoint-2 curve and fixpoint4-curve are sometimes in phase and sometimes not, depending on the "eich-value" x for h=0.

I was searching for some special property of such x. and the striking finding was, that if I used the imaginary height of Pi*I, beginning at x=1, then the curve of differences is nicely symmetric and "in phase".

Practically it is simply to negate the value of the schröder-function at x=1.

Example:

If we use the fixpoint-2-function, compute

Code:

`´x0=1`

a0 = schr(x0/t0-1)

a1 = - a0

x1 = t0*(1+schr°-1(a1))

we get some (real!) value x1 in the segment s1, which taken as norm for that segment x=x1, f(0)=x1 gives a very accurate symmetric image for the curve of differences.

However - I first thought here I got the perfect match, but this was only a near-match.

Instead I needed a binary search to find an even better norm-value, where symmetry was perfect; this could be seen, when not only the integer iterates but also the half-iterates f_2(0.5) and f_4(0.5) in that segment were equal. Then also the quarter-iterates f_2(0.25) - f_4(0.25) give the maximal difference dmax and using that value for the definition of the amplitude of a sinus-function amp=dmax one can relate the two functions by

Code:

`´find optimal eich-value for x, set this for h=0`

then

amp = f_4(0.25)-f_2(0.25)

and the functional relation

f_2(h) = f_4(h+ amp*sin(h*2*Pi))

This is a very nice functional relation.

...

This *were* a very nice functional relation...

However - again this is not the end. There is again a (smaller though) difference of the curves, and again not perfectly in phase.

May be one can continue this using fourier-analysis to some reasonable extend; unfortunately I've no clue about this matter.

So - my idea was to propose one eich-value x for the segments; I didn't actually succeed but the consideration may not be without worth. I think, the idea to use the value at the height I*Pi iterated from 1 has some charme, maybe someone finds an idea how to realize that.

Gottfried

Gottfried Helms, Kassel