*Update, perhaps the key for the general solution, see marked paragrap below*
A funny aftermath to my previous thread.
Let base \( b \) be on the Shell-Thron-boundary, such that with \( \small b=\exp(u\cdot\exp(-u)) \) we choose some real \( \small c \) such that \( \small u=\exp(2 \cdot \pi \cdot I/c) \) .
I used \( \small c= \) golden ratio \( \small\approx1.61 \)
We observed in the previous thread that with \( \small z_0=a+0 I \) and \( \small 0.5<a<1 \) all iterates lay on a closed curve (of the shape between a circle and a rough cut throug a potato).
So I speculated, that maybe the half-iterate (and of course all fractional iterates) should lay on this curve as well.
Now: can we find the approximation to the half-iterate \( \small h=0.5 \)using some high integer \( \small h \) - perhaps supported by the convergents of the continued fraction (which gives height-indexes for best-approximating natural iterates)?
After some numerical experiments it seems that this speculation is meaningful.
---------------------------
Let us assume, that the Schröder-mechanism indeed gives "the" best half-iterate. So to compute some version of the half-iterate, I consider the Schröder-function \( \small \Psi(z) \) and the inverse \( \small \Psi^{-1}(z) \).
Let for an example \( \small z_0=0.7 \), then the Schröder's half-iterate is \( \small z_{0.5}=\Psi^{-1 }(u^{0.5}\;\cdot \Psi(z))\approx 0.881010-0.539753\cdot I \). Next we check, whether this Schröder-half-iterate \( z_{0.5} \) is asymptotically on that curve of the orbit from \( \small z_0 \). Let's find that iteration-heights, which are approximating that value. If we come arbitrarily near to it, we can meaningfully define that Schröder-half-iterate as the limit of what we might provisorically call "Siegel-disc/cont-frac-half-iterate". Let's see the progress of approximation using increasing heights \(h\) , where we just document that \(h\) where the approximation has local optima.
This gave the following table:
The differences of the iteration-heights are actually from the convergents of the cont-frac of c (read first row), and in steps of 3 (read star-markers):
So we get the surprising approximation using
\( h(m)= 1 + \sum_{k=0}^m p\[3k+2\] \)
where \( p\[k\] \) is the numerator of the k'th convergent of the continued fraction of \( c \) and then
\( \lim_{m\to \infty} |\exp_b^{0.5}(z_0)-\exp_b^{h(m)}(z_0)|=0 \)
*Update* The key for the observation is likely that because of the circular behave of powers of \(u\) we have a modulo-situation in the background and that evaluates to \( h=0.5 \)
\( \hspace{96}\small\lim_{m\to \infty} h(m)\equiv0.5\hspace{48}\pmod{\frac1c} \)
It's similar when I used \( \small z_0=0.8 \) and I'd like to know whether this can be generalized to other fractional heights ...
For instance here is the approximation to the \(h=1/4\) fractional height:
This gave the following table. I've not yet an idea how to adapt the beginning such that the *diff(i_cv)* harmonize meaningfully:
Well, if this works more generally, then I'd conclude: the properties of the Siegel-disc with that type of exponential bases supports the meaningfulness of the Schröder-mechanism for the computation of the fractional iterates in that cases.
(my time is reduced at the moment, perhaps I can come back to this later)
A funny aftermath to my previous thread.
Let base \( b \) be on the Shell-Thron-boundary, such that with \( \small b=\exp(u\cdot\exp(-u)) \) we choose some real \( \small c \) such that \( \small u=\exp(2 \cdot \pi \cdot I/c) \) .
I used \( \small c= \) golden ratio \( \small\approx1.61 \)
We observed in the previous thread that with \( \small z_0=a+0 I \) and \( \small 0.5<a<1 \) all iterates lay on a closed curve (of the shape between a circle and a rough cut throug a potato).
So I speculated, that maybe the half-iterate (and of course all fractional iterates) should lay on this curve as well.
Now: can we find the approximation to the half-iterate \( \small h=0.5 \)using some high integer \( \small h \) - perhaps supported by the convergents of the continued fraction (which gives height-indexes for best-approximating natural iterates)?
After some numerical experiments it seems that this speculation is meaningful.
---------------------------
Let us assume, that the Schröder-mechanism indeed gives "the" best half-iterate. So to compute some version of the half-iterate, I consider the Schröder-function \( \small \Psi(z) \) and the inverse \( \small \Psi^{-1}(z) \).
Let for an example \( \small z_0=0.7 \), then the Schröder's half-iterate is \( \small z_{0.5}=\Psi^{-1 }(u^{0.5}\;\cdot \Psi(z))\approx 0.881010-0.539753\cdot I \). Next we check, whether this Schröder-half-iterate \( z_{0.5} \) is asymptotically on that curve of the orbit from \( \small z_0 \). Let's find that iteration-heights, which are approximating that value. If we come arbitrarily near to it, we can meaningfully define that Schröder-half-iterate as the limit of what we might provisorically call "Siegel-disc/cont-frac-half-iterate". Let's see the progress of approximation using increasing heights \(h\) , where we just document that \(h\) where the approximation has local optima.
Code:
\\ code for Pari/GP , realpresision at least 40 dec digits
c=(sqrt(5)+1)/2 \\ golden ratio, later we'll try different c
[u = exp(2*Pi*I/c) , t=exp(u) , bl=u/t , b=exp(bl) ] \\ b is base here, t is fixpoint
z0 = 0.7
s0 = schr ( z0 ) \\ compute schroeder-value s0 for startvalue z0 (using well known power series for schroeder)
\\ s0 ~ 0.306477 + 0.902543*I
z_05 = schrI ( s0 * u^0.5 ) \\ compute half-iterate for z0 by inverse Schroeder function
\\ z_05 ~ 0.881010 - 0.539753*I
\\ now display the improving approximations of consecutive h'th iterates to z_05
{ w = z0 ; \\ = 0.7
mindist = 9 ; minh = -1 ;
for(h=1,1 000 000,
w = exp( bl*w );
d = abs(w - z_05);
if( d >mindist , next() );
mindist = d; minh = h; minw = w;
print( [ minh, minw, mindist ] );
) }This gave the following table:
Code:
h z_h as approx to z_05 distance difference to next h
-------------------------------------------------------------------------
[ 1, 0.0994504 - 0.793112*I, 0.821600] + 2
[ 3, 0.886975 - 0.365889*I, 0.173966] + 8
[ 11, 0.869856 - 0.584506*I, 0.0461214] + 34
[ 45, 0.883019 - 0.529344*I, 0.0106015] + 144
[ 189, 0.880502 - 0.542220*I, 0.00251854] + 610
[ 799, 0.881128 - 0.539172*I, 0.000593661] + 2584
[ 3383, 0.880982 - 0.539891*I, 0.000140194] + 10946
[ 14329, 0.881017 - 0.539721*I, 0.0000330925] + 46368
[ 60697, 0.881009 - 0.539761*I, 0.00000781224] +196418
[257115, 0.881011 - 0.539752*I, 0.00000184421]The differences of the iteration-heights are actually from the convergents of the cont-frac of c (read first row), and in steps of 3 (read star-markers):
Code:
cvgts of cont-frac of c
* * * * * * * * *
[1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 10946 17711 28657 46368 75025 121393 196418 317811 514229 832040]
[0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 10946 17711 28657 46368 75025 121393 196418 317811 514229]So we get the surprising approximation using
\( h(m)= 1 + \sum_{k=0}^m p\[3k+2\] \)
where \( p\[k\] \) is the numerator of the k'th convergent of the continued fraction of \( c \) and then
\( \lim_{m\to \infty} |\exp_b^{0.5}(z_0)-\exp_b^{h(m)}(z_0)|=0 \)
*Update* The key for the observation is likely that because of the circular behave of powers of \(u\) we have a modulo-situation in the background and that evaluates to \( h=0.5 \)
\( \hspace{96}\small\lim_{m\to \infty} h(m)\equiv0.5\hspace{48}\pmod{\frac1c} \)
It's similar when I used \( \small z_0=0.8 \) and I'd like to know whether this can be generalized to other fractional heights ...
For instance here is the approximation to the \(h=1/4\) fractional height:
Code:
z_025=schrI(s0 * u^0.25 )
\\ z_025= 0.843362 - 0.211533*I
{w=0.7;
mindist=9;minh=-1;oldminh=0;
for(h=1,1 000 000,
w=exp(bl*w);
d=abs(w-z_025);
if( d>mindist,next());
mindist=d;minh=h;minw=w;
print([minh,minw,mindist,minh-oldminh]);
oldminh=minh) }This gave the following table. I've not yet an idea how to adapt the beginning such that the *diff(i_cv)* harmonize meaningfully:
Code:
h approx to z_025 dist diff(h) i_cv diff(i_cv) /// here i_cv= index of diff(h) in cvgts-numerators
--------------------------------------------------------------------------
[ 1, 0.0994504 - 0.793112*I, 0.944266, 1] 1 1
[ 2, 0.187985 + 0.00909452*I, 0.691517, 1] 2 3
[ 3, 0.886975 - 0.365889*I, 0.160399, 5] 5 1
[ 8, 0.792226 - 0.114578*I, 0.109614, 8] 6 3
[ 16, 0.860617 - 0.255881*I, 0.0475870, 34] 9 2
[ 50, 0.846561 - 0.219068*I, 0.00818594, 89] 11 1
[ 139, 0.840795 - 0.205664*I, 0.00640565, 144] 12 3
[ 283, 0.844382 - 0.213908*I, 0.00258456, 610] 15 2
[ 893, 0.843542 - 0.211950*I, 0.000454229, 1597] 17 1
[ 2490, 0.843220 - 0.211204*I, 0.000358160, 2584] 18 3
[ 5074, 0.843419 - 0.211665*I, 0.000143838, 10946] 21 2
[ 16020, 0.843372 - 0.211556*I, 0.0000253073, 28657] 23 1
[ 44677, 0.843354 - 0.211515*I, 0.0000199633, 46368] 24 3
[ 91045, 0.843365 - 0.211540*I, 0.00000801522, 196418] 27 2
[287463, 0.843363 - 0.211534*I, 0.00000141031, 514229] 29 1
[801692, 0.843362 - 0.211532*I, 0.00000111253, ]
cvgts
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 index k
* * * * * * * * * * * * * * * * marked
[1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 10946 17711 28657 46368 75025 121393 196418 317811 514229 832040] numerator
[0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 10946 17711 28657 46368 75025 121393 196418 317811 514229] denominatorWell, if this works more generally, then I'd conclude: the properties of the Siegel-disc with that type of exponential bases supports the meaningfulness of the Schröder-mechanism for the computation of the fractional iterates in that cases.
(my time is reduced at the moment, perhaps I can come back to this later)
Gottfried Helms, Kassel
