Tetration Forum
Approximation to half-iterate by high indexed natural iterates (base on ShlThrb) - Printable Version

+- Tetration Forum (https://tetrationforum.org)
+-- Forum: Tetration and Related Topics (https://tetrationforum.org/forumdisplay.php?fid=1)
+--- Forum: Mathematical and General Discussion (https://tetrationforum.org/forumdisplay.php?fid=3)
+--- Thread: Approximation to half-iterate by high indexed natural iterates (base on ShlThrb) (/showthread.php?tid=1233)

Approximation to half-iterate by high indexed natural iterates (base on ShlThrb) - Gottfried - 08/21/2019

*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.

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] denominator


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)

RE: Approximation to half-iterate by high indexed natural iterates (base on ShlThrb) - tommy1729 - 09/09/2019

I am confused by your sum formula and use of continued fractions.

I Will post a thread in a few min to explain how I do or see dynamics of a Siegel disk.

Maybe you can clarity and compare.

Regards 

Tommy1729