+- Tetration Forum (https://math.eretrandre.org/tetrationforum) +-- Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1) +--- Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/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 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