Fractals/Iterations in the complex plane/Fatou coordinate 1

G HolmsEdit

Fractional iteration of the function f(x) = 1/(1+x) by [1]

Will JagyEdit

Below is an example by Will Jagy[2]

"First, an example. Begin with

 

which has derivative 1 at   but, along the positive real axis, is slightly less than   when  .

We want to find a Fatou coordinate, which Milnor (page 107)[3] denotes   that is infinite at   and otherwise solves what is usually called the Abel functional equation,[4]

 

There is only one holomorphic Fatou coordinate up to an additive constant. We take

 

To get fractional iterates   of  , with real   we take

 

and finally

 

The desired semigroup homomorphism holds,

 

with   and  

Alright, the case of   emphasizing the positive real axis is not terribly different, as long as we restrict to the interval   For any such   define   and in general   This sequence approaches 0, and in fact does so for any   in a certain open set around the interval   that is called a petal.

Now, given a specific   with   and   it is a result of Jean Ecalle at Orsay that we may take  

Note that   actually is defined on   with   but the symmetry also means that the inverse function returns to the interval  

Before going on, the limit technique in the previous paragraph is given in pages 346-353 of *Iterative Functional Equations* by Marek Kuczma, Bogdan Choczewski, and Roman Ger. The solution is specifically Theorem 8.5.8 of subsection 8.5D, bottom of page 351 to top of page 353. Subsection 8.5A, pages 346-347, about Julia's equation, is part of the development.

As before, we define ( at least for  ) the parametrized interpolating functions,  

In particular  

I calculated all of this last night. First, by the kindness of Daniel Geisler, I have a pdf of the graph of this at:

http://zakuski.math.utsa.edu/~jagy/sine_half.pdf

Note that we use the evident symmetries   and  

The result gives an interpolation of functions   ending at   but beginning at the continuous periodic sawtooth function,   for   then   for   continue with period   We do get   plus the holomorphicity and symmetry of   show that   is analytic on the full open interval  

    • EDIT, TUTORIAL**: Given some   in the complex plane in the interior of the equilateral triangle with vertices at   take   in general   and   It does not take long to show that   stays within the triangle, and that   as  

Second, say   is a true Fatou coordinate on the triangle,   although we do not know any specific value. Now,   Also   Induction, given   we have  

So, given   we have  

Third , let  . This is a sort of asymptotic expansion (at 0) for   the error is   It is unlikely that putting more terms on   leads to a convergent series, even in the triangle.

Fourth, given some   in the triangle. We know that  . So   Or   finally   Thus the limit being used is appropriate.

Fifth, there is a bootstrapping effect in use. We have no actual value for   but we can write a formal power series for the solution of a Julia equation for   that is   The formal power series for   begins (KCG Theorem 8.5.1) with   the first term in the power series of   after the initial   We write several more terms,   We find the formal reciprocal,   Finally we integrate term by term,   and truncate where we like,  

Numerically, let me give some indication of what happens, in particular to emphasize

 
       x      alpha(x)      f(x)       f(f(x))     sin x       f(f(x))- sin x
   1.570796   2.089608    1.140179    1.000000    1.000000      1.80442e-11
   1.560796   2.089837    1.140095    0.999950    0.999950      1.11629e-09
   1.550796   2.090525    1.139841    0.999800    0.999800      1.42091e-10
   1.540796   2.091672    1.139419    0.999550    0.999550      3.71042e-10
   1.530796   2.093279    1.138828    0.999200    0.999200      1.97844e-10
   1.520796   2.095349    1.138070    0.998750    0.998750      -2.82238e-10
   1.510796   2.097883    1.137144    0.998201    0.998201      -7.31867e-10
   1.500796   2.100884    1.136052    0.997551    0.997551      -1.29813e-09
   1.490796   2.104355    1.134794    0.996802    0.996802      -1.14504e-09
   1.480796   2.108299    1.133372    0.995953    0.995953      9.09416e-11
   1.470796   2.112721    1.131787    0.995004    0.995004      1.57743e-09
   1.460796   2.117625    1.130040    0.993956    0.993956      5.63618e-10
   1.450796   2.123017    1.128133    0.992809    0.992809      -3.00337e-10
   1.440796   2.128902    1.126066    0.991562    0.991562      1.19926e-09
   1.430796   2.135285    1.123843    0.990216    0.990216      2.46512e-09
   1.420796   2.142174    1.121465    0.988771    0.988771      -2.4357e-10
   1.410796   2.149577    1.118932    0.987227    0.987227      -1.01798e-10
   1.400796   2.157500    1.116249    0.985585    0.985585      -1.72108e-10
   1.390796   2.165952    1.113415    0.983844    0.983844      -2.31266e-10
   1.380796   2.174942    1.110434    0.982004    0.982004      -4.08812e-10
   1.370796   2.184481    1.107308    0.980067    0.980067      1.02334e-09
   1.360796   2.194576    1.104038    0.978031    0.978031      3.59356e-10
   1.350796   2.205241    1.100627    0.975897    0.975897      2.36773e-09
   1.340796   2.216486    1.097077    0.973666    0.973666      -1.56162e-10
   1.330796   2.228323    1.093390    0.971338    0.971338      -5.29822e-11
   1.320796   2.240766    1.089569    0.968912    0.968912      8.31102e-10
   1.310796   2.253827    1.085616    0.966390    0.966390      -2.91373e-10
   1.300796   2.267522    1.081532    0.963771    0.963771      -5.45974e-10
   1.290796   2.281865    1.077322    0.961055    0.961055      -1.43066e-10
   1.280796   2.296873    1.072986    0.958244    0.958244      -1.58642e-10
   1.270796   2.312562    1.068526    0.955336    0.955336      -3.14188e-10
   1.260796   2.328950    1.063947    0.952334    0.952334      3.20439e-10
   1.250796   2.346055    1.059248    0.949235    0.949235      4.32107e-10
   1.240796   2.363898    1.054434    0.946042    0.946042      1.49412e-10
   1.230796   2.382498    1.049505    0.942755    0.942755      3.42659e-10
   1.220796   2.401878    1.044464    0.939373    0.939373      4.62813e-10
   1.210796   2.422059    1.039314    0.935897    0.935897      3.63659e-11
   1.200796   2.443066    1.034056    0.932327    0.932327      3.08511e-09
   1.190796   2.464924    1.028693    0.928665    0.928665      -8.44918e-10
   1.180796   2.487659    1.023226    0.924909    0.924909      6.32892e-10
   1.170796   2.511298    1.017658    0.921061    0.921061      -1.80822e-09
   1.160796   2.535871    1.011990    0.917121    0.917121      3.02818e-10
   1.150796   2.561407    1.006225    0.913089    0.913089      -3.52346e-10
   1.140796   2.587938    1.000365    0.908966    0.908966      9.35707e-10
   1.130796   2.615498    0.994410    0.904752    0.904752      -2.54345e-10
   1.120796   2.644121    0.988364    0.900447    0.900447      -6.20484e-10
   1.110796   2.673845    0.982228    0.896052    0.896052      -7.91102e-10
   1.100796   2.704708    0.976004    0.891568    0.891568      -1.62699e-09
   1.090796   2.736749    0.969693    0.886995    0.886995      -5.2244e-10
   1.080796   2.770013    0.963297    0.882333    0.882333      -8.63283e-10
   1.070796   2.804543    0.956818    0.877583    0.877583      -2.85301e-10
   1.060796   2.840386    0.950258    0.872745    0.872745      -1.30496e-10
   1.050796   2.877592    0.943618    0.867819    0.867819      -2.82645e-10
   1.040796   2.916212    0.936899    0.862807    0.862807      8.81083e-10
   1.030796   2.956300    0.930104    0.857709    0.857709      -7.70554e-10
   1.020796   2.997914    0.923233    0.852525    0.852525      1.0091e-09
   1.010796   3.041114    0.916288    0.847255    0.847255      -4.96194e-10
   1.000796   3.085963    0.909270    0.841901    0.841901      6.71018e-10
   0.990796   3.132529    0.902182    0.836463    0.836463      -9.28187e-10
   0.980796   3.180880    0.895023    0.830941    0.830941      -1.45774e-10
   0.970796   3.231092    0.887796    0.825336    0.825336      1.26379e-09
   0.960796   3.283242    0.880502    0.819648    0.819648      -1.84287e-10
   0.950796   3.337412    0.873142    0.813878    0.813878      5.84829e-10
   0.940796   3.393689    0.865718    0.808028    0.808028      -2.81364e-10
   0.930796   3.452165    0.858230    0.802096    0.802096      -1.54149e-10
   0.920796   3.512937    0.850679    0.796084    0.796084      -8.29982e-10
   0.910796   3.576106    0.843068    0.789992    0.789992      3.00744e-10
   0.900796   3.641781    0.835396    0.783822    0.783822      8.10903e-10
   0.890796   3.710076    0.827666    0.777573    0.777573      -1.23505e-10
   0.880796   3.781111    0.819878    0.771246    0.771246      5.31326e-10
   0.870796   3.855015    0.812033    0.764842    0.764842      2.26584e-10
   0.860796   3.931924    0.804132    0.758362    0.758362      3.97021e-10
   0.850796   4.011981    0.796177    0.751806    0.751806      -7.84946e-10
   0.840796   4.095339    0.788168    0.745174    0.745174      -3.03503e-10
   0.830796   4.182159    0.780107    0.738469    0.738469      2.63202e-10
   0.820796   4.272614    0.771994    0.731689    0.731689      -7.36693e-11
   0.810796   4.366886    0.763830    0.724836    0.724836      -1.84604e-10
   0.800796   4.465171    0.755616    0.717911    0.717911      3.22084e-10
   0.790796   4.567674    0.747354    0.710914    0.710914      -2.93204e-10
   0.780796   4.674617    0.739043    0.703845    0.703845      1.58448e-11
   0.770796   4.786234    0.730686    0.696707    0.696707      -8.89497e-10
   0.760796   4.902777    0.722282    0.689498    0.689498      2.40592e-10
   0.750796   5.024513    0.713833    0.682221    0.682221      -3.11017e-10
   0.740796   5.151728    0.705339    0.674876    0.674876      7.32554e-10
   0.730796   5.284728    0.696801    0.667463    0.667463      -1.73919e-10
   0.720796   5.423842    0.688221    0.659983    0.659983      -1.66422e-10
   0.710796   5.569419    0.679599    0.652437    0.652437      5.99509e-10
   0.700796   5.721838    0.670935    0.644827    0.644827      -2.45424e-10
   0.690796   5.881501    0.662231    0.637151    0.637151      -6.29884e-10
   0.680796   6.048843    0.653487    0.629412    0.629412      1.86262e-10
   0.670796   6.224333    0.644704    0.621610    0.621610      -5.04285e-10
   0.660796   6.408471    0.635883    0.613746    0.613746      -6.94697e-12
   0.650796   6.601802    0.627025    0.605820    0.605820      -3.81152e-10
   0.640796   6.804910    0.618129    0.597834    0.597834      4.10222e-10
   0.630796   7.018428    0.609198    0.589788    0.589788      -1.91816e-10
   0.620796   7.243040    0.600231    0.581683    0.581683      -4.90592e-10
   0.610796   7.479486    0.591230    0.573520    0.573520      4.29742e-10
   0.600796   7.728570    0.582195    0.565300    0.565300      -1.38719e-10
   0.590796   7.991165    0.573126    0.557023    0.557023      -4.05081e-10
   0.580796   8.268218    0.564025    0.548690    0.548690      -5.76379e-10
   0.570796   8.560763    0.554892    0.540302    0.540302      1.49155e-10
   0.560796   8.869925    0.545728    0.531861    0.531861      1.0459e-11
   0.550796   9.196935    0.536533    0.523366    0.523366      -1.15537e-10
   0.540796   9.543137    0.527308    0.514819    0.514819      -2.84462e-10
   0.530796   9.910004    0.518054    0.506220    0.506220      6.24335e-11
   0.520796   10.299155    0.508771    0.497571    0.497571      -9.24078e-12
   0.510796   10.712365    0.499460    0.488872    0.488872      8.29491e-11
   0.500796   11.151592    0.490122    0.480124    0.480124      3.31769e-10
   0.490796   11.618996    0.480757    0.471328    0.471328      2.27307e-10
   0.480796   12.116964    0.471366    0.462485    0.462485      3.06434e-10
   0.470796   12.648140    0.461949    0.453596    0.453596      4.77846e-11
   0.460796   13.215459    0.452507    0.444662    0.444662      1.53162e-10
   0.450796   13.822186    0.443041    0.435682    0.435682      -2.87541e-10
   0.440796   14.471963    0.433551    0.426660    0.426660      -5.20332e-11
   0.430796   15.168860    0.424037    0.417595    0.417595      -8.17951e-11
   0.420796   15.917436    0.414501    0.408487    0.408487      -4.6788e-10
   0.410796   16.722816    0.404944    0.399340    0.399340      3.70729e-10
   0.400796   17.590771    0.395364    0.390152    0.390152      -6.97547e-11
   0.390796   18.527825    0.385764    0.380925    0.380925      -2.45522e-10
   0.380796   19.541368    0.376143    0.371660    0.371660      4.09758e-10
   0.370796   20.639804    0.366503    0.362358    0.362358      1.15221e-10
   0.360796   21.832721    0.356843    0.353019    0.353019      -4.75977e-11
   0.350796   23.131092    0.347165    0.343646    0.343646      -4.27696e-10
   0.340796   24.547531    0.337468    0.334238    0.334238      2.12743e-10
   0.330796   26.096586    0.327755    0.324796    0.324796      4.06133e-10
   0.320796   27.795115    0.318024    0.315322    0.315322      -2.71476e-10
   0.310796   29.662732    0.308276    0.305817    0.305817      -3.74988e-10
   0.300796   31.722372    0.298513    0.296281    0.296281      -1.50491e-10
   0.290796   34.000986    0.288734    0.286715    0.286715      2.17798e-11
   0.280796   36.530413    0.278940    0.277121    0.277121      4.538e-10
   0.270796   39.348484    0.269132    0.267499    0.267499      5.24261e-11
   0.260796   42.500432    0.259311    0.257850    0.257850      7.03059e-11
   0.250796   46.040690    0.249475    0.248175    0.248175      -1.83863e-10
   0.240796   50.035239    0.239628    0.238476    0.238476      4.06119e-10
   0.230796   54.564668    0.229768    0.228753    0.228753      -2.56253e-10
   0.220796   59.728239    0.219896    0.219007    0.219007      -7.32657e-11
   0.210796   65.649323    0.210013    0.209239    0.209239      3.43103e-11
   0.200796   72.482783    0.200120    0.199450    0.199450      -1.20351e-10
   0.190796   80.425131    0.190216    0.189641    0.189641      1.07544e-10
   0.180796   89.728726    0.180303    0.179813    0.179813      9.93221e-11
   0.170796   100.721954    0.170380    0.169967    0.169967      2.63903e-10
   0.160796   113.838454    0.160449    0.160104    0.160104      6.74095e-10
   0.150796   129.660347    0.150510    0.150225    0.150225      4.34057e-10
   0.140796   148.983681    0.140563    0.140332    0.140332      -2.90965e-11
   0.130796   172.920186    0.130610    0.130424    0.130424      4.02502e-10
   0.120796   203.060297    0.120649    0.120503    0.120503      -1.85618e-11
   0.110796   241.743576    0.110683    0.110570    0.110570      4.2044e-11
   0.100796   292.525678    0.100711    0.100626    0.100626      -1.73504e-11
   0.090796   361.023855    0.090734    0.090672    0.090672      2.88887e-10
   0.080796   456.537044    0.080752    0.080708    0.080708      -2.90848e-10
   0.070796   595.371955    0.070767    0.070737    0.070737      4.71103e-10
   0.060796   808.285844    0.060778    0.060759    0.060759      -3.90636e-10
   0.050796   1159.094719    0.050785    0.050774    0.050774      3.01403e-11
   0.040796   1798.677124    0.040791    0.040785    0.040785      3.77092e-10
   0.030796   3159.000053    0.030794    0.030791    0.030791      2.4813e-10
   0.020796   6931.973789    0.020796    0.020795    0.020795      2.95307e-10
   0.010796   25732.234731    0.010796    0.010796    0.010796      1.31774e-10
       x       alpha(x)        f(x)        f(f(x))     sin x       f(f(x))- sin x

ReferencesEdit

  1. Fractional iteration of the function f(x) = 1/(1+x) by Gottfried Helms
  2. mathoverflow question (45608) : does-the-formal-power-series-solution-to-ffx-sin-x-converge
  3. Dynamics in one complex variable: introductory lectures John W. Milnor
  4. wikipedia: Abel equation
 [1]: http://oskicat.berkeley.edu/record=b14897585~S1