Fractals/Iterations in the complex plane/Fatou coordinate 1

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G Holms[edit]

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

Will Jagy[edit]

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

References[edit]

  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