Fractals/Iterations in the complex plane/Fatou coordinate 1

=G Holms= Fractional iteration of the function f(x) = 1/(1+x) by

=Will Jagy= Below is an example by Will Jagy

"First, an example. Begin with

$$f(z) = \frac{z}{1 + z}$$

which has derivative 1 at $$z=0$$ but, along the positive real axis, is slightly less than $$x$$ when $$x > 0$$.

We want to find a Fatou coordinate, which Milnor (page 107) denotes $$\alpha$$ that is infinite at $$0$$ and otherwise solves what is usually called the Abel functional equation,

$$ \alpha(f(z)) = \alpha(z) + 1$$

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

$$ \alpha(z)= \frac{1}{ z}$$

To get fractional iterates $$f_s(z)$$ of $$f(z)$$, with real $$0 \leq s \leq 1$$ we take

$$ f_s (z) = \alpha^{-1} \left( s + \alpha(z)   \right) $$

and finally

$$f_s(z) = \frac{z}{1 + s z}$$

The desired semigroup homomorphism holds,

$$ f_s(f_t(z)) = f_{s + t}(z) $$

with $$f_0(z) = z$$ and $$f_1(z) = f(z)$$

Alright, the case of $$\sin z$$ emphasizing the positive real axis is not terribly different, as long as we restrict to the interval $$ 0 < x \leq \frac{\pi}{2}.$$ For any such $$x,$$ define $$x_0 = x, \; x_1 = \sin x, \; x_2 = \sin \sin x,$$ and in general $$ x_{n+1} = \sin x_n.$$ This sequence approaches 0, and in fact does so for any $$z$$ in a certain open set around the interval $$ 0 < x \leq \frac{\pi}{2}$$ that is called a petal.

Now, given a specific $$x$$ with $$x_1 = \sin x$$ and $$ x_{n+1} = \sin x_n$$ it is a result of Jean Ecalle at Orsay that we may take $$ \alpha(x) = \lim_{n \rightarrow \infty} \; \; \; \frac{3}{x_n^2} \; + \; \frac{6 \log x_n}{5}  \; + \; \frac{79  x_n^2}{1050}   \; + \; \frac{29  x_n^4}{2625}  \; - \; n.$$

Note that $$\alpha$$ actually is defined on $$ 0 < x < \pi$$ with $$\alpha(\pi - x) = \alpha(x),$$ but the symmetry also means that the inverse function returns to the interval $$ 0 < x \leq \frac{\pi}{2}.$$

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 $$ 0 < x \leq \frac{\pi}{2}$$) the parametrized interpolating functions, $$ f_s (x) = \alpha^{-1} \left( s + \alpha(x)   \right) $$

In particular $$ f_{1/2} (x) = \alpha^{-1} \left( \frac{1}{2} + \alpha(x)   \right) $$

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 $$ f_{1/2} (-x) = - f_{1/2} (x)$$ and $$ f_{1/2} (\pi -x) =  f_{1/2} (x)$$

The result gives an interpolation of functions $$f_s(x)$$ ending at $$ f_1(x)=\sin x$$ but beginning at the continuous periodic sawtooth function, $$x$$ for $$ -\frac{\pi}{2} \leq x \leq \frac{\pi}{2},$$ then $$\pi - x$$ for $$ \frac{\pi}{2} \leq x \leq \frac{3\pi}{2},$$ continue with period $$2 \pi.$$ We do get $$ f_s(f_t(z)) = f_{s + t}(z), $$ plus the holomorphicity and symmetry of $$\alpha$$ show that $$f_s(x)$$ is analytic on the full open interval $$ 0 < x < \pi.$$


 * EDIT, TUTORIAL**: Given some $$z$$ in the complex plane in the interior of the equilateral triangle with vertices at $$0, \sqrt 3 + i, \sqrt 3 - i,$$ take $$z_0 = z, \; \; z_1 = \sin z, \; z_2 = \sin \sin z,$$ in general $$z_{n+1} = \sin z_n$$ and $$z_n = \sin^{[n]}(z).$$ It does not take long to show that $$z_n$$ stays within the triangle, and that $$z_n \rightarrow 0$$ as $$n \rightarrow \infty.$$

Second, say $$\alpha(z)$$ is a true Fatou coordinate on the triangle, $$\alpha(\sin z) = \alpha(z) + 1,$$ although we do not know any specific value. Now, $$\alpha(z_1) - 1 = \alpha(\sin z_0) - 1 = \alpha(z_0) + 1 - 1 = \alpha(z_0).$$ Also $$\alpha(z_2) - 2 = \alpha(\sin(z_1)) - 2 = \alpha(z_1) + 1 - 2 = \alpha(z_1) - 1 = \alpha(z_0).$$ Induction, given $$\alpha(z_n) - n = \alpha(z_0),$$ we have $$\alpha(z_{n+1}) - (n+1) = \alpha(\sin z_n) - n - 1 = \alpha(z_n) + 1 - n - 1 = \alpha(z_0).$$

So, given $$z_n = \sin^{[n]}(z),$$ we have $$\alpha(z_n) - n = \alpha(z).$$

Third, let $$L(z) = \frac{3}{z^2}+ \frac{6 \log z}{5} + \frac{79 z^2}{ 1050} + \frac{29 z^4}{2625}$$. This is a sort of asymptotic expansion (at 0) for $$\alpha(z),$$ the error is $$| L(z) - \alpha(z) | < c_6 |z|^6.$$ It is unlikely that putting more terms on $$L(z)$$ leads to a convergent series, even in the triangle.

Fourth, given some $$ z =z_0$$ in the triangle. We know that $$z_n \rightarrow 0$$. So $$| L(z_n) - \alpha(z_n) | < c_6 |z_n|^6.$$ Or $$| (L(z_n) - n ) - ( \alpha(z_n) - n) | < c_6 |z_n|^6 ,$$ finally $$ | (L(z_n) - n ) - \alpha(z)  |  < c_6 |z_n|^6 .$$ Thus the limit being used is appropriate.

Fifth, there is a bootstrapping effect in use. We have no actual value for $$\alpha(z),$$ but we can write a formal power series for the solution of a Julia equation for $$\lambda(z) = 1 / \alpha'(z),$$ that is $$\lambda(\sin z ) = \cos z \; \lambda(z).$$ The formal power series for $$\lambda(z)$$ begins (KCG Theorem 8.5.1) with  $$- z^3 / 6,$$ the first term in the power series of $$\sin z$$ after the initial $$z.$$ We write several more terms, $$\lambda(z) \asymp - \frac{z^3}{6} - \frac{z^5}{30} - \frac{41 z^7}{3780} - \frac{4 z^9}{945} \cdots.$$ We find the formal reciprocal, $$\frac{1}{\lambda(z)} = \alpha'(z) \asymp -\frac{6}{z^3} + \frac{6}{5 z} + \frac{79 z}{525} + \frac{116 z^3}{2625} + \frac{91543 z^5}{6063750}\cdots.$$ Finally we integrate term by term, $$\alpha(z) \asymp \frac{3}{z^2} + \frac{6 \log z }{5} + \frac{79 z^2}{1050} + \frac{29 z^4}{2625} + \frac{91543 z^6}{36382500}\cdots.$$ and truncate where we like, $$\alpha(z) = \frac{3}{z^2} + \frac{6 \log z }{5} + \frac{79 z^2}{1050} + \frac{29 z^4}{2625} + O(z^6)$$

Numerically, let me give some indication of what happens, in particular to emphasize $$ f_{1/2} (\pi/2) = 1.140179\ldots.$$

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=

[1]: http://oskicat.berkeley.edu/record=b14897585~S1