G Holms

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

Will Jagy

Below is an example by Will Jagy^[2]

"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)^[3] denotes $\alpha$ that is infinite at $0$ and otherwise solves what is usually called the Abel functional equation,^[4]

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

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

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

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 $\alpha (x)=\lim _{n\rightarrow \infty }\;\;\;{\frac {3}{x_{n}^{2}}}\;+\;{\frac {6\log x_{n}}{5}}\;+\;{\frac {79x_{n}^{2}}{1050}}\;+\;{\frac {29x_{n}^{4}}{2625}}\;-\;n.$

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 $L(z)={\frac {3}{z^{2}}}+{\frac {6\log z}{5}}+{\frac {79z^{2}}{1050}}+{\frac {29z^{4}}{2625}}$ . 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, $\lambda (z)\asymp -{\frac {z^{3}}{6}}-{\frac {z^{5}}{30}}-{\frac {41z^{7}}{3780}}-{\frac {4z^{9}}{945}}\cdots .$ We find the formal reciprocal, ${\frac {1}{\lambda (z)}}=\alpha '(z)\asymp -{\frac {6}{z^{3}}}+{\frac {6}{5z}}+{\frac {79z}{525}}+{\frac {116z^{3}}{2625}}+{\frac {91543z^{5}}{6063750}}\cdots .$ Finally we integrate term by term, $\alpha (z)\asymp {\frac {3}{z^{2}}}+{\frac {6\log z}{5}}+{\frac {79z^{2}}{1050}}+{\frac {29z^{4}}{2625}}+{\frac {91543z^{6}}{36382500}}\cdots .$ and truncate where we like, $\alpha (z)={\frac {3}{z^{2}}}+{\frac {6\log z}{5}}+{\frac {79z^{2}}{1050}}+{\frac {29z^{4}}{2625}}+O(z^{6})$

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

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

[2] thoverflow 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

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