Dear Dr. Gosper,
what a terrific idea rotating a sine wave! I imagine that more interesting functions can be constructed this way. Thanks for sharing!
Your post ended with a question:
We're (quadratically?) approaching the step function - Why?
Well (if I understand this question correctly), is iteration not basically equivalent to a search for attractive fixed points?
f = # + Sin[#] &;
g = InverseFunction[# - Sin[#] &];
tiltedSinwave = f@*g;
startx1 = 6.28;
startx2 = 6.4;
pts1 = Flatten[
BlockMap[With[{x = First[#], y = Last[#]}, {{x, y}, {y, y}}] &,
NestList[tiltedSinwave, startx1, 3], 2, 1], 1];
pts2 = Flatten[
BlockMap[With[{x = First[#], y = Last[#]}, {{x, y}, {y, y}}] &,
NestList[tiltedSinwave, startx2, 3], 2, 1], 1];
PrependTo[pts1, {startx1, 0}];
PrependTo[pts2, {startx2, 0}];
What I mean is of course the usual picture:
Plot[{x, tiltedSinwave[x]}, {x, 0, 10}, AspectRatio -> Automatic, ImageSize -> 400,
Epilog -> {{Thick, Red, Arrow[pts1]}, {Thick, Green, Arrow[pts2]}}, PlotStyle -> {Dashed, {Thick, Black}}]
The iteration actually reads: f @* g @* f @* g @* f @* g @* f @* g @* f @* g ...
But because of the above mentioned mechanism each single one of these functions after iteration will end up in a step function (albeit much slower):
Plot[{x, Nest[f, x, 3], Nest[g, x, 3]}, {x, -10, 10}, AspectRatio -> Automatic, ImageSize -> 400,
PlotStyle -> {Dashed, Thick, Thick}]
[All this is just meant as an additional demo!]
