# Iterating tiltedsinewave discretizes vehemently

Posted 6 months ago
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Posted 6 months ago
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Posted 6 months ago
 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!]