Running Out of Bits

Iterate a chaotic map:

In[1]:=

NestList[4#(1-#)&,N[1/3,40],70]

Out[1]=

{0.3333333333333333333333333333333333333333,0.888888888888888888888888888888888888889,0.395061728395061728395061728395061728395,0.95595183661027282426459381191891479957,0.16843169076687629703549313314712170527,0.5602498252491506199445347255740894253,0.985479834229787249749298058946920761,0.057237322224873168879945173218028430,0.21584484467759683550677451476907180,0.67702339081480373015172092445305984,0.8746508764177170493755567062784498,0.438546883197746027552549282684748,0.984894057741154121983362312085481,0.05951101106927316923647876580405,0.22387780252314406223522437984770,0.6950261282422086895855927508860,0.847859237211414280582683172436,0.515975804346772041146047337653,0.99897909470189463822276310479,0.00407945220191064871855883370,0.0162512410865719003132739164,0.063948552998872070696564970,0.239436542272890082360982538,0.72842673798917041181318333,0.79128490148650756402671382,0.6606124246639823340670616,0.896814596174226396895002,0.370152705052342517507175,0.93255871997910417448342,0.25157181508015577230863,0.7531337477497267242295,0.743693223000710882310,0.762454452254103186390,0.72447064196799468346,0.79845172357790537561,0.6437062747735102420,0.917394026362881495,0.303128907026328749,0.84496709100540836,0.5239908244930652,0.9976977613605718,0.009187753346701,0.036413354140566,0.14034968712320,0.4826066097905,0.9987898799081,0.004834622805,0.019244996910,0.07549850802,0.2791939332,0.8049787235,0.627951913,0.934513232,0.24479301,0.7394776,0.7706020,0.707098,0.828441,0.56851,0.9812,0.0737,0.273,0.794,0.65,0.9,0.3,0.×

-1

10

,0.×

-1

10

,0``-0.41943292797263654,0``-1.4409258472732356,0``-3.4839116858744332}

This starts from precision 40, but because of sensitive dependence on initial conditions, no correct bits are left after slightly less than 70 iterations.

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