Keith Briggs

This page was last modified 2013-05-12  


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The Yoccoz U function

These pictures relate to the Yoccoz U function. See: J.-C. Yoccoz, Astérisque 231 (1995), page 70; and Stefano Marmi, An introduction to small divisors problems (2000), page 19 for more details.

The function u(l) is defined by u0=1/2; un+1=un-ln/2 un2 (n=0,1,2,...)

The series begins u(l) = 1/2-l/8-l2/8+....

I computed 1000 terms of the power series of U by exact rational arithmetic, summed the series with high-precision real arithmetic, and created the graphics with R.

Radial limit of U

Radial limit plot

arg(U) inside unit circle

arg(u) plot

abs(U) inside unit circle

abs(U) plot
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