Tuesday, October 29, 2013

Lattès Example

I gave a DynaLite on the Lattès example last week and given the shortage of information on the web I'm posting my notes. This is a more algebraic geometry perspective than those offered in Beardon and Carleson/Gamelin, it somewhat follows insight from Milnor. Here are my notes which are somewhat out of order, and here is a helpful picture of a torus generated with the following tikz code. The clip function is really handy, and you can put things outside the clipped box by just inserting the code for them prior to the clip command.

      \clip (-3,-3) rectangle (3,3);
      \foreach \x in {-2,-1,...,2}
 \draw[-] (3.000000,\x) -- (-3.000000,\x);
      %Slants slope = .61825
      \foreach \x in {-5,-4,...,5}
 \draw[-] (2+\x,2*1.61825) -- (-2+\x,2*-1.61825);
      \fill[green!20!white] (0,0) -- (1,0) --   (1.61825,1) --(.61825,1) --cycle;
      \fill[blue!20!white] (0,0) -- (-1,0) --   (-1.61825,-1) --(-.61825,-1) --cycle;

      \draw[<->][red] (3,0) -- (-3,0);
      \draw[<->][red] (0,3) -- (0,-3);


Monday, October 28, 2013

Diophantine Numbers

I spent a while today writing up this little primer on Diophantine numbers. It's mostly the coverage Milnor gives in chapter 11 of Dynamics in One Complex Variable, but I've added some details and included the solution to 11-a.

It's probably riddled with typos and logical errors, feel free to correct in the comments.