A MISGUIDING EXAMPLE
                       ====================


The following presents a MATLAB sequence of commands generating a random
orbit of the transformation of the interval [0,1), taking each point x
to 2x (mod 1). The example (along with many more behaving in the same
manner) would convince anyone that all orbits under this transformation
are finite and are, moreover, 0 from some place on. As elaborated in class,
this is very far from being true.

>> x=rand; y=x; for i=1:60, x=frac_part(2*x); y=[y,x]; end, y

y =

  Columns 1 through 7 

    0.2190    0.4379    0.8758    0.7517    0.5033    0.0067    0.0134

  Columns 8 through 14 

    0.0268    0.0536    0.1071    0.2142    0.4284    0.8568    0.7137

  Columns 15 through 21 

    0.4273    0.8546    0.7092    0.4185    0.8369    0.6739    0.3478

  Columns 22 through 28 

    0.6955    0.3911    0.7821    0.5642    0.1284    0.2568    0.5137

  Columns 29 through 35 

    0.0274    0.0547    0.1095    0.2190    0.4379    0.8758    0.7517

  Columns 36 through 42 

    0.5033    0.0067    0.0134    0.0268    0.0536    0.1071    0.2142

  Columns 43 through 49 

    0.4285    0.8569    0.7139    0.4277    0.8555    0.7109    0.4219

  Columns 50 through 56 

    0.8438    0.6875    0.3750    0.7500    0.5000         0         0

  Columns 57 through 61 

         0         0         0         0         0

>> quit

 120 flops.

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function y=frac_part(x)
y=x-floor(x);
end;