Fixed Points, Continued

Write the address of the point x as b1b2b3... .

Then x is a fixed point of f0(f1) means f0(f1(x)) = x.

From the effect of f0 and f1 on addresses we see f0(f1(x)) = x means

01b1b2b3... = b1b2b3b4...

Consequently, b1b2b3b4... = 0101... and x = 1/4 + 1/16 + 1/64 + ... = 1/3 by summing the geometric series.

The same result is easy to obtain algebraically: x = f0(f1(x)) = (1/2)*(x/2 + 1/2) = x/4 + 1/4 and so x = 1/3.

Exercise Show the fixed point of f1(f0) is x = 2/3.

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