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Unformatted text preview: n and x , F n ( x )x is not an integer. Therefore we can write k n < F n ( x )x < k n + 1 , (1) for some integer k n that may depend on n . Using this fact, show that the rotation number cannot be a rational number of the form (integer) /n . Since n can be any positive integer, this proves that the absence of periodic point leads to irrational rotation number. 4. Consider the map F : S 1 S 1 given by F ( x ) = 2 x mod 1 (expansion map). Show that all the rational points are eventually periodic points (i.e. preimages of periodic points). 1...
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 Spring '09
 Math

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