math04hw1

# math04hw1 - n and x F n x-x is not an integer Therefore we...

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Homework 1: Due April 12th, Monday, 2010 1. Prove that sup n N (sin n ) = 1. 2. Find the asymptotic frequency of 0 being the second leading digit of power of 3. 3. For a circle map f : S 1 S 1 that is homeomorphic and orientation preserving, let us denote its lift on R by F : R R . We have shown in the class that the existence of periodic point of period q leads to the rotation number ρ ( x ) = lim n →∞ F n ( x ) n being a rational number of the form p/q for some integer p . We will prove the inverse statement, that is, if there is no periodic point, then the rotation number must be irrational. If there is no periodic point, then for all
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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. pre-images of periodic points). 1...
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