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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online