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math04hw2 - Homework 2 Due April 19th Monday 2010 1 Let f[a...

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Homework 2: Due April 19th, Monday, 2010 1. Let f : [ a, b ] [ a, b ] be differentiable with | f 0 ( x ) | ≤ 1 for all x [ a, b ]. Prove that there is no periodic points of period greater than 2. 2. Prove that a homeomorphism cannot have eventually periodic points (meaning that its orbit cannot contain some non-periodic portion before settling down into periodic phase). 3. (I have repeated the description that were given in the class. The problem itself is very short.) For the logistic map f ( x ) = μx (1 - x ) with μ > 2 + 5, the invariant set Λ was defined by the set of points that remains in [0 , 1] after infinite iterations. We have shown that Λ is contained inside the two intervals, [0 , a 1 ] and [ a 2 , 1] for some a 1 < 1 / 2 < a 2 (where a j ’s are the two solutions of f ( x ) = 1). And its “itinerary” ( x, f ( x ) , f 2 ( x ) , f 3 ( x ) , .... ) can maps to a sequence of, for instance, ( left , right , right , left , ... ), where j th entry is determined by the interval on which f j - 1 ( x ) is located. For convenience, we will write 0 for left and 1 for right . We have also defined the sequence space Σ 2 = { ( s 0 s 1 s 2 ...
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