math04hw2 - Homework 2: Due April 19th, Monday, 2010 1. Let...

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Unformatted text preview: Homework 2: Due April 19th, Monday, 2010 1. Let f : [a, b] → [a, b] be differentiable with |f (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, a1 ] and [a2 , 1] for some a1 < 1/2 < a2 (where aj ’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 Σ2 = {(s0 s1 s2 ...) | sj = 0 or 1}. We defined the metric by | s −t | d(s, t) = ∞ j2j j for any s, t ∈ Σ2 . j =0 Above, we have just described a mapping between F : Λ → Σ2 . To prove that it is homeomorphism, we must show that • F is 1-to-1. (If x = y ∈ Λ then F (x) and F (y ) must be different.) • F is onto. (There exists a preimage for any sequence in Σ2 .) • F is continuous. • F −1 is continuous. We have sketched the first two. Prove the last two. 4. A point p is a non-wandering point for f , if, for any open interval J containing p, there exists x ∈ J and n > 0 such that f n (x) ∈ J . Let Ω(f ) denote the set of non-wandering points of f . • Prove that Ω(f ) is a closed set. √ • If Fµ is the quadratic map with µ > 2 + 5, show that Ω(Fµ ) = Λ. (One can use the properties that we proved through the sequence space.) • Identify Ω(Fµ ) for each 0 < µ < 3. 1 ...
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This document was uploaded on 07/19/2010.

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