Math 301 Problem Set 4 Solutions

Math 301 Problem Set 4 Solutions - Math 301/ENAS 513:...

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Math 301/ENAS 513: Homework 4 Solutions to Selected Problems Triet Le October 11, 2007 1. Page 44: 10. Prove that M [0 , ) = ± 0 , 1 4 ² , where M is the Mandelbrot set. Solution 1 . To show [0 , 1 / 4] M [0 , ): Suppose c [0 , 1 / 4]. Claim the sequence { z n } is bounded by 1 / 2, where z 0 = 0, z n +1 = z 2 n + c . Clearly, z 0 1 / 2. Suppose that z n 1 / 2, then z n +1 = z 2 n + c (1 / 2) 2 + 1 / 4 = 1 / 2. Thus by induction, { z n } is bounded, and c M , for all c [0 , 1 / 4]. To show M [0 , ) [0 , 1 / 4]: Suppose c > 1 / 4. Claim: the sequence { z n } is strictly increasing and unbounded. This implies c / M [0 , ). We have z 1 = c > z 0 = 0. Suppose that z n > z n - 1 , then z n +1 = z 2 n + c > z 2 n - 1 + c = z n . Thus { z n } is strictly increasing. Suppose { z n } is bounded. This implies that there exists a real number z > 1 / 4 such that z n z . Hence, z = lim n →∞ z n +1 = lim n →∞ z 2 n + c = z 2 + c.
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Math 301 Problem Set 4 Solutions - Math 301/ENAS 513:...

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