Given a complex number cE C, we consider the function F(z) = z2 + c. For each zo

Question

Given a complex number cE C, we consider the function F(z) = z2 + c. For each zo E C, consider the sequence n1 Fn) for n 20. The filled Julia Set is the set of values zo such that the corresponding sequence remains bounded. We are going to approximate this set as follows: etc) and the intellectual content of the course itself are protected by United States Federal Copyright Law, and the California Civil Code. The UC Policy 102.23 expressly prohibits students (and all other persons) from recording written permission of the instructor (a) Fix a complex number c. (b) Define a grid in the domain z-a+ib, a E [-2,2, beI-1,2 in the complex plane, with 1000 in each direction (c) For each complex number in the grid generate 100 points of the sequence defined by F. (d) At the end of the iteration check to see if lzl S maxflel,2). If so, store the point in a file. (e) Plot the file using Matlab or other visualization software.

Explanation / Answer

For step (a), observe that by multiplying the relation 0 < c < 1 by c^n we get 0<cn+1<cn0<cn+1<cnand hence taking n(n+1) th root on both sides we obtain c1n<c1n+1c1n<c1n+1 . so X_n is increasing. But c<1c1n<11n=1c<1c1n<11n=1 . so X_n is bounded above by 1. Any increasing sequence which is bounded above is convergent.

For step (b), let us denote the limit by x. Since X_n is positive and increasing, so is x. But then any subsequence will also converge to x. In particular choose the subsequence X_{2n}. then limX2n=limc1/2n=limc1/n=x1/2.limX2n=limc1/2n=limc1/n=x1/2.So we have x1/2=xx=1,since x>0.

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