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The stable set is commonly called the Fatou set because of Pierre Fatou’s role in developing the theory of complex function iteration in 1917 (see [12, 13, 14]). In honor of their pioneering work in the field, the dynamics of complex analytic functions is often called the Fatou-Julia theory. 57. z/ D 3z. Try it out! 58. z/ defined on C. Try to formulate what it means for orbits to be drastically or significantly different. Try it out! g/ for the Julia set of g. A on page 78. These should be read and compared with your own definitions, but it is not necessary to know these formal definitions well to continue on in the text; an intuitive understanding is enough.

This is called linearizing the map f near 0. It can always be done when 0 < ja1 j ¤ 1, but only sometimes when ja1 j D 1. , [1, 3, 24]). 24. ˛Cˇ/ 1, respectively, and then analyze the simpler map obtained by conjugation. z/ D z 2 . z/ D z 2 . z/ ! ˛. z/j < 1. Since jgn . z/j2 ! 0, we have gn . z// ! 0. gn . z/// ! 0/ D ˛. ˛/. 31. ˇ/. Try it out! 3, called a commutative diagram because the maps ı F and g ı from the upper left to the bottom right are equal. 0; 1/ in the bottom pictures. 1//. Let’s return to the question of whether Newton’s method can fail in this example.

A on page 78. These should be read and compared with your own definitions, but it is not necessary to know these formal definitions well to continue on in the text; an intuitive understanding is enough. 0; 1/. 0; 1/. To see this, consider z0 D e i written in polar form. z00 /j > jz0 z00 j. z00 / along the unit circle is 2ˇ. z00 / along the unit circle is 2n ˇ as long as 2n ˇ < . , no matter how small ˇ is, corresponding orbit points will eventually be far apart. z00 /j > 1. 0; 1/ do not have corresponding orbit values that forever stay close.

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