By Beattie J. A.

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Try it out! 25, it appears that the boundary between them is a straight line; points on one side look closer to the root of f on that side than to the other root. 3) of the line segment from ˛ to ˇ. This is true (pictures, however, can sometimes be misleading) and we can prove it using the technique of global conjugation. 6 Global Conjugation We learn in linear algebra that a change of basis can facilitate calculations and procedures, and lead to better understanding. Similarity of matrices plays a key role (matrices A and B are called similar when there is an invertible matrix P such that A D PBP 1 ).

Similarity of matrices plays a key role (matrices A and B are called similar when there is an invertible matrix P such that A D PBP 1 ). An analog often used in dynamics is a type of change of coordinates provided by what we call conjugation. 28. z/ D azCb where ad bc ¤ 0, noting it is czCd a bijection of C onto itself. When f and g are rational maps (quotients of two polynomials) such that g D ı f ı 1 , we say f and g are globally conjugate by the map . Often the point of conjugating a map f to a map g is that g is easier to work with than f .

0:13 C 0:73i / to verify that it has what we could call an attracting 3-cycle. Try it out! z/ D z 2 C c having an attracting 2-cycle, 3-cycle, 4-cycle, and so on. We will be able to make some progress, but ✐ ✐ ✐ ✐ ✐ ✐ “ECA˙Book” — 2012/8/28 — 13:58 — page 30 — #48 ✐ ✐ 30 Chapter 1. Complex Dynamics: Chaos, Fractals, the Mandelbrot Set, and More it will be limited because our methods require us to find the roots of polynomials of high degree. We begin by defining cycles and then we see how to classify them as attracting, repelling, or indifferent by using the derivative (or more precisely the multiplier).