Complex Analysis: Theory and Applications · Problem 4.2.12

Sequences, Series, and Fractals — Howell & Matthews

Mathematics — Complex Analysis · attracting-n-cycles

Problem

Suppose that {z0,z1}\{z_0, z_1\} is a 2-cycle for ff. (a) Show that, if z0z_0 is attracting for g2(z)g_2(z), then so is the point z1z_1. Hint: Differentiate g2(z)=f(f(z))g_2(z) = f\bigl(f(z)\bigr) using the chain rule, and show that g2′(z0)=g2′(z1)g_2'(z_0) = g_2'(z_1). (b) Generalize part (a) to nn-cycles.

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