By Lennart Carleson

Complex dynamics is at the present time greatly a spotlight of curiosity. even though numerous nice expository articles have been on hand, by means of P. Blanchard and by means of M. Yu. Lyubich specifically, till lately there has been no unmarried resource the place scholars may well locate the fabric with proofs. For somebody in our place, amassing and organizing the cloth required loads of paintings dealing with preprints and papers and from time to time even discovering an explanation. we are hoping that the result of our efforts could be of aid to others who plan to benefit approximately complicated dynamics and even perhaps lecture. in the meantime books within the box a. re commencing to look. The Stony Brook path notes of J. Milnor have been quite welcome and worthwhile. nonetheless we are hoping that our certain emphasis at the analytic aspect will fulfill a necessity. This publication is a revised and accelerated model of notes in accordance with lectures of the 1st writer at UCLA over a number of \Vinter Quarters, rather 1986 and 1990. We owe Chris Bishop loads of gratitude for supervising the creation after all notes, including new fabric, and making desktop photos. now we have used his computing device photos, and we'll additionally seek advice from the horny colour pix within the well known remedy of H. -O. Peitgen and P. Richter. we've benefited from discussions with a couple of colleagues, and from feedback of scholars in either our courses.

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1), and let V = {Rez > A} be a half-plane such that g(V) C V. Let L be the vertical line bounding V. Then L' = g(L) is a smooth curve which is approximately the translation of L by 1 to the right. Let S denote the vertical strip in the z- plane bounded by Land L'. In the ( = + iry plane, let ~ be the honest strip {O < < 1}. We construct a diffeomorphism h : ~ --+ S by setting h(iry) = A + iry and h(1 + iry) = g(A + iry) and then filling in these boundary values in a smooth way, uniformly as 1m ( --+ ±oo.

L. 1) is solvable. L for all integers p and q, q satisfies i= O. e. real number O. L . L) 00 ~ O. Q=n In particular, almost all real numbers are Diophantine. By a theorem of J. t > m. 4 (Siegel). 1), that is, f can be conjugated near 0 to multiplication by e27riO • Proof. We want to solve h(Az) = f(h(z)). If we define j and h by fez) = AZ + j(z) and h(z) = z + h(z), then the equation can be written h(AZ) - Ah(z) = j(h(z)). 3) Siegel's original method was to expand both sides in power series 44 II.

The definition implies the iterates In and gn are also conjugate, gn =