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Please someone suggest me some reference on the topic "Complex Dynamics". I want a brief geometric treatment from the root level. I have graduate level background on complex analysis, riemannian geometry and topology.

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With your background, the book by Milnor or by Beardon looks like a good choice. In addition to the references already suggested, there also are:

MR1224235 Steinmetz, Norbert:
Rational iteration. Complex analytic dynamical systems. de Gruyter Studies in Mathematics, 16. Walter de Gruyter & Co., Berlin, 1993. x+189 pp. ISBN: 3-11-013765-8

and

MR1747010 Morosawa, S.; Nishimura, Y.; Taniguchi, M.; Ueda, T. Holomorphic dynamics. Translated from the 1995 Japanese original and revised by the authors. Cambridge Studies in Advanced Mathematics, 66. Cambridge University Press, Cambridge, 2000. xii+338 pp. ISBN: 0-521-66258-3

(the latter also introduces some aspects of dynamics in two complex variables).

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Milnor's dynamics in one complex variable seems to be a popular choice.

Mathscinet

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Dynamics in One Complex Variable. (AM-160): Third edition (Annals of Mathematics Studies) Jan 22, 2006 by John Milnor

Complex Dynamics (Universitext / Universitext: Tracts in Mathematics) Feb 2, 1996 by Lennart Carleson and Theodore Gamelin

Iteration of Rational Functions: Complex Analytic Dynamical Systems (Graduate Texts in Mathematics) Oct 1, 1991by Alan F. Beardon

An Introduction to Chaotic Dynamical Systems, 2nd Edition (Studies in Nonlinearity) Feb 7, 2003 by Robert Devaney

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