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.
3 Answers
Dynamics in One Complex Variable. (AM160): 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

$\begingroup$ Just an update that a third edition of Devaney was published in 2021: routledge.com/AnIntroductionToChaoticDynamicalSystems/… $\endgroup$– J WCommented Oct 15, 2022 at 17:49
Milnor's dynamics in one complex variable seems to be a popular choice.
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: 3110137658
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: 0521662583
(the latter also introduces some aspects of dynamics in two complex variables).