http://www.elsevier.com/locate/aim Advances in Mathematics 184 (2004) 207–267 Real laminations and the topological dynamics of complex polynomials Jan Kiwi 1 Facultad de Matema´ticas, Pontificia Universidad Cato´lica, Casilla 306, Correo 22, Santiago, Chile Received 25 March 2002; accepted 28 March 2003 Communicated by R.D. Mauldin Abstract We characterize the laminations associated to complex polynomials with connected Julia sets and without irrationally neutral cycles. r 2003 Elsevier Science (USA). All rights reserved. MSC: 37F20 Keywords: Julia sets; Laminations 1. Introduction The main purpose of this paper is to study the topological dynamics of polynomials f : C-C with connected Julia sets and without irrationally neutral cycles. Inspired by the work of Levin [18] and by classical results in one real- dimensionaldynamics(e.g.see [5]),weconsiderthedynamicalsystemobtainedafter collapsing the wandering connected sets contained in the Julia set J ð f Þ of such a polynomial f : Themainresultofthispaperistogiveacompletedescriptionofthe dynamical systems which arise from this collapsing procedure. To be more precise we fix a degree d X2 monic polynomial f : C-C with connected Julia set J ð f Þ and without irrationally neutral cycles. We consider the topological space X f which is the quotient of J ð f Þ by the equivalence relation (see Remark5.4)thatidentifiestwodistinctpointsifandonlyiftheylieinawandering ARTICLE IN PRESS E-mail address: jkiwi@mat.puc.cl. 1 Supported by FONDECYT Grant #1990436. 0001-8708/03/$-see front matter r 2003 Elsevier Science (USA). All rights reserved. doi:10.1016/S0001-8708(03)00144-0