C. McMullen. The classification of conformal dynamical systems. In Current Developments in Mathematics, 1995, pages 323--360. International Press, 1995. Home/Search Document Details and Download
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Renormalization Ideas In Conformal Dynamics - Lyubich (1996) (2 citations) (Correct)
....convergence of the sequence R n f to a map f independent of f (from some class of similar maps) means that all maps of this class have in small scales a universal geometry represented by f . A striking phenomenon of this kind is the Feigenbaum Coullet Tresser Universality Law ( CT, F] see [McM1], x6) It deals with the class of sufficiently smooth unimodal maps of an interval I with the critical point 0 of a given type jxj d ( unimodal means: with one critical point ) Under some combinatorial assumptions on the positions of the first four iterates of the critical point, the interval ....
....same geometry in small scales. A similar picture is observed not only for the doubling renormalization but for other periods as well. We have here a kind of the rigidity phenomenon: Combinatorics of an object determines its geometry. Compare it with the Rigidity Conjecture discussed by McMullen [McM1]. The latter is concerned with a finitely dimensional family of globally defined objects, rational maps. The rigidity conclusion is also global: the geometry of the whole Julia set is determined by combinatorics. In the Feigenbaum Coullet Tresser Based on the talk given at the Cambridge seminar ....
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C. McMullen. The classification of conformal dynamical systems. Paper in this volume.
Kleinian groups and John domains - McMullen (1998) Self-citation (Mcmullen) (Correct)
....of the domain of discontinuity is a John domain (x3) and also when it is uniformly connected (x4) Our results are motivated by the analogies between iterated Research partially supported by the NSF. 1 rational maps and Kleinian groups that have emerged in the past decade; see [Sul2] and [Mc1] for part of the dictionary. In x5 we provide examples and computer images illustrating the results below. We also amplify on the distinction between limit sets and Julia sets, by giving examples where both are dendrites, but of radically different geometry. Statement of results. Let Gamma be a ....
C. McMullen. The classification of conformal dynamical systems. In Current Developments in Mathematics, 1995, pages 323--360. International Press, 1995.
Elder Siblings and the Taming of Hyperbolic 3-Manifolds - Freedman, McMullen (1998) Self-citation (Mcmullen) (Correct)
....conjecture of Thurston proposes a complete isometric classification of hyperbolic 3 manifolds with finitely generated fundamental group, in terms of topology, a combinatorial lamination and the Riemann surface at infinity. For a more detailed account of work towards this classification, see [Mc]. 2 Seeing wildness in an orbit of Gamma Let M = H 3 = Gamma be a hyperbolic 3 manifold with 1 (M) finitely generated. The manifold M is determined by the finitely generated Kleinian group Gamma = 1 (M) ae Isom(H 3 ) We say M is tame if it is homeomorphic to the interior of a compact ....
C. McMullen. The classification of conformal dynamical systems. In Current Developments in Mathematics, 1995, pages 323--360. International Press, 1995.
Hausdorff dimension and conformal dynamics II: Geometrically.. - McMullen (1998) (2 citations) Self-citation (Mcmullen) (Correct)
....The first equality in Theorem 1. 1 is due to Denker, Urba nski and Przytycki [11] 35] The second was observed independently in [45] Basic references for the dynamics of rational maps include [3] 7] 30] and [39] For the dictionary between rational maps and Kleinian groups, see [40] and [25]. Several sections below include an exposition and consolidation of results known to experts, with references and remarks collected in notes at the end of each section. We hope the present systematic treatment will provide a useful contribution to the foundations of the field. Part III of this ....
C. McMullen. The classification of conformal dynamical systems. In Current Developments in Mathematics, 1995, pages 323--360. International Press, 1995.
Rigidity and Inflexibility in Conformal Dynamics - McMullen (1998) Self-citation (Mcmullen) (Correct)
....qn to f qn s . The dictionary. Table 3 summarizes the parallels which emerge between hyperbolic manifolds, quadratic like maps on the interval, critical circle maps and Siegel disks. This table can be seen as a contribution to Sullivan s dictionary between conformal dynamical systems [Sul2] [Mc1]. 7 Surface groups and their geometric limits For a complete classification of conformal dynamical systems, one must go beyond the bounded geometry of the preceding examples, and confront short geodesics, Documenta Mathematica Delta Extra Volume ICM 1998 Delta II Delta 841 855 Conformal ....
....Corollary 7.2 Each Bers slice of AH(S) is bounded by a Jordan curve naturally parameterized by R [ 1, with rational points corresponding to cusps. Corollary 7.3 Geometrically finite manifolds are dense in AH(S) Theorem 7. 1 establishes a special case of Thurston s ending lamination conjecture [Mc1, x4]. We remark that E is not a homeomorphism, and indeed AH(S) is not even a topological manifold with boundary [Mc3, Appendix] The proof of Theorem 7.1 from [Min] can be illustrated in the case E(M) with 2 H and 2 R an irrational number with continued fraction [a 1 ; a 2 ; By ....
C. McMullen. The classification of conformal dynamical systems. In Current Developments in Mathematics, 1995, pages 323--360. International Press, 1995.
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