C. McMullen and D. Sullivan, Quasiconformal homeomorphisms and dynamics III: The Teichmuller space of a holomorphic dynamical system, Adv. Math. 135 (1998), 351--395.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....structure is the pull back of a standard Riemann surface structure on Y . Conjugating by f yields a new conformal dynamical system on Y . The deformation theory of Kleinian groups is founded on this construction [Bers4] Mask2] Sul1] and a parallel theory can be developed for rational maps [Sul3]. Despite its power, this deformation theory is difficult to control; the geometry of the new dynamical system is typically hard to predict. In this regard, a fundamental problem is to estimate the algebraic effect of a Research partially supported by an NSF Postdoctoral Fellowship. 1 ....

D. Sullivan. Quasiconformal homeomorphisms and dynamics III: Topological conjugacy classes of analytic endomorphisms. Preprint.

....10] We will give an argument below that is custom tailored to our maps. For rational maps, as well as for entire holomorphic maps with only finitely many singular values (known as entire maps of finite type ) there are no domains at infinity and no wandering domains; this is Sullivan s Theorem [McMullen and Sullivan 1998] in the extension of Eremenko and Lyubich [1992] Unfortunately, the entire maps we are looking at cannot possibly be of finite type. We will be able to exclude domains at infinity, and we will show that a diverging integer for the 3n 1 problem must sit in a simply connected wandering domain. 3. ....

C. T. McMullen and D. P. Sullivan, "Quasiconformal homeomorphisms and dynamics, III: The Teichmuller space of a holomorphic dynamical system", Adv. Math. 135:2 (1998), 351-- 395. See http://math.harvard.edu/ ~ ctm/papers.html.

....A and B are topologically conjugate on the unit circle S 1 , and the conjugacy h is unique once we have chosen a pair 1 of fixed points (a; b) for A and B such that h(a) b. Moreover h is quasisymmetric; this is a general property of conjugacies between expanding conformal dynamical systems [Sul]. Now glue two copies of the disk together by h and transport the dynamics of A and B to the resulting Riemann surface, which is a sphere. We obtain in this way an expanding (i.e. hyperbolic) rational map f(A; B) The Julia set J of f(A; B) is a quasicircle, and f is holomorphically conjugate to ....

D. Sullivan. Quasiconformal homeomorphisms and dynamics III: Topological conjugacy classes of analytic endomorphisms. Preprint.

....and g on neighbourhoods of X and Y satisfying hj X = h 0 j X . Notice that TCE is a topological property, so the assumption that (X; f) is TCE immediately implies that (Y; g) is TCE, too. Theorem A for rational maps f and g that are expanding on their Julia sets is due to McMullen and Sullivan, [McS]. It is not hard to modify h 0 to become quasiconformal (qc for short) off X. The main idea of our proof of Theorem A is to show that for every x 2 X there is a sequence of discs centered at x, of radii converging to 0, that are mapped under h to boundedly distorted topological discs. Indeed, TCE ....

....holomorphic repellers which are topologically conjugate by an orientation preserving homeomorphism h 0 , then there exist neighbourhoods WX ; W Y of X;Y respectively and a homeomorphism h : WX W Y conjugating f and g , such that h is qc on WX n X and h is equal to h 0 on X. Proof. Compare [McS] in the rational expanding case. We can assume that V is small enough that (V n X) Crit(f) and that V is contained in the domain of h 0 : Take WX = W from Lemma 3.1. By a small change we can assume that WX has smooth boundary 0 . Let H : 0; 1] Theta 0 C be a homotopy from Hj ....

C. McMullen, D. Sullivan, Quasiconformal homeomorphisms and dynamics III: The Teichmuller space of a holomorphic dynamical systems, to appear in Adv. Math.

....proper subset. The action of a rational map on the complement of its Julia set (the Fatou set) again has a kind of regular behaviour, and there is even the analogue of a fundamental domain in the hyperbolic case, that is, when the Fatou set is made up of basins of attraction of periodic orbits [Sullivan 1984]. Munzner and Rasch [Rasch 1988; Munzner and Rasch 1991] have shown that the same dynamical dichotomy exists on the space of orbits of an algebraic correspondence under one way iteration, and that much of the classical Fatou Julia theory extends to this situation. Our interest here is quite ....

....between Kleinian group and rational map actions and indicate some directions for further study. Our motivation for undertaking this investigation was the striking series of results of Sullivan obtained by applying quasiconformal deformation theory to both rational maps and Kleinian groups [Sullivan 1984; 1985a; 1985b] Our hope was that by studying iterated correspondences we could obtain further insight into how these classes of dynamical systems are related. The results outlined in this paper are a step in that direction: we believe the examples also have considerable interest in their own ....

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D. Sullivan, Quasiconformal homeomorphisms and dynamics III (preprint), IHES, 1984.

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C. McMullen and D. Sullivan, Quasiconformal homeomorphisms and dynamics III: The Teichmuller space of a holomorphic dynamical system, Adv. Math. 135 (1998), 351--395.

....of Theorem 3.1 is elementary apart from the use of: Theorem 3.2 (Ahlfors Finiteness Theorem) If Gamma is a finitely generated Kleinian group with domain of discontinuity Omega , then Omega = Gamma is a finite union of hyperbolic Riemann surfaces of finite area. See [Ah] Gre] Bers1] [McS]. It is worth noting that Gamma U is almost determined by U . Indeed, let Aut(U) be the group of all Mobius transformations stabilizing U . Suppose a component U of Omega is not a round disk; then Aut(U) is discrete, and it contains Gamma U with finite index because U= Gamma U covers U= ....

C. McMullen and D. Sullivan. Quasiconformal homeomorphisms and dynamics III: The Teichmuller space of a holomorphic dynamical system. Adv. Math., to appear.

....Julia set . 40 10 Example: Quadratic polynomials . 41 2 1 Introduction This paper completes in a definitive way the picture of rational mappings begun in [30] It also provides new proofs for and expands upon an earlier version [46] from the early 1980s. In the meantime the methods and conclusions have become widely used, but rarely stated in full generality. We hope the present treatment will provide a useful contribution to the foundations of the field. Let f : b C b C be a rational map on the Riemann sphere b C , with ....

....map carries no measurable 3 invariant line field on its Julia set. The proof depends on the Teichmuller theory developed below and the instability of Herman rings [29] In the last section we illustrate the general theory with the case of quadratic polynomials. An early version of this paper [46] was circulated in preprint form in 1983. The present work achieves the foundations of the Teichmuller theory of general holomorphic dynamical systems in detail, recapitulates a self contained account of Ahlfors finiteness theorem (x4.3) the no wandering domains theorem (x6.3) and the density of ....

D. Sullivan. Quasiconformal homeomorphisms and dynamics III: The Teichmuller space of a holomorphic dynamical system. Preprint, 1983.

.... Ahlfors finiteness theorem and the Ahlfors measure zero problem appear in [Ah] the number of triply punctured spheres is bounded in [Gre] Sullivan s no wandering domains theorem appears in [Sul4] see also [Bers3] and the Teichmuller space of a holomorphic dynamical system is developed in [McS]. Bers area theorem and Shishikura s bounds are in [Bers1] and [Shi1] x3 Expanding dynamics. A candidate 3 dimensional object for a rational maps is constructed in [LM] A prototype for the conjecture on the density of convex cocompact groups is [Bers2, Conjecture II] The relation of ....

.... 3 dimensional object for a rational maps is constructed in [LM] A prototype for the conjecture on the density of convex cocompact groups is [Bers2, Conjecture II] The relation of hyperbolicity to structural stability is discussed for Kleinian groups in [Sul5] and for rational maps in [MSS] [McS] and [Mc4] Theorem 3.6 and its Corollary can be found in [McS] Density of hyperbolicity has been announced for the family of entire functions f (z) tanz [KK] x4 Topology of hyperbolic 3 manifolds. Marden s work on the topology of geometrically finite 3 manifolds appears in [Mar] The basic ....

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C. McMullen and D. Sullivan. Quasiconformal homeomorphisms and dynamics III: The Teichmuller space of a holomorphic dynamical system. To appear, Adv. Math.

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C. McMullen and D. Sullivan. Quasiconformal homeomorphisms and dynamics III: The Teichmuller space of a holomorphic dynamical system. Adv. Math., To appear.

....0 ) are locally constant. It can be shown that U 1 is also open and dense, and the conjugacy OE can be extended to the grand orbits of the critical points over U 1 . Finally general results on holomorphic motions [ST] BR] prolong OE to a conjugacy on the whole sphere. For details see [MSS] [McS]. In smooth dynamics, the notion of structural stability goes back at least to the work of Andronov and Pontryagin in 1937, and the problem of the density of structurally stable systems was known for some time. In 1965, Smale showed that structural stability is not dense, by giving a ....

....line field. Conjecture NILF. A rational map f carries no invariant line field on its Julia set, except when f is double covered by an integral torus endomorphism. This conjecture is stronger than the density of hyperbolic dynamics: Theorem 3.1 (Ma n e, Sad, Sullivan) NILF = HD. See [MSS] [McS]. One attractive feature of Conjecture NILF is that it shifts the focus of study from the family of all rational maps to the ergodic theory of a single rational map. In support of this Conjecture, and hence of the density of hyperbolic dynamics, we state a parallel result for degree one rational ....

C. McMullen and D. Sullivan. Quasiconformal homeomorphisms and dynamics III: The Teichmuller space of a holomorphic dynamical system. Preprint.

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Dennis P. Sullivan. Quasiconformal homeomorphisms and dynamics iii. preprint IHES/M/83/1.

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Dennis P. Sullivan. Quasiconformal homeomorphisms and dynamics iii. preprint IHES/M/83/1.

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