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39
Local connectivity of Julia sets: expository lectures
 The Mandelbrot Set, Theme and Variations
, 1992
"... An exposition of results of Yoccoz, Branner, Hubbard and Douady concerning polynomial Julia sets. The contents are as follows: 1. Local connectivity of quadratic Julia sets (following Yoccoz) . 2 2. Polynomials for which all but one of the critical orbits escape (following Branner and Hubbard) . ..."
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Cited by 53 (0 self)
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An exposition of results of Yoccoz, Branner, Hubbard and Douady concerning polynomial Julia sets. The contents are as follows: 1. Local connectivity of quadratic Julia sets (following Yoccoz) . 2 2. Polynomials for which all but one of the critical orbits escape (following Branner and Hubbard) . . . . . . . . . . . . 21 3. An innitely renormalizable non locally connected Julia set (following Douady and Hubbard) . . . . . . . . . . . . 34 Appendix: Lengthareamodulus inequalities . . . . . . . . . 41 References . . . . . . . . . . . . . . . . . . . . . . . . 47 Introduction The following notes provide an introduction to work of Branner, Hubbard, Douady, and Yoccoz on the geometry of polynomial Julia sets. They are an expanded version of lectures given in Stony Brook in Spring 1992. Section 1 describes unpublished work by J.C. Yoccoz on local connectivity of quadratic Julia sets. (Compare [Hu3].) It presents only the \easy" part of his theory, in the sense that it considers...
Dynamics of quadratic polynomials: Complex bounds for real maps
 Ann. Inst. Fourier (Grenoble
, 1997
"... Abstract. We extend Sullivan’s complex a priori bounds to real quadratic polynomials with essentially bounded combinatorics. Combined with the previous results of the first author, this yields complex bounds for all real quadratics. Local connectivity of the corresponding Julia sets follows. 1. ..."
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Cited by 23 (6 self)
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Abstract. We extend Sullivan’s complex a priori bounds to real quadratic polynomials with essentially bounded combinatorics. Combined with the previous results of the first author, this yields complex bounds for all real quadratics. Local connectivity of the corresponding Julia sets follows. 1.
On the topology of deformation spaces of Kleinian groups
"... Let M be a compact, hyperbolizable 3manifold with nonempty incompressible boundary and let AH(π1(M)) denote the space of (conjugacy classes of) discrete fathful representations of π1(M) into PSL2(C). The components of the interior MP(π1(M)) of AH(π1(M)) (as a subset of the appropriate representatio ..."
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Cited by 21 (8 self)
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Let M be a compact, hyperbolizable 3manifold with nonempty incompressible boundary and let AH(π1(M)) denote the space of (conjugacy classes of) discrete fathful representations of π1(M) into PSL2(C). The components of the interior MP(π1(M)) of AH(π1(M)) (as a subset of the appropriate representation variety) are enumerated by the space A(M) of marked homeomorphism types of oriented, compact, irreducible 3manifold homotopy equivalent to M. In this paper, we give a topological enumeration of the components of the closure of MP(π1(M)) and hence a conjectural topological enumeration of the components of AH(π1(M)). We do so by characterizing exactly which changes of marked homeomorphism type can occur in the algebraic limit of a sequence of isomorphic freely indecomposable Kleinian groups. We use this enumeration to exhibit manifolds M for which AH(π1(M)) has infinitely many components.
On Fibers and Local Connectivity of Mandelbrot and Multibrot Sets
, 1998
"... We give new proofs that the Mandelbrot set is locally connected at every Misiurewicz point and at every point on the boundary of a hyperbolic component. The idea is to show "shrinking of puzzle pieces" without using specific puzzles. Instead, we introduce fibers of the Mandelbrot set (see Definit ..."
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Cited by 18 (8 self)
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We give new proofs that the Mandelbrot set is locally connected at every Misiurewicz point and at every point on the boundary of a hyperbolic component. The idea is to show "shrinking of puzzle pieces" without using specific puzzles. Instead, we introduce fibers of the Mandelbrot set (see Definition 3.2) and show that fibers of certain points are "trivial", i.e., they consist of single points. This implies local connectivity at these points. Locally, triviality of fibers is strictly stronger than local connectivity. Local connectivity proofs in holomorphic dynamics often actually yield that fibers are trivial, and this extra knowledge is sometimes useful. We include the proof that local connectivity of the Mandelbrot set implies density of hyperbolicity in the space of quadratic polynomials (Corollary 3.6). We write our proofs more generally for Multibrot sets, which are the loci of connected Julia sets for polynomials of the form z 7! z d + c. Although this paper is a c...
Local connectivity of Julia sets for unicritical polynomials
 Preprint IMS at Stony Brook, # 3
, 2005
"... Abstract. We prove that the Julia set J(f) of at most finitely renormalizable unicritical polynomial f: z ↦ → z d + c with all periodic points repelling is locally connected. (For d = 2 it was proved by Yoccoz around 1990.) It follows from a priori bounds in a modified Principle Nest of puzzle piece ..."
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Cited by 10 (5 self)
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Abstract. We prove that the Julia set J(f) of at most finitely renormalizable unicritical polynomial f: z ↦ → z d + c with all periodic points repelling is locally connected. (For d = 2 it was proved by Yoccoz around 1990.) It follows from a priori bounds in a modified Principle Nest of puzzle pieces. The proof of a priori bounds makes use of new analytic tools developed in [KL] that give control of moduli of annuli under maps of high degree. 1.
Is the Mandelbrot set computable?
 MATH. LOGIC QUART
, 2005
"... We discuss the question whether the Mandelbrot set is computable. The computability notions which we consider are studied in computable analysis and will be introduced and discussed. We show that the exterior of the Mandelbrot set, the boundary of the Mandelbrot set, and the hyperbolic components sa ..."
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Cited by 10 (0 self)
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We discuss the question whether the Mandelbrot set is computable. The computability notions which we consider are studied in computable analysis and will be introduced and discussed. We show that the exterior of the Mandelbrot set, the boundary of the Mandelbrot set, and the hyperbolic components satisfy certain natural computability conditions. We conclude that the two–sided distance function of the Mandelbrot set is computable if the hyperbolicity conjecture is true. We formulate the question whether the distance function of the Mandelbrot set is computable also in terms of the escape time.
On Multicorns and Unicorns I: Antiholomorphic Dynamics, Hyperbolic Components, and Real Cubic Polynomials
, 2000
"... We investigate the dynamics and the bifurcation diagrams of iterated antiholomorphic polynomials: these are complex conjugates of ordinary polynomials. Their second iterates are holomorphic polynomials, but dependence on parameters is only realanalytic. The parameter space of degree two antihol ..."
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Cited by 9 (1 self)
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We investigate the dynamics and the bifurcation diagrams of iterated antiholomorphic polynomials: these are complex conjugates of ordinary polynomials. Their second iterates are holomorphic polynomials, but dependence on parameters is only realanalytic. The parameter space of degree two antiholomorphic polynomials features distorted copies to the Mandelbrot set, as well as parts where local connectivity and pathwise connectivity fails. This space arises naturally in the context of iterated real cubic (holomorphic) polynomials, which we investigate as well.
Mating Siegel quadratic polynomials
 J. Amer. Math. Soc
"... Abstract. Let F be a quadratic rational map of the sphere which has two fixed Siegel disks with bounded type rotation numbers θ and ν. Using a new degree 3 Blaschke product model for the dynamics of F and an adaptation of complex a priori bounds for renormalization of critical circle maps, we prove ..."
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Cited by 9 (4 self)
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Abstract. Let F be a quadratic rational map of the sphere which has two fixed Siegel disks with bounded type rotation numbers θ and ν. Using a new degree 3 Blaschke product model for the dynamics of F and an adaptation of complex a priori bounds for renormalization of critical circle maps, we prove that F can be realized as the mating of two Siegel quadratic polynomials with the corresponding rotation numbers θ and ν.
A PRIORI BOUNDS FOR SOME INFINITELY RENORMALIZABLE QUADRATICS: II. DECORATIONS.
, 2006
"... 2. Yoccoz puzzle, decorations, and the Modified Principal Nest 4 3. Pseudoquadraticlike maps and pseudopuzzle 10 ..."
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Cited by 7 (2 self)
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2. Yoccoz puzzle, decorations, and the Modified Principal Nest 4 3. Pseudoquadraticlike maps and pseudopuzzle 10