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77
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 (s ..."
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Cited by 37 (15 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...
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 33 (13 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.
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 29 (7 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.
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 22 (6 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.
Combinatorial rigidity for unicritical polynomials
 Annals Math
"... Abstract. We prove that any unicritical polynomial fc: z 7! zd + c which is at most nitely renormalizable and has only repelling periodic points is combinatorially rigid. It implies that the connectedness locus (the \Multibrot set") is locally connected at the corresponding parameter values. It ..."
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Cited by 19 (4 self)
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Abstract. We prove that any unicritical polynomial fc: z 7! zd + c which is at most nitely renormalizable and has only repelling periodic points is combinatorially rigid. It implies that the connectedness locus (the \Multibrot set") is locally connected at the corresponding parameter values. It generalizes Yoccoz's Theorem for quadratics to the higher degree case. 1.
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 19 (4 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.
Periodic points and dynamic rays of exponential maps’, preprint. Dierk Schleicher School of Engineering and
 Science International University Bremen Postfach 750 561 D28725 Bremen Germany dierk@iubremen.de Johannes Zimmer Division of Engineering and Applied Science Mail Stop 10444 California Institute of Technology Pasadena CA 91125 USA zimmer@caltech.edu
"... We investigate the dynamics of exponential maps z 7 → λez; the goal is a description by means of dynamic rays. We discuss landing properties of dynamic rays and show that in many important cases, repelling and parabolic periodic points are landing points of periodic dynamic rays. For postsingularly ..."
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Cited by 18 (4 self)
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We investigate the dynamics of exponential maps z 7 → λez; the goal is a description by means of dynamic rays. We discuss landing properties of dynamic rays and show that in many important cases, repelling and parabolic periodic points are landing points of periodic dynamic rays. For postsingularly finite exponential maps, we use spider theory to show that a dynamic ray lands at the singular value.
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 18 (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.