Hausdorff dimension and conformal dynamics, III: Computation of dimension

@article{McMullen1998HausdorffDA,
  title={Hausdorff dimension and conformal dynamics, III: Computation of dimension},
  author={C. McMullen},
  journal={American Journal of Mathematics},
  year={1998},
  volume={120},
  pages={691 - 721}
}
  • C. McMullen
  • Published 1998
  • Mathematics
  • American Journal of Mathematics
<abstract abstract-type="TeX"><p>This paper presents an eigenvalue algorithm for accurately computing the Hausdorff dimension of limit sets of Kleinian groups and Julia sets of rational maps. The algorithm is applied to Schottky groups, quadratic polynomials and Blaschke products, yielding both numerical and theoretical results. Dimension graphs are presented for (a) the family of Fuchsian groups generated by reflections in 3 symmetric geodesics; (b) the family of polynomials <i>f<sub>c</sub… Expand
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Hausdorff Dimension and Conformal Dynamics I: Strong Convergence of Kleinian Groups
This paper investigates the behavior of the Hausdorff dimensions of the limit sets Λn and Λ of a sequence of Kleinian groups Γn → Γ, where M = H/Γ is geometrically finite. We show if Γn → Γ strongly,Expand
Hausdorff dimension and conformal dynamics II: Geometrically finite rational maps
Abstract. This paper investigates several dynamically defined dimensions for rational maps f on the Riemann sphere, providing a systematic treatment modeled on the theory for Kleinian groups.¶WeExpand
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Addressing researchers and graduate students in the active meeting ground of analysis, geometry, and dynamics, this book presents a study of renormalization of quadratic polynomials and a rapidExpand
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Let X n+a denote the real hyperbolic space of dimension n+ l . We will make use of both the ball and upper half space models of X n+~. The ball model is Bn+t={xERn+I; Ix[<l} with the line elementExpand
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Algorithms to compute self-similar measures associated to iterated function systems, and more general self-replicating measures that include Hausdorff measure on the attractor of a nonlinear i.f.s, are described and an intriguing structure associated to these ratios is found that is called density diagrams. Expand
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This paper deals with the Hausdorff dimension of the Julia set of quadratic polynomials. It is divided in two parts. The first aims to compute good numerical approximations of the dimension forExpand
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We show that the Hausdorff dimension of the limit set is a real analytic function on the deformation space of a class of convex co-compact Kleinian groups which includes all convex co-compactExpand
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Using a classical method from physics called Rayleigh’s cutting method, we prove the conjecture of Phillips and Sarnak that there is a universal lower bound L2 > 0 for the lowest eigenvalue of theExpand
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The purpose of this note is to prove a conjecture of D. Sullivan that when the Julia set J of a rational function f is hyperbolic, the Hausdorff dimension of J depends real analytically on f . WeExpand
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