Hausdorff Dimension and Conformal Dynamics III: Computation of Dimension

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Hausdorff Dimension and Conformal Dynamics III: Computation of Dimension

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Title: Hausdorff Dimension and Conformal Dynamics III: Computation of Dimension
Author: McMullen, Curtis T.
Citation: McMullen, Curtis T. 1998. Hausdorff dimension and conformal dynamics, III: Computation of dimension. American Journal of Mathematics 120(4): 691-721. Revised 2003.
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Abstract: 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 \(f_c(z) = z^2 +c, c \in [-1, \frac{1}{2}]\); and (c) the family of rational maps \(ft(z) = \frac{z}{t} + \frac{1}{z}, t \in (0, 1]\). We also calculate \(H. dim (\wedge) \approx 1.305688\) for the Apollonian gasket, and \(H. dim (J( f)) \approx 1.3934\) for Douady’s rabbit, where \(f(z) = z^2 + c\) satisfies \(f^3(0) = 0\).
Published Version: doi:10.1353/ajm.1998.0031
Other Sources: http://www.math.harvard.edu/~ctm/papers/index.html
Terms of Use: This article is made available under the terms and conditions applicable to Other Posted Material, as set forth at http://nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of-use#LAA
Citable link to this page: http://nrs.harvard.edu/urn-3:HUL.InstRepos:3445973

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  • FAS Scholarly Articles [7594]
    Peer reviewed scholarly articles from the Faculty of Arts and Sciences of Harvard University
 
 

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