# Hausdorff Dimension and Conformal Dynamics III: Computation of Dimension

 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. Full Text & Related Files: McMullen_HausdorffComputatDimension.pdf (1.197Mb; PDF) 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 Downloads of this work:

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