Hausdorff Dimension and Conformal Dynamics I: Strong Convergence of Kleinian Groups

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Hausdorff Dimension and Conformal Dynamics I: Strong Convergence of Kleinian Groups

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Title: Hausdorff Dimension and Conformal Dynamics I: Strong Convergence of Kleinian Groups
Author: McMullen, Curtis T.
Citation: McMullen, Curtis T. 1999. Hausdorff dimension and conformal dynamics I: Strong convergence of Kleinian groups. Journal of Differential Geometry 51(3): 471–515.
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Abstract: This paper investigates the behavior of the Hausdorff dimensions of the limit sets \(\Lambda_n\) and \(\Lambda\) of a sequence of Kleinian groups \(\Gamma_n \rightarrow \Gamma\), where \(M = \mathbb{H}^3/\Gamma\) is geometrically finite. We show if \(\Gamma_n \rightarrow \Gamma\) strongly, then: (a) \(M_n = \mathbb{H}^3/\Gamma_n\) is geometrically finite for all \(n \gg 0\), (b) \(\Lambda_n \rightarrow \Lambda\) in the Hausdorff topology, and (c) \(H. dim(\Lambda_n) \rightarrow H. dim(\Lambda)\), if \(H. dim(\Lambda) \geq 1\). On the other hand, we give examples showing the dimension can vary discontinuously under strong limits when \(H. dim(\Lambda) < 1\). Continuity can be recovered by requiring that accidental parabolics converge radially. Similar results hold for higher-dimensional manifolds. Applications are given to quasifuchsian groups and their limits.
Published Version: http://www.intlpress.com/journals/JDG/archive/vol.51/issue3/3_3.pdf
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:9871959

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  • FAS Scholarly Articles [7078]
    Peer reviewed scholarly articles from the Faculty of Arts and Sciences of Harvard University
 
 

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