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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dc.contributor.author McMullen, Curtis T.
dc.date.accessioned 2012-11-06T17:03:45Z
dc.date.issued 1999
dc.identifier.citation McMullen, Curtis T. 1999. Hausdorff dimension and conformal dynamics I: Strong convergence of Kleinian groups. Journal of Differential Geometry 51(3): 471–515. en_US
dc.identifier.issn 0022-040X en_US
dc.identifier.issn 1945-743X en_US
dc.identifier.uri http://nrs.harvard.edu/urn-3:HUL.InstRepos:9871959
dc.description.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. en_US
dc.description.sponsorship Mathematics en_US
dc.language.iso en_US en_US
dc.publisher International Press en_US
dc.relation.isversionof http://www.intlpress.com/journals/JDG/archive/vol.51/issue3/3_3.pdf en_US
dash.license LAA
dc.title Hausdorff Dimension and Conformal Dynamics I: Strong Convergence of Kleinian Groups en_US
dc.type Journal Article en_US
dc.description.version Author's Original en_US
dc.relation.journal Journal of Differential Geometry en_US
dash.depositing.author McMullen, Curtis T.
dc.date.available 2012-11-06T17:03:45Z

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  • FAS Scholarly Articles [7106]
    Peer reviewed scholarly articles from the Faculty of Arts and Sciences of Harvard University

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