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

 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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