Hausdorff Dimension and Conformal Dynamics I: Strong Convergence of Kleinian Groups
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| dc.contributor.author |
McMullen, Curtis T.
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| dc.date.accessioned |
2012-11-06T17:03:45Z |
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| dc.date.issued |
1999 |
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| 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 |
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| dc.identifier.uri |
http://nrs.harvard.edu/urn-3:HUL.InstRepos:9871959 |
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| 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. |
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| dc.description.sponsorship |
Mathematics |
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| dc.language.iso |
en_US |
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| dc.publisher |
International Press |
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| dc.relation.isversionof |
http://www.intlpress.com/journals/JDG/archive/vol.51/issue3/3_3.pdf |
en_US |
| dash.license |
LAA |
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| 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.
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| dc.date.available |
2012-11-06T17:03:45Z |
|
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FAS Scholarly Articles [5137]
Peer reviewed scholarly articles from the Faculty of Arts and Sciences of Harvard University
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