Hausdorff Dimension and Conformal Dynamics II: Geometrically Finite Rational Maps
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| dc.contributor.author |
McMullen, Curtis T.
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| dc.date.accessioned |
2009-12-21T19:29:34Z |
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| dc.date.issued |
2000 |
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| dc.identifier.citation |
McMullen, Curtis T. 2000. Hausdorff dimension and conformal dynamics II:
Geometrically finite rational maps. Commentarii Mathematici Helvetici 75(4): 535–593. Revised 2003. |
en_US |
| dc.identifier.issn |
0010-2571 |
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| dc.identifier.uri |
http://nrs.harvard.edu/urn-3:HUL.InstRepos:3445996 |
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| dc.description.abstract |
This paper investigates several dynamically defined dimensions for rational maps \(f\) on the Riemann sphere, providing a systematic treatment modeled on the theory for Kleinian groups. We begin by defining the radial Julia set \(J_{rad}(f)\), and showing that every rational map satisfies \(H. dimJ_{rad}(f) = \alpha(f)\) where \(\alpha(f)\) is the minimal dimension of an \(f\)-invariant conformal density on the sphere. A rational map \(f\) is geometrically finite if every critical point in the Julia set is preperiodic. In this case we show \(H. dimJ_{rad}(f) = H. dimJ(f) = \delta(f)\), where \(\delta(f)\) is the critical exponent of the Poincar´e series; and \(f\) admits a unique normalized invariant density \(\mu\) of dimension \(\delta(f)\). Now let \(f\) be geometrically finite and suppose \(f_n \rightarrow f\) algebraically, preserving critical relations. When the convergence is horocyclic for each parabolic point of \(f\), we show \(fn\) is geometrically finite for \(n \gg 0\) and \(J(f_n) \rightarrow J(f)\) in the Hausdorff topology. If the convergence is radial, then in addition we show \(H. dim J(f_n) \rightarrow H. dimJ(f).\) We give examples of horocyclic but not radial convergence where \(H. dim J(f_n) \rightarrow 1 > H. dim J(f) = \frac{1}{2} + \epsilon \). We also give a simple demonstration of Shishikura’s result that there exist \(fn(z) = z^2 + c_n \) with \(H. dimJ(f_n) \rightarrow 2\). The proofs employ a new method that reduces the study of parabolic points to the case of elementary Kleinian groups. |
en_US |
| dc.description.sponsorship |
Mathematics |
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| dc.language.iso |
en_US |
en_US |
| dc.publisher |
Birkhäuser Basel |
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| dc.relation.isversionof |
doi:10.1007/s000140050140 |
en_US |
| dc.relation.hasversion |
http://www.math.harvard.edu/~ctm/papers/index.html |
en_US |
| dash.license |
LAA |
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| dc.subject |
complex dynamics |
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| dc.subject |
iterated rational maps |
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| dc.subject |
Julia sets |
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| dc.subject |
Hausdorff dimension |
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| dc.title |
Hausdorff Dimension and Conformal Dynamics II: Geometrically Finite Rational Maps |
en_US |
| dc.type |
Journal Article |
en_US |
| dc.description.version |
Author's Original |
en_US |
| dc.relation.journal |
Commentarii Mathematici Helvetici |
en_US |
| dash.depositing.author |
McMullen, Curtis T.
|
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| dc.date.available |
2009-12-21T19:29:34Z |
|
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FAS Scholarly Articles [5171]
Peer reviewed scholarly articles from the Faculty of Arts and Sciences of Harvard University
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