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dc.contributor.authorMcMullen, Curtis T.
dc.date.accessioned2009-12-21T19:29:34Z
dc.date.issued2000
dc.identifier.citationMcMullen, 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.issn0010-2571en_US
dc.identifier.urihttp://nrs.harvard.edu/urn-3:HUL.InstRepos:3445996
dc.description.abstractThis 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.sponsorshipMathematicsen_US
dc.language.isoen_USen_US
dc.publisherBirkhäuser Baselen_US
dc.relation.isversionofdoi:10.1007/s000140050140en_US
dc.relation.hasversionhttp://www.math.harvard.edu/~ctm/papers/index.htmlen_US
dash.licenseLAA
dc.subjectcomplex dynamicsen_US
dc.subjectiterated rational mapsen_US
dc.subjectJulia setsen_US
dc.subjectHausdorff dimensionen_US
dc.titleHausdorff Dimension and Conformal Dynamics II: Geometrically Finite Rational Mapsen_US
dc.typeJournal Articleen_US
dc.description.versionAuthor's Originalen_US
dc.relation.journalCommentarii Mathematici Helveticien_US
dash.depositing.authorMcMullen, Curtis T.
dc.date.available2009-12-21T19:29:34Z
dc.identifier.doi10.1007/s000140050140*
dash.contributor.affiliatedMcMullen, Curtis


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