dc.contributor.author | McMullen, Curtis T. | |
dc.date.accessioned | 2009-12-21T19:29:34Z | |
dc.date.issued | 2000 | |
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 | en_US |
dc.identifier.uri | http://nrs.harvard.edu/urn-3:HUL.InstRepos:3445996 | |
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 | en_US |
dc.language.iso | en_US | en_US |
dc.publisher | Birkhäuser Basel | en_US |
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 | |
dc.subject | complex dynamics | en_US |
dc.subject | iterated rational maps | en_US |
dc.subject | Julia sets | en_US |
dc.subject | Hausdorff dimension | en_US |
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. | |
dc.date.available | 2009-12-21T19:29:34Z | |
dc.identifier.doi | 10.1007/s000140050140 | * |
dash.contributor.affiliated | McMullen, Curtis | |