2007/12/27 by Lasse Rempe · 2 citations
Mathematics · #Analytic and geometric function theory #Holomorphic and Operator Theory #Mathematical Dynamics and Fractals #math.CV #math.DS #msc:30D05 #msc:37F10 #msc:37F35
paper · pdf · doi:10.1090/s0002-9939-08-09650-0
published as Proc. Amer. Math. Soc. 137 (2009), 1411-1420. · 11 pages
arxiv created 2007/12/27 · openalex publication_date 2008/11/03 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
We survey the definition of the radial Julia set <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J Subscript r Baseline left-parenthesis f right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>J</mml:mi> <mml:mi>r</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">Jr(f)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of a meromorphic function (in fact, more generally, any <italic>Ahlfors islands map</italic> ), and give a simple proof that the Hausdorff dimension of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper J Subscript r Baseline left-parenthesis f right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>J</mml:mi> <mml:mi>r</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">Jr(f)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the hyperbolic dimension <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="dimension Subscript h y p Baseline left-parenthesis f right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>dim</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>hyp</mml:mi> </mml:mrow> </mml:msub> <mml:mo> </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">dim _\operatorname hyp(f)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> always coincide.