Topology and measure of buried points in Julia sets

Volume 222 / 2013

Clinton P. Curry, John C. Mayer, E. D. Tymchatyn Fundamenta Mathematicae 222 (2013), 1-17 MSC: Primary 37F20; Secondary 54F50. DOI: 10.4064/fm222-1-1


It is well-known that the set of \emph {buried points} of a Julia set of a rational function (also called the residual Julia set) is topologically “fat” in the sense that it is a dense $G_\delta $ if it is non-empty. We show that it is, in many cases, a full-measure subset of the Julia set with respect to conformal measure and the measure of maximal entropy. We also address Hausdorff dimension of buried points in the same cases, and discuss connectivity and topological dimension of the set of buried points. Finally, we present a non-dynamical example of a plane continuum whose set of buried points is a dense and hereditarily disconnected (components are points) $G_\delta $, but not totally disconnected (not all quasi-components are points).


  • Clinton P. CurryDepartment of Mathematics
    Huntingdon College
    Montgomery, AL 36106, U.S.A.
  • John C. MayerDepartment of Mathematics
    University of Alabama at Birmingham
    Birmingham, AL 35294-1170, U.S.A.
  • E. D. TymchatynDepartment of Mathematics
    University of Saskatchewan
    Saskatoon, SK, S7N 5E6, Canada

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