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## Fundamenta Mathematicae

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## Topology and measure of buried points in Julia sets

### Tom 222 / 2013

Fundamenta Mathematicae 222 (2013), 1-17 MSC: Primary 37F20; Secondary 54F50. DOI: 10.4064/fm222-1-1

#### Streszczenie

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).

#### Autorzy

• Clinton P. CurryDepartment of Mathematics
Huntingdon College
Montgomery, AL 36106, U.S.A.
e-mail
• John C. MayerDepartment of Mathematics
University of Alabama at Birmingham
Birmingham, AL 35294-1170, U.S.A.
e-mail
• E. D. TymchatynDepartment of Mathematics