Topology and measure of buried points in Julia sets

Clinton P. Curry ¹   ; John C. Mayer ²   ; E. D. Tymchatyn ³

¹ Department of Mathematics Huntingdon College Montgomery, AL 36106, U.S.A.
² Department of Mathematics University of Alabama at Birmingham Birmingham, AL 35294-1170, U.S.A.
³ Department of Mathematics University of Saskatchewan Saskatoon, SK, S7N 5E6, Canada

Fundamenta Mathematicae, Tome 222 (2013) no. 1, pp. 1-17

Résumé

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).

Détail

DOI : 10.4064/fm222-1-1

Keywords: well known set emph buried points julia set rational function called residual julia set topologically fat sense dense delta non empty many cases full measure subset julia set respect conformal measure measure maximal entropy address hausdorff dimension buried points cases discuss connectivity topological dimension set buried points finally present non dynamical example plane continuum whose set buried points dense hereditarily disconnected components points delta totally disconnected quasi components points