It is well-known that the set of 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 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) , but not totally disconnected (not all quasi-components are points).
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm222-1-1,
author = {Clinton P. Curry and John C. Mayer and E. D. Tymchatyn},
title = {Topology and measure of buried points in Julia sets},
journal = {Fundamenta Mathematicae},
volume = {220},
year = {2013},
pages = {1-17},
zbl = {1280.37042},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm222-1-1}
}
Clinton P. Curry; John C. Mayer; E. D. Tymchatyn. Topology and measure of buried points in Julia sets. Fundamenta Mathematicae, Tome 220 (2013) pp. 1-17. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm222-1-1/