Topology and measure of buried points in Julia sets

Clinton P. Curry, John C. Mayer, and E. D. Tymchatyn. "Topology and measure of buried points in Julia sets." Fundamenta Mathematicae 222.1 (2013): 1-17. <http://eudml.org/doc/283291>.

@article{ClintonP2013,
abstract = {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 $G_\{δ\}$ 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) $G_\{δ\}$, but not totally disconnected (not all quasi-components are points).},
author = {Clinton P. Curry, John C. Mayer, E. D. Tymchatyn},
journal = {Fundamenta Mathematicae},
keywords = {Julia set; holomorphic dynamics; Fatou domain; complex dynamics; buried point; residual Julia set; Suslinian},
language = {eng},
number = {1},
pages = {1-17},
title = {Topology and measure of buried points in Julia sets},
url = {http://eudml.org/doc/283291},
volume = {222},
year = {2013},
}

TY - JOUR
AU - Clinton P. Curry
AU - John C. Mayer
AU - E. D. Tymchatyn
TI - Topology and measure of buried points in Julia sets
JO - Fundamenta Mathematicae
PY - 2013
VL - 222
IS - 1
SP - 1
EP - 17
AB - 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 $G_{δ}$ 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) $G_{δ}$, but not totally disconnected (not all quasi-components are points).
LA - eng
KW - Julia set; holomorphic dynamics; Fatou domain; complex dynamics; buried point; residual Julia set; Suslinian
UR - http://eudml.org/doc/283291
ER -