Buried points in Julia sets

Abstract

Dedicated to Robert Devaney on the occasion of his 60th birthday An introduction to buried points in Julia sets and a list of questions about buried points, written to encourage aficionados of topology and dynamics to work on these questions.

Plane continua and totally disconnected sets of buried points

“…Curry, Mayer and Tymchatyn proposed to study it from a purely topological point of view. For more details, see [1,2]. …”

Topology and measure of buried points in Julia sets

“…We are particularly interested in the set of buried points of J(R), defined to be the set of points of J(R) which are not on the boundary of any Fatou domain of R. (Note that, while every point of J(R) will be on the boundary of the Fatou set of R, it is not the case that every point of J(R) will be on the boundary of a component of the Fatou set.) For learning about buried points of Julia sets (also called the residual Julia set), we recommend [CM10], and especially [DF08] for transcendental functions.…”

“…Topology of buried points. In [CM10] we asked, among others, the following two questions, to which we provide partial answers in this paper. Other recent papers have addressed Question 1 from a continuum-theoretic viewpoint [vMT, vMTTV].…”