Buried Points in Julia Sets

We give 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.

The plane fixed point problem

In this paper we present proofs of basic results, including those developed so far by H. Bell, for the plane fixed point problem. Some of these results had been announced much earlier by Bell but without accessible proofs. We define the concept of the variation of a map on a simple closed curve and relate it to the index of the map on that curve: Index = Variation + 1. We develop a prime end theory through hyperbolic chords in maximal round balls contained in the complement of a non-separating plane continuum $X$. We define the concept of an {em outchannel} for a fixed point free map which carries the boundary of $X$ minimally into itself and prove that such a map has a emph{unique} outchannel, and that outchannel must have variation $=-1$. We also extend Bells linchpin theorem for a foliation of a simply connected domain, by closed convex subsets, to arbitrary domains in the sphere. We introduce the notion of an oriented map of the plane. We show that the perfect oriented maps of the plane coincide with confluent (that is composition of monotone and open) perfect maps of the plane. We obtain a fixed point theorem for positively oriented, perfect maps of the plane. This generalizes results announced by Bell in 1982 (see also cite{akis99}). It follows that if $X$ is invariant under an oriented map $f$, then $f$ has a point of period at most two in $X$.

Any counterexample to Makienkos conjecture is an indecomposable continuum

Makienkos conjecture, a proposed addition to Sullivans dictionary, can be stated as follows: The Julia set of a rational function R has buried points if and only if no component of the Fatou set is completely invariant under the second iterate of R. We prove Makienkos conjecture for rational functions with Julia sets that are decomposable continua. This is a very broad collection of Julia sets; it is not known if there exists a rational functions whose Julia set is an indecomposable continuum.

Characterizing indecomposable plane continua from their complements

We show that a plane continuum X is indecomposable iff X has a sequence (U_n) of not necessarily distinct complementary domains satisfying what we call the double-pass condition: If one draws an open arc A_n in each U_n whose ends limit into the boundary of U_n, one can choose components of U_n minus A_n whose boundaries intersected with the continuum (which we call shadows) converge to the continuum.