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By studying laminations of the unit disk, we can gain insight into the structure of Julia sets of polynomials and their dynamics in the complex plane. The polynomials of a given degree, $d$, have a parameter space. The hyperbolic components of such parameter spaces are in correspondence to rotational polygons, or classes of "rotational sets", which we study in this paper. By studying the count of such rotational sets, and therefore the underlying structure behind these rotational sets and polygons, we can gain insight into the interrelationship among hyperbolic components of the parameter space of these polynomials. These rotational sets are created by uniting rotational orbits, as we define in this paper. The number of such sets for a given degree $d$, rotation number $\frac pq$, and cardinality $k$ can be determined by analyzing the potential placements of pre-images of zero on the unit circle with respect to the rotational set under the $d$-tupling map. We obtain a closed-form formula for the count. Though this count is already known based upon some sophisticated results, our count is based upon elementary geometric and combinatorial principles, and provides an intuitive explanation., Comment: 12 pages, 5 figures

Quadratic laminations of the unit disk were introduced by Thurston as a vehicle for understanding the (connected) Julia sets of quadratic polynomials and the parameter space of quadratic polynomials. The "Central Strip Lemma" plays a key role in Thurston's classification of gaps in quadratic laminations, and in describing the corresponding parameter space. We generalize the notion of {\em Central Strip} to laminations of all degrees $d\ge2$ and prove a Central Strip Lemma for degree $d\ge2$. We conclude with applications of the Central Strip Lemma to {\em identity return polygons} that show it may play a role similar to Thurston's lemma for higher degree laminations., Comment: 31 pages and 19 figures

We present proofs of basic results, including those developed by Harold Bell, for the plane fixed point problem: does every map of a non-separating plane continuum have a fixed point? 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. A fixed point theorem for positively oriented, perfect maps of the plane is obtained. This generalizes results announced by Bell in 1982. A continuous map of an interval to the real line which sends the endpoints in opposite directions has a fixed point. We generalize this to maps on non-invariant continua in the plane under positively oriented maps of the plane (with appropriate boundary conditions). These methods imply that in some cases non-invariant continua in the plane are degenerate. This has important applications in complex dynamics. E.g., a special case of our results shows that if $X$ is a non-separating invariant subcontinuum of the Julia set of a polynomial $P$ containing no fixed Cremer points and exhibiting no local rotation at all fixed points, then $X$ must be a point. It follows that impressions of some external rays to polynomial Julia sets are degenerate., Comment: 107 pages, 10 figures; the preprint expands and combines several earlier preprints; to appear in the Memoirs of the American Mathematical Society

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., Comment: 10 pages, 2 figures

Makienko's conjecture, a proposed addition to Sullivan's 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 Makienko's 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., Comment: 10 pages

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., Comment: 11 pages, 3 figures. To appear in Proceedings of the American Mathematical Society

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 Bell's 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 $X$., Comment: 50 pages, 4 figures Version 2: 52 pages 5 figures. Corrected authors, added picture, expanded and simplified some arguments