John C. Mayer

An explosion point for the set of endpoints of the Julia set of λ exp ( z )

The Julia set J λ of the complex exponential function E λ : z → λ e z for a real parameter λ(0 e ) is known to be a Cantor bouquet of rays extending from the set A λ of endpoints of J λ to ∞. Since A λ contains all the repelling periodic points of E λ , it follows that J λ = Cl ( A λ ). We show that A λ is a totally disconnected subspace of the complex plane ℂ, but if the point at ∞ is added, then is a connected subspace of the Riemann sphere . As a corollary, A λ has topological dimension 1. Thus, ∞ is an explosion point in the topological sense for Âλ. It is remarkable that a space with an explosion point occurs ‘naturally’ in this way.

A Topological Characterization of R-Trees

R-trees arise naturally in the study of groups of isometries of hyperbolic space. An R-tree is a uniquely arcwise connected metric space in which each arc is isometric to a subarc of the reals R. Actions on R-trees can be viewed as ideal points in the compactification of groups of isometries. As such they have applications to the study of hyperbolic manifolds. Our concern in this paper, however, is with the topological characterization of R-trees. Our main theorem is the following: Let (X, p) be a metric space. Then X is uniquely arcwise connected and locally arcwise connected if, and only iJf X admits a compatible metric d such that (X, d) is an R-tree. Essentially, we show how to put a convex metric on a uniquely arcwise connected, locally arcwise connected, metrizable space.

Universal spaces for R-trees

R-trees arise naturally in the study of groups of isometries of hyperboIic space. An R-tree is a uniquely arcwise connected metric space in which each arc is isometric to a subarc of the reals. It follows that an R-tree is locally arcwise connected, contractible, and one-dimensional. Unique and local arcwise connectivity characterize R-trees among metric spaces. A universal R-tree would be of interest in attempting to classify the actions of groups of isometries on R-trees. It is easy to see that there is no universal R-tree. However, we show that there is a universal separable R-tree T ℵ 0

Indecomposable continua and the Julia sets of polynomials

We find several necessary and sufficient conditions for the Julia set J of a polynomial of degree d ≥ 2 to be an indecomposable continuum. One condition that may be easier to check than others is the following: Suppose J is connected; then J is an indecomposable continuum iff the impression of some prime end of the unbounded complementary domain of J has interior in J

Characterization of Separable Metric R-Trees

An R-tree (X, d) is a uniquely arcwise connected metric space in which each arc is isometric to a subarc of the reals. R-trees arise naturally in the study of groups of isometries of hyperbolic space. Two of the authors had previously characterized R-trees topologically among metric spaces. The purpose of this paper is to provide a simpler proof of this characterization for separable metric spaces. The main theorem is the following : Let (X, r) be a separable metric space. Then the following are equivalent : (1) X admits an equivalent metric d such that (X, d) is an R-tree. (2) X is locally arcwise connected and uniquely arcwise connected

Recurrent critical points and typical limit sets for conformal measures

An apparatus for facilitating manipulation of the body of a human patient for the treatment of vertebral disorders comprises an upright support from which a pivotal arm is cantilevered and has at its ends a suspended traverse. The traverse is provided with knee-supporting shells or saddles from which the patient may be suspended head downward to permit manipulation. The traverse may also carry a roll or drum adapted to carry the patient in a bent-over condition to permit manipulation of the spine.

Buried points in Julia sets

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.

Any counterexample to Makienko's 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 ) =i of not necessarily distinct complementary domains satisfying the double-pass condition: for any sequence (A n )jj l of open arcs, with An C U n and An \ An C dU n , there is a sequence of shadows (S n )jj =l , where each S n is a shadow of An, such that lim S n = X. Such an open arc divides U n into disjoint subdomains V n,i and V n,2 , and a shadow (of An) is one of the sets <9V n,i n 9U.

f-1 – Continuum Theory

Publisher Summary Continuum theory includes the study of all continua (that is, all compact and connected spaces) and the maps between them. The vitality of an area is often determined by the existence of major open problems that are either natural or connected to other areas of mathematics. This chapter focuses on these problems with a brief description of each, and is restricted mainly to metric continua. All continua naturally belong to one of the two important types. A decomposable continuum is the union of two proper subcontinua and is indecomposable otherwise. All locally connected continua are decomposable while all indecomposable continua are nowhere locally connected. Hence, decompasable continua include the class of continua with characteristic local properties, while indecomposable continua include the class with complicated local structure. It is well known that all locally connected continua are arcwise connected and hence between every pair of points there is a very simple irreducible subcontinuum containing them. Each one-dimensional continuum can be represented as an inverse limit on graphs and are called graph-like. Special classes of one dimensional continua are the tree-like, arc-like, and circle-like continua.