Lex G. Oversteegen

The geometry of Julia sets

The long term analysis of dynamical systems inspired the study of the dynamics of families of mappings. Many of these investigations led to the study of the dynamics of mappings on Cantor sets and on intervals. Julia sets play a critical role in the understanding of the dynamics of families of mappings. In this paper we introduce another class of objects (called hairy objects) which share many properties with the Cantor set and the interval: they are topologically unique and admit only one embedding in the plane. These uniqueness properties explain the regular occurrence of hairy objects in pictures of Julia sets-hairy objects are ubiquitous. Hairy arcs will be used to give a complete topological description of the Julia sets of many members of the exponential family

On the dimension of certain totally disconnected spaces

It is well known that there exist separable, metrizable, totally disconnected spaces of all dimensions. In this note we introduce the notion of an almost 0-dimensional space and prove that every such space is a totally disconnected subspace of an R-tree and, hence, at most 1-dimensional. As applications we prove that the spaces of homeomorphisms of the universal Menger continua are 1-dimensional and that hereditarily locally connected spaces have dimension at most two.

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

Homogeneous weak solenoids

A (generalized) weak solenoid is an inverse limit space over manifolds with bonding maps that are covering maps. If the covering maps are regular, then we call the inverse limit space a strong solenoid. By a theorem of M.C. McCord, strong solenoids are homogeneous. We show conversely that homogeneous weak solenoids are topologically equivalent to strong solenoids. We also give an example of a weak solenoid that has simply connected path-components, but which is not homogeneous.

Fixed-point-free maps on tree-like continua

Abstract A fixed-point-free homeomorphism on a tree-like continuum is described. The tree-like continuum is obtained as inverse limit of trees, and the homeomorphism is obtained as an induced map of the inverse limit space.

On minimal maps of 2-manifolds

We prove that a minimal self-mapping of a compact 2-manifold has tree-like fibers (i.e. all points have preimages which are connected, at most one-dimensional and with trivial shape). We also prove that the only 2-manifolds (compact or not) which admit minimal maps are either finite unions of tori, or finite unions of Klein bottles.

On almost one-to-one maps

A continuous map f: X → Y of topological spaces X, Y is said to be almost 1-to-1 if the set of the points x ∈ X such that f -1 (f(x)) = {x} is dense in X; it is said to be light if pointwise preimages are zero dimensional. We study almost 1-to-1 light maps of some compact and σ-compact spaces (e.g., n-manifolds or dendrites) and prove that in some important cases they must be homeomorphisms or embeddings. In a forthcoming paper we use these results and show that if f is a minimal self-mapping of a 2-manifold M, then point preimages under f are tree-like continua and either M is a union of 2-tori, or M is a union of Klein bottles permuted by f.

Backward stability for polynomial maps with locally connected Julia sets

We study topological dynamics on unshielded planar continua with weak expanding properties at cycles for which we prove that the absence of wandering continua implies backward stability. Then we deduce from this that a polynomial f with a locally connected Julia set is backward stable outside any neighborhood of its attracting and neutral cycles. For a conformal measure μ this easily implies that one of the following holds: 1. for μ-a.e. x E J(f), ω(x) = J(f); 2. for μ-a.e. x ∈ J(f), ω(x) = ω(c(x)) for a critical point c(x) depending on x.

The dynamics of the Sierpiński curve

The Sierpinski curve X admits a homeomorphism with a dense orbit. However, X is not minimal and does not admit an expansive homeo- morphism. 1. Statement of the theorem