Part 1. Basic Theory Chapter 2. Preliminaries and outline of Part 1 2.1. Index 2.2. Variation 2.3. Classes of maps 2.4. Partitioning domains Chapter 3. Tools 3.1. Stability of Index 3.2. Index and variation for finite partitions 3.3. Locating arcs of negative variation 3.4. Crosscuts and bumping arcs 3.5. Index and Variation for Carathéodory Loops 3.6. Prime Ends 3.7. Oriented maps 3.8. Induced maps of prime ends Chapter 4. Partitions of domains in the sphere 4.1. Kulkarni-Pinkall Partitions 4.2. Hyperbolic foliation of simply connected domains 4.3. Schoenflies Theorem 4.4. Prime ends Part 2. Applications of basic theory Chapter 5. Description of main results of Part 2 5.1. Outchannels 5.2. Fixed points in invariant continua 5.3. Fixed points in non-invariant continua -the case of dendrites 5.4. Fixed points in non-invariant continua -the planar case 5.5. The polynomial case Chapter 6. Outchannels and their properties 6.1. Outchannels v vi CONTENTS 6.2. Uniqueness of the Outchannel Chapter 7. Fixed points 7.1. Fixed points in invariant continua 7.2. Dendrites 7.3. Non-invariant continua and positively oriented maps of the plane 7.4. Maps with isolated fixed points 7.5. Applications to complex dynamics Bibliography Index

Abstract. 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 O-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.

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 1-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.

In this paper we solve the following problem in the affirmative: Let Z be a continuum in the plane ‫ރ‬ and suppose that h W Z OE0; 1 ! ‫ރ‬ is an isotopy starting at the identity. Can h be extended to an isotopy of the plane? We will provide a new characterization of an accessible point in a planar continuum Z and use it to show that accessibility of a point is preserved during the isotopy. We show next that the isotopy can be extended over small hyperbolic crosscuts which are shown to remain small under the isotopy. The proof makes use of the notion of a metric external ray, which mimics the notion of a conformal external ray, but is easier to control during an isotopy. It also relies on the existence of a partition of a hyperbolic, simply connected domain U in the sphere, into hyperbolically convex subsets, which have limited distortion under conformal maps to the unit disk.

On homogeneous totally disconnected 1-dimensional spaces by Kazuhiro K a w a m u r a (Saskatoon), Lex G. O v e r s t e e g e n (Birmingham, Ala.) and E. D. T y m c h a t y n (Saskatoon)