Abstract. 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 pathcomponents, but which is not homogeneous.

Abstract.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.

Let P be a polynomial with a connected Julia set J. We use continuum theory to show that it admits a finest monotone map ϕ onto a locally connected continuum J ∼ P , i.e. a monotone map ϕ : J → J ∼ P such that for any other monotone map ψ : J → J ′ there exists a monotone map h with ψ = h • ϕ. Then we extend ϕ onto the complex plane C (keeping the same notation) and show that ϕ monotonically semiconjugates P | C to a topological polynomial g : C → C. If P does not have Siegel or Cremer periodic points this gives an alternative proof of Kiwi's fundamental results on locally connected models of dynamics on the Julia sets, but the results hold for all polynomials with connected Julia sets. We also give a characterization and a useful sufficient condition for the map ϕ not to collapse all of J into a point.

Abstract. Thurston defined invariant laminations, i.e. collections of chords of the unit circle S (called leaves) that are pairwise disjoint inside the open unit disk and satisfy a few dynamical properties. To be directly associated to a polynomial, a lamination has to be generated by an equivalence relation with specific properties on S; then it is called a q-lamination. Since not all laminations are q-laminations, then from the point of view of studying polynomials the most interesting are those of them which are limits of q-laminations. In this paper we introduce an alternative definition of an invariant lamination, which involves only conditions on the leaves (and avoids gap invariance). The new class of laminations is slightly smaller than that defined by Thurston and is closed. We use this notion to elucidate the connection between invariant laminations and invariant equivalence relations on S.

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