Fixed Point Theorems for Maps With Local and Pointwise Contraction Properties

This paper constitutes a comprehensive study of ten classes of self-maps on metric spaces ⟨X, d⟩ with the pointwise (i.e., local radial) and local contraction properties. Each of these classes appeared previously in the literature in the context of fixed point theorems.We begin with an overview of these fixed point results, including concise self contained sketches of their proofs. Then we proceed with a discussion of the relations among the ten classes of self-maps with domains ⟨X, d⟩ having various topological properties that often appear in the theory of fixed point theorems: completeness, compactness, (path) connectedness, rectifiable-path connectedness, and d-convexity. The bulk of the results presented in this part consists of examples of maps that show non-reversibility of the previously established inclusions between these classes. Among these examples, the most striking is a differentiable auto-homeomorphism f of a compact perfect subset X of ℝ with f′ ≡ 0, which constitutes also a minimal dynamical system. We finish by discussing a few remaining open problems on whether the maps with specific pointwise contraction properties must have the fixed points.

On the structure of perfect sets in various topologies associated with tree forcings

We prove that the Ellentuck, Hechler and dual Ellentuck topologies are perfect isomorphic to one another. This shows that the structure of perfect sets in all these spaces is the same. We prove this by finding homeomorphic embeddings of one space into a perfect subset of another. We prove also that the space corresponding to eventually different forcing cannot contain a perfect subset homeomorphic to any of the spaces above.

On Intersections of Small Perfect Sets

For a not empty perfect subset of the unit circle C there is a perfect subset of C measure zero which being rotated to every position intersects the first set on a nonempty perfect set. This result may be stated in terms of set of distances between pairs of points from these two sets. A generalization of this result to a product of tori is suggested.

Schrödinger Operators with Fairly Arbitrary Spectral Features

It is shown, using methods of inverse-spectral theory, that there exist Schrödinger operators on the line with fairly general spectral features. Thus, for instance, it follows from the main theorem, that if Σ is any perfect subset of (-∞, 0], then there exist potentials qj, j = 1, 2 such that the associated Schrödinger operators Hj are self-adjoint and satisfy: σ(Hj)=Σ∪[0, ∞), σ ac (Hj)=[0, ∞), σ pp (H1)=σ sc (H2) =Σ. The main result also implies the existence of states with interesting transport properties.

Mansfield and Solovay type results on covering plane sets by lines

F. van Engelen, K. Kunen, and A. W. Miller proved, in [EKM], that for every analytic set A on the plane, either A can be covered by a countable family of lines or else there is a perfect subset P of A such that no three points of P are collinear. In this paper, we present some generalizations of their result. In particular, a question which was raised by van Engelen et al. in the last paragraph of [EKM] is answered (see Section 3).

The perfect set theorem and definable wellorderings of the continuum

Let Γ be a collection of relations on the reals and let M be a set of reals. We call M a perfect set basis for Γ if every set in Γ with parameters from M which is not totally included in M contains a perfect subset with code in M. A simple elementary proof is given of the following result (assuming mild regularity conditions on Γ and M): If M is a perfect set basis for Γ, the field of every wellordering in Γ is contained in M. An immediate corollary is Mansfield's Theorem that the existence of a Σ21 wellordering of the reals implies that every real is constructible. Other applications and extensions of the main result are also given.

The Reproductive Behaviour of Gynandromorphic Drosophila melanogaster

Reproductive behaviour was studied in 192 gynandromorphs with female genitalia and reproductive system, produced by ring-X chromosome loss. Male and female behaviour patterns were frequently found to coexist in the same individuals, and male courtship behaviour, when it occurred, retained its characteristic hierarchical organisation. Sexually receptive individuals were found to be an almost perfect subset of those ovipositing, and the control of both of these behaviours mapped to the head, as did male orientation (courtship). High rates of wing flicking, a response of males to courtship, mapped rather to the thorax, although a quantitative analysis demonstrated that the frequency of flicking behaviour was also influenced by male tissue in the head. In non-ovipositing individuals mature oocytes were retained in the ovary. An egg held in the uterus is not deposited by a fly without female tissue in the head and all but one sexually receptive individuals laid eggs. It is therefore concluded that both of these behaviours depend upon closely related neural circuitry operating the genital musculature under control from the brain.

Some results about Markov processes on subsets of the state space

Kingman [5] proved a formula that expresses the joint distribution of the processes where b is a regular point in the state space of a Hunt process. We give an extension of this formula, as well as several interesting facts related to it, for the case when Φ is any finely perfect subset of the state space. We also establish some connections between this result and results on last-exit decompositions.

Some results about Markov processes on subsets of the state space

Kingman [5] proved a formula that expresses the joint distribution of the processes where b is a regular point in the state space of a Hunt process. We give an extension of this formula, as well as several interesting facts related to it, for the case when Φ is any finely perfect subset of the state space. We also establish some connections between this result and results on last-exit decompositions.