Generic Properties of Nonexpansive Mappings

The project Generic properties of nonexpansive mappings is concerned with the study of properties of functions which do no increase distances. These functions appear naturally in various areas of mathematics, for example in optimization, where they play an important role. In this project, we do not consider properties which are shared by all of these functions but with properties which are typical for them. In this context being typical means that most of the functions do have this property. Since there are infinitely many nonexpansive mappings, we cannot decide what is typical by counting the functions with a certain property and compare their number with the number of all nonexpansive mappings but we have to use a different approach. One possibility of describing this approach is to look at the points of a disk in the plane: there are infinitely many such points but it seems clear that most of them do not lie on the boundary of the disc. A formal way of checking this statement is to see that arbitrarily close to every point on the boundary of the disk, there is a smaller disk which lies entirely outside the circle. The characterization we use for typical nonexpansive mappings is very similar to the corresponding characterization of the points in the interior of the disk. More precisely, we want to consider the question of finding geometric conditions such that the typical nonexpansive mapping has a fixed point, i.e. a point which is not changed by this mapping. Another property we are interested in the behavior of the typical slope of such a function. In some situations, e.g. when taking into account uncertainties, it makes sense to consider functions whose values do not consists of single points but of certain small sets. For such functions, if their values are compact, we want to consider methods to find fixed points in an iterative fashion. Also in this setting, it turns out that these methods do not work for all nonexpansive mapping and we consider the question of whether they work for typical such functions. A focus of this project is to study the connection between the typical behavior of nonexpansive mappings and the geometry of the underlying space. For the problems described above this is a completely new approach.

Nonexpansive mappings are functions which do no increase distances. These mappings play an important role in various areas of applied mathematics, e.g. optimisation. In addition they are of interest for purely theoretical questions in functional analysis. This project was concerned with typical properties of these mappings. In this context a property is called typical if the set of mappings enjoying it is a "large" set. We have been looking at a number of different notions of largeness and the connected notions of smallness. One possibility of describing one approach to largeness is to look at the points of a disk in the plane: there are infinitely many such points but it seems clear that most of them do not lie on the boundary of the disc. A formal way of checking this statement is to see that arbitrarily close to every point on the boundary of the disk, there is a smaller disk which lies entirely outside the circle. The characterization we use for typical nonexpansive mappings is very similar to the corresponding characterization of the points in the interior of the disk. We have been investigating how geometric properties of the space on which the mappings are defined influence the typical behaviour of the mappings. We managed to show that certain curvature conditions on the space allow us to conclude that the typical nonexpansive mapping has a fixed point, i.e. a point it does not change. For two classes of set-valued functions, i.e. functions whose valued are not points but sets, we were able to show that certain simple algorithms for finding fixed points work for the typical mapping. It has been know already that for mappings on unbounded domains in so called Hilbert spaces the behaviour is drastically different from the behaviour of mappings defined on bounded sets. We were able to extend these results to a much more general setting. In order to work on questions of large and small sets in abstract spaces, we also worked on obtaining a better understanding of so called Haar null sets. These form an important class of small sets. We were able to obtain a characterisation of Haar null sets in a certain class of sets.