Markov-Kakutani Theorem on Hyperspace of a Banach Space

Suppose $X$ is a Banach space and $K$ is a compact convex subset of $X$. Let $\mathcal{F}$ be a commutative family of continuous affine mappings of $K$ into $K$. It follows from Markov-Kakutani Theorem that $\mathcal{F}$ has a common fixed point in $K$. Suppose now $(CC(X), h)$ is the corresponding hyperspace of $X$ containing all compact, convex subsets of $X$ endowed with Hausdorff metric $h$. We shall prove the above version of Markov-Kakutani Theorem is valid on the hyperspace $(CC(X), h)$.

An abstract common fixed point principle and its applications

We establish a common fixed point principle for a commutative family of self-maps on an abstract set. This principle easily yields the Markoff-Kakutani theorem for affine maps, Kirk's theorem for nonexpansive maps and Cano's theorem for maps on the unit interval. As another application we obtain a new common fixed point theorem for a commutative family of maps on an arbitrary interval, which generalises an earlier result of Mitchell.

Local product structure for group actions

A differentiable G-space is introduced, for a Lie group G, into which every countably separated Borel G-space can be imbedded. The imbedding can be a continuous map if the space is a separable metric space. Such a G-space is called a universal G-space. This universal G-space has a local product structure for the action of G. That structure is inherited by invariant subspaces, giving a local product structure on general G-spaces. This information is used to prove that G-spaces are stratified by the subsets consisting of points whose orbits have the same dimension, to prove that G-spaces with stabilizers of constant dimension are foliated, to give a short proof that closed subgroups of Lie groups are Lie groups, to give a new proof and a stronger version of the Ambrose—Kakutani Theorem and to give a new proof of the existence of near-slices at points having compact stabilizers and hence of the existence of slices for Cartan G-spaces.

Partitioning the plane into denumerably many sets without repeated distances

Ceder(1) proved (assuming the axiom of choice, as we do throughout this paper) that the Euclidean plane can be partitioned into ℵ0 sets none of which contains an equilateral triangle; indeed he proved that given any denumerable set of triangles, the plane can be partitioned into ℵ0 sets, none containing a triangle similar to one of the given triangles. Erdős and Kakutani(2) proved that the continuum hypothesis implies that the real line can be partitioned into ℵ0, rationally independent sete, and that the existence of such a partition implies the continuum hypothesis. Erdős asked (private communication) whether there is a partition of the plane into ℵ0 sets not containing an isosceles triangle, or more generally in which any four points determine six different distances. Assuming the continuum hypothesis, it will be shown here (Theorem 1) that a, partition of the latter kind does exist. (I communicated this result to Erdős and others some years ago, but subsequently noticed that the argument was incomplete.) Conversely (Theorem 2) the existence of such a partitition, even for the line, implies the continuum hypothesis. This strengthening of the converse half of the Erdős-Kakutani theorem is proved by what is essentially their method (actually in a rather simplified form).

A Mapping Problem and Jp-Index. I

Indices of normal spaces with countable basis for equivariant mappings have been investigated by Bourgin [4; 6] and by Wu [11; 12] in the case where the transformation groups are of prime order p. One of us has extended the concept to the case where the transformation group is a cyclic group of order pt and discussed its applications to the Kakutani Theorem (see [10]). In this paper we will define the Jp-index of a normal space with countable basis in the case where the transformation group is a cyclic group of order n, where n is divisible by p. We will decide, by means of the spectral sequence technique of Borel [1; 2], the Jp-index of SO(n) where n is an odd integer divisible by p. The method used in this paper can be applied to find the Jp-index of a classical group G whose cohomology ring over Jp has a system of universally transgressive generators of odd degrees.