In this paper we further study links between concentration of measure in topological transformation groups, existence of fixed points, and Ramsey-type theorems for metric spaces. We prove that whenever the group Iso (U) of isometries of Urysohn’s universal complete separable metric space U, equipped with the compact-open topology, acts upon an arbitrary compact space, it has a fixed point. The ...

Let $p:Xlo B$ be a locally trivial principal G-bundle and $wt{p}:wt{X}lo B$ be a locally trivial principal $wt{G}$-bundle. In this paper, by using the structure of principal bundles according to transition functions, we show that $wt{G}$ is a covering group of $G$ if and only if $wt{X}$ is a covering space of $X$. Then we conclude that a topological space $X$ with non-simply connected universal...

Part 1. Analogies 3 1. Poincaré recurrence vs. Birkhoff’s recurrence 3 1.1. Poincaré recurrence theorem and topological recurrence 3 1.2. The existence of Borel cross-sections 4 1.3. Recurrence sequences and Poincaré sequences 5 2. The equivalence of weak mixing and continuous spectrum 7 3. Disjointness: measure vs. topological 10 4. Mild mixing: measure vs. topological 12 5. Distal systems: to...

Part 1. Analogies 3 1. Poincaré recurrence vs. Birkhoff’s recurrence 3 1.1. Poincaré recurrence theorem and topological recurrence 3 1.2. The existence of Borel cross-sections 4 1.3. Recurrence sequences and Poincaré sequences 5 2. The equivalence of weak mixing and continuous spectrum 7 3. Disjointness: measure vs. topological 10 4. Mild mixing: measure vs. topological 12 5. Distal systems: to...

Introduction. A topological group © is said to act on a topological space 1{ if the elements of © are homeomorphisms of 'R. onto itself and if the mapping (a{p) of ©X^ onto %_ is continuous. Familiar examples include the rotation group acting on the Cartesian plane and the Euclidean group acting on Euclidean space. The set ®(p) (that is, the set of all <r(p) where <r(E®) is called the o...

In this article we introduce the concept of $z^circ$-filter on a topological space $X$. We study and investigate the behavior of $z^circ$-filters and compare them  with corresponding ideals, namely, $z^circ$-ideals of $C(X)$,  the ring of real-valued continuous functions on a completely regular Hausdorff space $X$. It is observed that $X$ is a compact space if and only if every $z^circ$-filter ...

We define and study a quantale-valued Wijsman structure on the hyperspace of all non-empty closed sets of a quantale-valued metric space. We show its admissibility and that the metrical coreflection coincides with the quantale-valued Hausdorff metric and that, for a metric space, the topological coreflection coincides with the classical Wijsman topology. We further define an index of compactnes...

We study the topological entropy of a particular class of dynamical systems: cellular automata. The topological entropy of a dynamical system (X; F) is a measure of the complexity of the dynamics of F over the space X . The problem of computing (or even approximating) the topological entropy of a given cellular automata is algorithmically undecidable (Ergodic Theory Dynamical Systems 12 (1992) ...

We are interested in the computability of the invariant measures in a computable dynamical system. We construct two counter-examples. The first one has a unique SRB measure, which is not computable. The second one has no computable invariant measure at all. The systems are topological, i.e. continuous transformations on compact spaces, so they admit invariant measures. A topological dynamical s...

Paul Garrett [email protected] http://www.math.umn.edu/ g̃arrett/ [This document is http://www.math.umn.edu/ ̃garrett/m/fun/choice functions.pdf] This note is essentially a rewrite of Appendix V of J. Dixmier’s von Neumann Algebras, North Holland, 1981. The point is to obtain the practical special case: [0.0.1] Corollary: Let X be a locally compact Hausdorff second-countable [1] topological sp...