Homogeneous spaces and transitive actions by Polish groups

We prove that for every homogeneous and strongly locally homogeneous Polish space X there is a Polish group admitting a transitive action on X. We also construct an example of a homogeneous Polish space which is not a coset space and on which no separable metrizable topological group acts transitively.     

On subgroups of minimal topological groups

A topological group is minimal if it does not admit a strictly coarser Hausdorff group topology. The Roelcke uniformity (or lower uniformity) on a topological group is the greatest lower bound of the left and right uniformities. A group is Roelcke-precompact if it is precompact with respect to the Roelcke uniformity. Many naturally arising non-Abelian topological groups are Roelcke-precompact and hence have a natural compactification. We use such compactifications to prove that some groups of isometries are minimal. In particular, if U₁ is the Urysohn universal metric space of diameter 1, the group Iso(U₁) of all self-isometries of U₁ is Roelcke-precompact, topologically simple and minimal. We also show that every topological group is a subgroup of a minimal topologically simple Roelcke-precompact group of the form Iso(M), where M is an appropriate non-separable version of the Urysohn space.     

Almost isometric embeddings into configuration-compacta

We show that a Polish metric space X which has ‘compact configuration spaces’ is free of almost isometric embeddings, i.e. given such an embedding into X one gets an isometric embedding. By assuming a continuous version of ℵ₀-homogeneity we get the converse statement, and also prove almost isometry uniqueness.     

Isometric actions on pseudo-Riemannian nilmanifolds

This work deals with the structure of the isometry group of pseudo-Riemannian 2-step nilmanifolds. We study the action by isometries of several groups and we construct examples showing substantial differences with the Riemannian situation; for instance, the action of the nilradical of the isometry group does not need to be transitive. For a nilpotent Lie group endowed with a left-invariant pseudo-Riemannian metric, we study conditions for which the subgroup of isometries fixing the identity element equals the subgroup of isometric automorphisms. This set equality holds for pseudo- $H$ -type Lie groups.     

Stationary probability measures and topological realizations

We establish the generic inexistence of stationary Borel probability measures for aperiodic Borel actions of countable groups on Polish spaces. Using this, we show that every aperiodic continuous action of a countable group on a compact Polish space has an invariant Borel set on which it has no σ-compact realization.     

Sensitivity and chaos of semigroup actions

We prove that if an action of a C-semigroup S on a Polish space is syndetic transitive, then the system is either minimal and equicontinuous, or sensitive. Additionally, we show that if an action of an abelian monoid S on a Polish space has a transitive point x and a periodic orbit O such that [`(Hx)]\overline{Hx} is perfect where H={s∈S:s| _O is an identity map}, then the system is chaotic.     

On the geometry of Urysohn's universal metric space

In recent years, much interest was devoted to the Urysohn space U and its isometry group; this paper is a contribution to this field of research. We mostly concern ourselves with the properties of isometries of U, showing for instance that any Polish metric space is isometric to the set of fixed points of some isometry φ. We conclude the paper by studying a question of Urysohn, proving that compact homogeneity is the strongest homogeneity property possible in U.     

Metric ends,fibers and automorphisms of graphs

Several results on the action of graph automorphisms on ends and fibers are generalized for the case of metric ends. This includes results on the action of the automorphisms on the end space, directions of automorphisms, double rays which are invariant under a power of an automorphism and metrically almost transitive automorphism groups. It is proved that the bounded automorphisms of a metrically almost transitive graph with more than one end are precisely the kernel of the action on the space of metric ends. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On sensitive sets and regionally proximal sets of group actions

Semigroup Forum - We introduce the concepts of S-sets and Q-sets for a flow (a group action on a compact metric space) and prove that a transitive flow is sensitive if and only if there exists an...     

Coupled-expanding maps and one-sided symbolic dynamical systems

This paper studies relationships between coupled-expanding maps and one-sided symbolic dynamical systems. The concept of coupled-expanding map is extended to a more general one: coupled-expansion for a transitive matrix. It is found that the subshift for a transitive matrix is strictly coupled-expanding for the matrix in certain disjoint compact subsets; the topological conjugacy of a continuous map in its compact invariant set of a metric space to a subshift for a transitive matrix has a close relationship with that the map is strictly coupled-expanding for the matrix in some disjoint compact subsets. A certain relationship between strictly coupled-expanding maps for a transitive matrix in disjoint bounded and closed subsets of a complete metric space and their topological conjugacy to the subshift for the matrix is also obtained. Dynamical behaviors of subshifts for irreducible matrices are then studied and several equivalent statements to chaos are obtained; especially, chaos in the sense of Li–Yorke is equivalent to chaos in the sense of Devaney for the subshift, and is also equivalent to that the domain of the subshift is infinite. Based on these results, several new criteria of chaos for maps are finally established via strict coupled-expansions for irreducible transitive matrices in compact subsets of metric spaces and in bounded and closed subsets of complete metric spaces, respectively, where their conditions are weaker than those existing in the literature.     

On completion of fuzzy metric spaces

Completions of fuzzy metric spaces (in the sense of George and Veeramani) are discussed. A complete fuzzy metric space Y is said to be a˜fuzzy metric completion of a˜given fuzzy metric space X if X is isometric to a˜dense subspace of Y. We present an example of a˜fuzzy metric space that does not admit any fuzzy metric completion. However, we prove that every standard fuzzy metric space has an (up to isometry) unique fuzzy metric completion. We also show that for each fuzzy metric space there is an (up to uniform isomorphism) unique complete fuzzy metric space that contains a˜dense subspace uniformly isomorphic to it.     

Devaney’s chaos implies distributional chaos in a sequence

Let X be a complete metric space without isolated points, and let f:X→X be a continuous map. In this paper we prove that if f is transitive and has a periodic point of period p, then f is distributionally chaotic in a sequence. Particularly, chaos in the sense of Devaney is stronger than distributional chaos in a sequence.     

Transitive Isometry Groups of Aspheric Riemannian Manifolds

We consider the isometry groups of Riemannian solvmanifolds and also study a wider class of homogeneous aspheric Riemannian spaces. We clarify the topological structure of these spaces (Theorem 1). We demonstrate that each Riemannian space with a maximally symmetric metric admits an almost simply transitive action of a Lie group with triangular radical (Theorem 2). We apply this result to studying the isometry groups of solvmanifolds and, in particular, solvable Lie groups with some invariant Riemannian metric.     

On the geometry of the space of oriented lines of Euclidean space

We prove that the space of all oriented lines of the n-dimensional Euclidean space admits a pseudo-Riemannian metric which is invariant by the induced transitive action of a connected closed subgroup of the group of Euclidean motions, exactly when n=3 or n=7 (as usual, we consider Riemannian metrics as a particular case of pseudo-Riemannian ones). Up to equivalence, there are two such metrics for each dimension, and they are of split type and complete. Besides, we prove that the given metrics are Kähler or nearly Kähler if n=3 or n=7, respectively.     

Classification of four-dimensional transitive local Lie groups of transformations of R 4 and their two-point invariants

In the theory of physical structures the classification of metric functions (both on a single set and on two ones) plays an important role. A metric function represents a two-point invariant of a certain local Lie transformation group. Moreover, one can uniquely restore this group with the help of the invariance condition. According to this theorem, in order to find all metric functions, it suffices to construct the complete classification of local Lie transformation groups. In this paper we classify Lie algebras of simply transitive local Lie groups of local transformations of a four-dimensional space, and then we define metric functions. The obtained results admit application in physics, in particular, in thermodynamics.     

Strongly Bounded Groups and Infinite Powers of Finite Groups

We define a group as strongly bounded if every isometric action on a metric space has bounded orbits. This latter property is equivalent to the so-called uncountable strong cofinality, recently initiated by Bergman.

Our main result is that G ^I is strongly bounded when G is a finite, perfect group and I is any set. This strengthens a result of Koppelberg and Tits. We also prove that ω₁-existentially closed groups are strongly bounded.     

Free continuous actions on zero-dimensional spaces

We show that every countably infinite group admits a free, continuous action on the Cantor set having an invariant probability measure. We also show that every countably infinite group admits a free, continuous action on a non-homogeneous compact metric space and the action is minimal (that is to say, every orbit is dense). In answer to a question posed by Giordano, Putnam and Skau, we establish that there is a continuous, minimal action of a countably infinite group on the Cantor set such that no free continuous action of any group gives rise to the same equivalence relation.     

Polish group actions: Dichotomies and generalized elementary embeddings

We prove that any Polish group which admits a complete left-invariant metric satisfies the Topological Vaught Conjecture. We also generalize some theorems of model theory from the logic actions to other Polish group actions.