We study the classification problem of Polish metric spaces up to isometry and
the isometry groups of Polish metric spaces. In the framework of the descriptive
set theory of definable equivalence relations, we determine the exact complexity of
various classification problems concerning Polish metric spaces. We start with the
class of all Polish metric spaces and prove that it is Borel bireducible to the universal
orbit equivalence relation induced by Borel actions of Polish groups. We then turn
to special classes of Polish metric spaces, including locally compact, ultrametric,
zero-dimensional, homogeneous, and ultrahomogeneous spaces. In the investigation
of the classification problems we also obtain characterizations for isometry groups
of various classes of Polish metric spaces.

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On the classification of Polish metric spaces up to isometry

In this communication we present some recent results on the classification of Polish metric spaces up to isometry and on the isometry groups of Polish metric spaces. A Polish metric space is a complete separable metric space (X,d). Our first goal is to determine the exact complexity of the classification problem of general Polish metric spaces up to isometry. This work was motivated by a paper of Vershik [1998], where he remarks (in the beginning of Section 2): "The classification of Polish spaces up to isometry is an enormous task. More precisely, this classification is not 'smooth' in the modern terminology." Our Theorem 2.1 below quantifies precisely the enormity of this task. After doing this, we turn to special classes of Polish metric spaces and investigate the classification problems associated with them. Note that these classification problems are in principle no more complicated than the general one above. However, the determination of their exact complexity is not necessarily easier. The investigation of the classification problems naturally leads to some interesting results on the groups of isometries of Polish metric spaces. We shall also present these results below.

/pdf/polish-metric-spaces-their-classification-and-isometry-2ntjmri397.pdf

Polish Metric Spaces: Their Classification and Isometry Groups

We show that if a (locally compact) group $G$ acts properly on a locallycompact $\sigma$-compact space $X$ then there is a family of $G$-invariantproper continuous finite-valued pseudometrics which induces the topology of$X$. If $X$ is furthermore metrizable then $G$ acts properly on $X$ if and onlyif there exists a $G$-invariant proper compatible metric on $X$.

/pdf/proper-actions-and-proper-invariant-metrics-1bgiodcolh.pdf

Proper actions and proper invariant metrics

The Poincare-Bendixson Theorem and the development of the theory are presented — from the papers of Poincare and Bendixson to modern results.

/pdf/the-poincare-bendixson-theorem-from-poincare-to-the-xxist-4rktk5bjfj.pdf

The Poincaré-Bendixson Theorem: from Poincaré to the XXIst century

http://www.maths.lse.ac.uk/Personal/adam/BOst13.pdf

Topological regular variation: I. Slow variation

In the present paper we study conditions under which the group of isometries on a locally compact metric space is locally compact, or acts properly.

/pdf/on-the-group-of-isometries-on-a-locally-compact-metric-space-48ws4g7j7t.pdf

On the group of isometries on a locally compact metric space.

Actions by isometries

1. The structure of limit sets is important to understand the phase portraits of flows. A serious step in this direction is the study of minimal sets, at least because every compact invariant set contains such a set. It is known that a compact minimal set of a C-differentiable dynamical System on a 2-manifold M is either a fixed point, a periodic orbit or eise all of M, in which case the system is an irrational flow on the torus, see A.J. Schwartz [17] (note that the compactness of the manifold is not necessary, see P. Hartman [12], Ch. VII, 12. 1). Such a minimal set will be called simple in the sequel. The assumption in Schwartz's theorem that the dynamical system is at least Cdifferentiable is essential, äs it had been shown by A. Denjoy [8].

On minimal sets in 2-manifolds.

We study D-stable flows on orientable 2-manifolds of nite genus in connection with the topology of the underlying phase spaces. The description of the phase portrait is used to prove that a connected orientable 2-manifold of nite genus supporting a non-minimalD-stable flow must be homeomorphic to an open subset of the 2-sphere or the 2-torus. In the case of the presence of singularities we necessarily have an open subset of the 2-sphere.

/pdf/a-class-of-flows-on-2-manifolds-with-simple-recurrence-4rhg62cybb.pdf

A class of flows on 2-manifolds with simple recurrence

https://msp.org/pjm/1977/70-1/pjm-v70-n1-p25-s.pdf

Polychronis Strantzalos

Papers

On the group of isometries on a locally compact metric space.

Actions by isometries

On minimal sets in 2-manifolds.

A class of flows on 2-manifolds with simple recurrence

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