Games orbits play and obstructions to Borel reducibility

TL;DR

This paper introduces a game-theoretic framework for analyzing orbit equivalence relations, providing new proofs and criteria that obstruct classification by orbits, with applications to various operator and set relations.

Contribution

It presents a novel game-theoretic approach to anti-classification, offers a new dynamical obstruction criterion, and applies these to key relations in descriptive set theory and operator algebras.

Findings

A short proof of Hjorth's turbulence theorem using the new framework

A dynamical criterion obstructing classification by CLI group orbits

Applications to equality of countable sets of reals and unitary conjugacy relations

Abstract

We introduce a new, game-theoretic approach to anti-classification results for orbit equivalence relations. Within this framework, we give a short conceptual proof of Hjorth's turbulence theorem. We also introduce a new dynamical criterion providing an obstruction to classification by orbits of CLI groups. We apply this criterion to the relation of equality of countable sets of reals, and the relations of unitary conjugacy of unitary and selfadjoint operators on the separable infinite-dimensional Hilbert space.