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Some aspects of topological dynamics of Polish groups (with an introduction to descriptive set theory)

Julien Melleray

Abstract

The first part of these notes give an introduction to the theory of Polish group actions on compact Hausdorff spaces, leading up to a proof of the Kechris-Pestov-Todorcevic correspondence and discussions of properties of universal minimal flows. The second part proveides some background on descriptive set theory and culminates with B. Miller's proof of the $\mathcal{G}_0$-dichotomy theorem due to Kechris, Solecki, and Todorcevic.

Some aspects of topological dynamics of Polish groups (with an introduction to descriptive set theory)

Abstract

The first part of these notes give an introduction to the theory of Polish group actions on compact Hausdorff spaces, leading up to a proof of the Kechris-Pestov-Todorcevic correspondence and discussions of properties of universal minimal flows. The second part proveides some background on descriptive set theory and culminates with B. Miller's proof of the -dichotomy theorem due to Kechris, Solecki, and Todorcevic.
Paper Structure (21 sections, 148 theorems, 187 equations)

This paper contains 21 sections, 148 theorems, 187 equations.

Table of Contents

  1. Topological dynamics of Polish groups
  2. Polish groups and their actions
  3. Fraïssé limits and their automorphism groups
  4. Uniform spaces
  5. Uniform structures on Polish groups
  6. Compactifications
  7. The greatest ambit and universal minimal flow
  8. A theorem of Kechris, Pestov and Todorčevic
  9. Polish groups with a metrizable universal minimal flow
  10. Descriptive set theory
  11. Polish spaces
  12. Schemes and transfer theorems
  13. Borel and analytic subsets of Polish spaces
  14. Universal (co)-analytic sets: trees and orderings on $\omega$
  15. Definability and Baire measurability
  16. ...and 6 more sections

Key Result

Proposition I.1.6

Let $(X,d)$ be a Polish metric space, and $G$ be its isometry group. Then $G$, endowed with the pointwise convergence topology, is a Polish group.

Theorems & Definitions (374)

  • Definition I.1.1
  • Definition I.1.2
  • Definition I.1.3
  • Example I.1.4
  • Remark I.1.5
  • Proposition I.1.6
  • proof
  • Theorem I.1.7
  • proof
  • Definition I.1.8
  • ...and 364 more