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Circular orders: Topology and continuous actions

Michael Megrelishvili

TL;DR

This work develops a self-contained topology of circular orders, introducing GCOTS and convex uniform structures that yield a robust framework for COP compactifications and $G$-actions. It connects Novak's regular completion with a uniformity perspective, showing a precise correspondence between COP compactifications and precompact convex uniformities, and it extends these ideas through inverse limits and split-space constructions. The authors generalize Helly-type selection, bounded-variation theory, and fragmentation to circular and generalized ordered spaces, establishing tameness for compact circularly ordered dynamical systems and representations on Rosenthal spaces. The results unify order-theoretic, topological, and dynamical perspectives to study tame dynamics arising from circular orders, with implications for G-compactifications, invariant function spaces, and dynamical representations. Overall, the paper advances both the theory of circular orders and its applications to topological dynamics by providing systematic methods to construct, analyze, and represent COP-compactifications and their actions.

Abstract

We study the topology of circularly ordered sets. While the algebraic notion is classical, the general topological theory has received comparatively little attention. In this work we provide a self-contained topological exposition and present several new directions and results: Initiate a systematic study of generalized circularly ordered topological spaces and of continuous group actions on them. Provide a convex uniform structure description of circularly ordered compactifications. This yields a topological analysis of Novak's regular completion and its minimal uniformity. Show that this uniform structures approach implies several new results in the theory of proper $G$-compactifications for topological group actions on abstract ordered spaces. Reexamine functions of bounded variation on circularly ordered sets and prove generalizations of Helly's selection theorem (for circular and linear orders). These developments and the systematic analysis of circular order topologies are motivated by recent applications in topological dynamics, particularly in joint works with Eli Glasner, which demonstrate that circularly ordered dynamical systems provide a natural class of tame dynamics.

Circular orders: Topology and continuous actions

TL;DR

This work develops a self-contained topology of circular orders, introducing GCOTS and convex uniform structures that yield a robust framework for COP compactifications and -actions. It connects Novak's regular completion with a uniformity perspective, showing a precise correspondence between COP compactifications and precompact convex uniformities, and it extends these ideas through inverse limits and split-space constructions. The authors generalize Helly-type selection, bounded-variation theory, and fragmentation to circular and generalized ordered spaces, establishing tameness for compact circularly ordered dynamical systems and representations on Rosenthal spaces. The results unify order-theoretic, topological, and dynamical perspectives to study tame dynamics arising from circular orders, with implications for G-compactifications, invariant function spaces, and dynamical representations. Overall, the paper advances both the theory of circular orders and its applications to topological dynamics by providing systematic methods to construct, analyze, and represent COP-compactifications and their actions.

Abstract

We study the topology of circularly ordered sets. While the algebraic notion is classical, the general topological theory has received comparatively little attention. In this work we provide a self-contained topological exposition and present several new directions and results: Initiate a systematic study of generalized circularly ordered topological spaces and of continuous group actions on them. Provide a convex uniform structure description of circularly ordered compactifications. This yields a topological analysis of Novak's regular completion and its minimal uniformity. Show that this uniform structures approach implies several new results in the theory of proper -compactifications for topological group actions on abstract ordered spaces. Reexamine functions of bounded variation on circularly ordered sets and prove generalizations of Helly's selection theorem (for circular and linear orders). These developments and the systematic analysis of circular order topologies are motivated by recent applications in topological dynamics, particularly in joint works with Eli Glasner, which demonstrate that circularly ordered dynamical systems provide a natural class of tame dynamics.

Paper Structure

This paper contains 19 sections, 85 theorems, 85 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Circular and linear orders: basic properties
  3. Partial and linear orders
  4. Topology of circular orders
  5. Convexity and generalized Helly space
  6. Generalized Helly space
  7. Completeness, compactifications and inverse limits
  8. Uniform structures and uniform covers
  9. Circularly ordered inverse limits and compactifications
  10. Split spaces and topological properties of COTS
  11. Generalized circularly ordered spaces and COP actions
  12. Group actions on GLOTS
  13. Novák's regular completion and COP compactifications
  14. Convex uniform structures on GCOTS
  15. Compactifications of order preserving group actions
  16. ...and 4 more sections

Key Result

Lemma 2.2

$$$\Rightarrow [a,x,b]$.

Figures (2)

  • Figure 1: c-ordered lexicographic product (from Wikipedia - Cyclic order)
  • Figure 3: loxodromic idempotent $p$

Theorems & Definitions (209)

  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Definition 2.5
  • Proposition 2.6
  • proof
  • Lemma 2.7
  • Lemma 2.8
  • proof
  • ...and 199 more