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Non-locally discrete actions on the circle with at most $N$ fixed points

Christian Bonatti, João Carnevale, Michele Triestino

Abstract

A subgroup of $\mathrm{Homeo}_+(\mathbb{S}^1)$ is Möbius-like if every element is conjugate to an element of $\mathrm{PSL}(2,\mathbb{R})$. In general, a Möbius-like subgroup of $\mathrm{Homeo}_+(\mathbb{S}^1)$ is not necessarily (semi-)conjugate to a subgroup of $\mathrm{PSL}(2,\mathbb{R})$, as discovered by N. Kovačević [Trans. Amer. Math. Soc. 351 (1999), 4823-4835]. Here we determine simple dynamical criteria for the existence of such a (semi-)conjugacy. We show that Möbius-like subgroups of $\mathrm{Homeo}_+(\mathbb{S}^1)$ which are elementary (namely, preserving a Borel probability measure), are semi-conjugate to subgroups of $\mathrm{PSL}(2,\mathbb{R})$. On the other hand, we provide an example of elementary subgroup of $\mathrm{Diff}^\infty_+(\mathbb{S}^1)$ satisfying that every non-trivial element fixes at most 2 points, which is not isomorphic to any subgroup of $\mathrm{PSL}(2,\mathbb{R})$. Finally, we show that non-elementary, non-locally discrete subgroups acting with at most $N$ fixed points are conjugate to a dense subgroup of some finite central extension of $\mathrm{PSL}(2,\mathbb{R})$.

Non-locally discrete actions on the circle with at most $N$ fixed points

Abstract

A subgroup of is Möbius-like if every element is conjugate to an element of . In general, a Möbius-like subgroup of is not necessarily (semi-)conjugate to a subgroup of , as discovered by N. Kovačević [Trans. Amer. Math. Soc. 351 (1999), 4823-4835]. Here we determine simple dynamical criteria for the existence of such a (semi-)conjugacy. We show that Möbius-like subgroups of which are elementary (namely, preserving a Borel probability measure), are semi-conjugate to subgroups of . On the other hand, we provide an example of elementary subgroup of satisfying that every non-trivial element fixes at most 2 points, which is not isomorphic to any subgroup of . Finally, we show that non-elementary, non-locally discrete subgroups acting with at most fixed points are conjugate to a dense subgroup of some finite central extension of .
Paper Structure (9 sections, 18 theorems, 41 equations)

This paper contains 9 sections, 18 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Topology of the circle and of the group of circle homeomorphisms
  4. Semi-conjugacy of group actions on the circle
  5. Elementary groups
  6. Basic results
  7. Möbius-like elementary groups
  8. A group of smooth diffeomorphisms which is not isomorphic to any Möbius group
  9. Non-locally discrete groups with at most $N$ fixed points

Key Result

Theorem 1

If $G< {\mathrm{Homeo}}_+(\mathbb{S}^1)$ is an elementary, Möbius-like subgroup, then $G$ is continuously semi-conjugate to an elementary subgroup of ${\mathrm{PSL}}(2,{\mathbb R})$ and, moreover, the corresponding morphism $G\to {\mathrm{PSL}}(2,{\mathbb R})$ is injective.

Theorems & Definitions (48)

  • Remark 1.2
  • Theorem 1
  • Theorem 1.3: Solodov
  • Theorem 1.4: Kovačević
  • Theorem 2
  • Definition 1.5
  • Theorem 3
  • Remark 1.6
  • Corollary 4
  • Remark 1.7
  • ...and 38 more