Invariant Radon measures and minimal sets for subgroups of $\text{Homeo}_+(\mathbb{R})$

TL;DR

This paper explores the relationship between invariant Radon measures, minimal sets, and the structure of subgroups of orientation-preserving homeomorphisms of the real line, providing new insights into their dynamical properties.

Contribution

It establishes equivalences among invariant measures, minimal sets, and tower structures for subgroups without crossed elements, and shows that certain smoothness conditions guarantee invariant measures and minimal sets.

Findings

Equivalence between invariant measures, minimal sets, and tower absence for subgroups without crossed elements.

Nilpotent subgroups with $C^{1+eta}$ regularity always have invariant measures and minimal sets.

Counterexamples are provided for $C^1$ commutative subgroups lacking these properties.

Abstract

Let $G$ be a subgroup of $\text{Homeo}_+(\mathbb{R})$ without crossed elements. We show the equivalence among three items: (1) existence of $G$-invariant Radon measures on $\mathbb R$; (2) existence of minimal closed subsets of $\mathbb R$; (3) nonexistence of infinite towers covering the whole line. For a nilpotent subgroup $G$ of $\text{Homeo}_+(\mathbb{R})$, we show that $G$ always has an invariant Radon measure and a minimal closed set if every element of $G$ is $C^{1+\alpha} (\alpha>0$); a counterexample of $C^1$ commutative subgroup of $\text{Homeo}_+(\mathbb{R})$ is constructed.