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

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.