Invariant Radon measures and minimal sets for subgroups of [formula omitted

Abstract

Let G be a subgroup of Homeo+(R) without crossed elements. We show the equivalence among three items: (1) existence of G-invariant Radon measures on R; (2) existence of minimal closed subsets of R; (3) nonexistence of infinite towers covering the whole line. For a nilpotent subgroup G of Homeo+(R), we show that G always has an invariant Radon measure and a minimal closed set if every element of G is C1+α(α>0); a counterexample of C1 commutative subgroup of Homeo+(R) is constructed.