Random actions of homeomorphisms of Cantor sets embedded in a line and Tits alternative

In 2000, Margulis proved that any group of homeomorphisms of the circle either preserves a probabilty measure on the circle or contains a free subgroup in two generators, which is reminiscent of the Tits alternatve for linear groups. In this article, we prove an analogous statement for groups of locally monotonic homeomorphisms of a compact subset of R.