Short proofs of theorems of Malyutin and Margulis

The Ghys-Margulis alternative asserts that a subgroup G of homeomorphisms of the circle which does not contain a free subgroup on two generators must admit an invariant probability measure. Malyutin’s theorem classifies minimal actions of G. We present a short proof of Malyutin’s theorem and then deduce Margulis’ theorem which confirms the G-M alternative. The basic ideas are borrowed from the original work of Malyutin, but our proof is considerably shorter.