"münchhausen Trick" and Amenability of Self-similar Groups

The structure of a self-similar group G naturally gives rise to a transformation which assigns to any probability measure μ on G and any word w in the alphabet of the group a new probability measure μ. If μ is a convex combination of μ and the δ-measure at the group identity, then the asymptotic entropy of the random walk (G,μ) vanishes; therefore, the random walk is Liouville and the group G is amenable. Using this method we prove amenability of several classes of self-similar groups.