  We show that amenability of a group acting by homeomorphisms can be deduced
from a certain local property of the action and recurrency of the orbital
Schreier graphs. This covers amenability of a wide class groups, the
amenability of which was an open problem, as well as unifies many known
examples to one general proof. In particular, this includes Grigorchuk's group,
Basilica group, the full topological group of Cantor minimal system, groups
acting on rooted trees by bounded automorphisms, groups generated by finite
automata of linear activity growth, groups that naturally appear in holomorphic
dynamics.
