  In this paper we show that the minimal value of Furstenberg entropy (along
all measures, not restricting to stationary ones) for any amenable action is
the same as for the action of the group on itself. Using the boundary
amenability result of Adams, this allows us to compute the minimal value of the
entropy over all the measure classes in the boundary of the free group. Similar
results are proved for the action of a hyperbolic group on its Gromov boundary.
Our main tool is an ultralimit realization of the Poisson boundary of a time
dependent matrix-valued random walk on the group. This extends and refines the
results and tools of previous paper of the author with Y. Shalom.
