Ultralimits, amenable actions and entropy

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 [Boundary amenability for word hyperbolic groups and an application to smooth dynamics of simple groups, Topology 33(4) (1994) 765–783, https://doi.org/10.1016/0040-9383(94)90007-8], 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 in [E. Sayag and Y. Shalom, Entropy, ultralimits and the Poisson boundary, preprint (2022), arXiv:2202.06607v2].