On the structure of invariant measures for set-valued maps

Properties of measures invariant with respect to set-valued maps are studied. It is shown that an absolutely continuous invariant measure for a set-valued map need not be unique, and the set of all invariant measures need not be a Choquet simplex. The problem concerning the existence of invariant measures with respect to set-valued maps parametrized by single-valued and set-valued maps of the circle having various smoothness classes is studied.
Bibliography: 13 titles.