  For a compact set of actions, an invariant of Kushnirenko's entropy type is
chosen in such a way that on this set it is equal to zero, but will be infinity
for typical actions. As a consequence, we show that typical measure-preserving
transformations are not isomorphic to geometric shape exchange transformations.
This problem arose in connection with the result of Chaika and Davis about the
atypical nature of IETs.
