NEISHTADT, A. and TRESCHEV, D., 2011. Polymorphisms and adiabatic chaos. Ergodic Theory and Dynamical Systems, 31(1), pp.259-284.
At the end of the last century Vershik introduced some dynamical systems, called
polymorphisms. Systems of this kind are multivalued self-maps of an interval, where
(roughly speaking) each branch has some probability. By definition, the standard Lebesgue
measure should be invariant. Unexpectedly, some class of polymorphisms appeared in
the problem of destruction of an adiabatic invariant after a multiple passage through a
separatrix. We discuss ergodic properties of polymorphisms from this class.