  Consider an arbitrary automorphism of an Enriques surface with its lift to
the covering K3 surface. We prove a bound of the order of the lift acting on
the anti-invariant cohomology sublattice of the Enriques involution. We use it
to obtain some mod 2 constraint on the original automorphism. As an
application, we give a necessary condition for Salem numbers to be dynamical
degrees on Enriques surfaces and obtain a new lower bound on the minimal value.
In the Appendix, we give a complete list of Salem numbers that potentially may
be the minimal dynamical degree on Enriques surfaces and for which the
existence of geometric automorphisms is unknown.
