  We prove the analog of Amitsur's conjecture on asymptotic behavior for
codimensions of several generalizations of polynomial identities for finite
dimensional associative algebras over a field of characteristic 0, including
G-identities for any finite (not necessarily Abelian) group G and H-identities
for a finite dimensional semisimple Hopf algebra H. In addition, we prove that
the Hopf PI-exponent of Sweedler's 4-dimensional algebra with the action of its
dual equals 4.
