Asymptotics of H-identities for associative algebras with an H-invariant radical

We prove the existence of the Hopf PI-exponent for finite dimensional associative algebras A with a generalized Hopf action of an associative algebra H   with 1 over an algebraically closed field of characteristic 0 assuming only the invariance of the Jacobson radical J(A)J(A) under the H  -action and the existence of the decomposition of A/J(A)A/J(A) into the sum of H-simple algebras. As a consequence, we show that the analog of Amitsurʼs conjecture holds for G-codimensions of finite dimensional associative algebras over a field of characteristic 0 with an action of an arbitrary group G by automorphisms and anti-automorphisms and for differential codimensions of finite dimensional associative algebras with an action of an arbitrary Lie algebra by derivations.