  We show that if $A$ is a finite dimensional associative $H$-module algebra
for an arbitrary Hopf algebra $H$, then the proof of the analog of Amitsur's
conjecture for $H$-codimensions of $A$ can be reduced to the case when $A$ is
$H$-simple. (Here we do not require that the Jacobson radical of $A$ is an
$H$-submodule.) As an application, we prove that if $A$ is a finite dimensional
associative $H$-module algebra where $H$ is a Hopf algebra $H$ over a field of
characteristic $0$ such that $H$ is constructed by an iterated Ore extension of
a finite dimensional semisimple Hopf algebra by skew-primitive elements (e.g.
$H$ is a Taft algebra), then there exists integer
$\mathop{\mathrm{PIexp}}^H(A)$. In order to prove this, we study the structure
of algebras simple with respect to an action of an Ore extension.
