Actions of Ore extensions and growth of polynomial $H$-identities

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.