Algebras simple with respect to a Taft algebra action

Algebras simple with respect to an action of a Taft algebra $H_{m^2}(\zeta)$ deliver an interesting example of $H$-module algebras that are $H$-simple but not necessarily semisimple. We describe finite dimensional $H_{m^2}(\zeta)$-simple algebras and prove the analog of Amitsur's conjecture for codimensions of their polynomial $H_{m^2}(\zeta)$-identities. In particular, we show that the Hopf PI-exponent of an $H_{m^2}(\zeta)$-simple algebra $A$ over an algebraically closed field of characteristic $0$ equals $\dim A$. The groups of automorphisms preserving the structure of an $H_{m^2}(\zeta)$-module algebra are studied as well.