Codimension growth of algebras with superautomorphism

Let A be a finite dimensional algebra endowed with a superautomorphism over a field of characteristic zero. In this paper we study the asymptotic behavior of the sequence of φ-codimensions $c_n^{\phi }\left(A\right)$, $n=1,2,\dots$. More precisely, we shall prove that ${\lim }_{n\to \infty }\sqrt[n]{c_n^{\phi }\left(A\right)}$ always exists and it is an integer related in an explicit way to the dimension of a suitable semisimple subalgebra of A. This result gives a positive answer to a conjecture of Amitsur in this setting. In the final part of the paper we characterize the algebras whose exponential growth is bounded by 2.