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 cnφ(A), n=1,2,…. More precisely, we shall prove that limn→∞⁡cnφ(A)n 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.