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 behaviour of the sequence of φ -codimensions c n φ ( A ) , n = 1 , 2 , … . More precisely, we shall prove that lim n → ∞ ⁡ c n φ ( 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.