Codimensions of algebras with additional structures

Let A be an associative algebra endowed with an automorphism or an antiautomorphism phi of order <= 2. One associates to A, in a natural way, a numerical sequence c(n)(phi)(A), n = 1, 2, ... , called the sequence of phi-codimensions of A which is the main tool for the quantitative investigation of the polynomial identities satisfied by A. In [13] it was proved that such a sequence is eventually nondecreasing in case phi is an antiautomorphism. Here we prove that it still holds in case phi is an automorphism and present some recent results about the asymptotics of c(n)(phi)(A).