Codimensions of algebras with pseudoautomorphism and their exponential growth

Let $F$ be a fixed field of characteristic zero containing an element $i$ such that $i^2=-1$. In this paper we consider finite dimensional superalgebras over $F$ endowed with a pseudoautomorphism $p$ and we investigate the asymptotic behaviour of the corresponding sequence of $p$-codimensions $c_n^p(A),$ $n=1,2,\dots$. First we give a positive answer to a conjecture of Amitsur in this setting: the $p$-exponent ${\exp }^p(A)={\lim }_{n\to \mathrm{\infty }}\sqrt[n]{c_n^p(A)}$ always exists and it is an integer. In the final part we characterize the algebras whose exponential growth is bounded by $2$.