Differential codimensions and exponential growth

Let $A$ be a finite dimensional associative algebra with derivations over a field of characteristic zero, i.e., an algebra whose structure is enriched by the action of a Lie algebra $L$ by derivations, and let $c_n^L(A),$ $n\ge 1,$ be its differential codimension sequence. Such sequence is exponentially bounded and ${\exp }^L(A)={\lim }_{n\to \mathrm{\infty }}\sqrt[n]{c_n^L(A)}$ is an integer that can be computed, called differential PI-exponent of $A$. In this paper we prove that for any Lie algebra $L$, ${\exp }^L(A)$ coincides with $\exp (A)$, the ordinary PI-exponent of $A$. Furthermore, in case $L$ is a solvable Lie algebra, we apply such result to classify varieties of $L$-algebras of almost polynomial growth, i.e., varieties of exponential growth such that any proper subvariety has polynomial growth.