On invariants of modular free Lie algebras

Suppose that $L(X)$ is a free Lie algebra of finite rank over a field of positive characteristic. Let $G$ be a nontrivial finite group of homogeneous automorphisms of $L(X)$. It is known that the subalgebra of invariants $H=L^G$ is infinitely generated. Our goal is to describe how big its free generating set is. Let $Y=\bigcup_{n=1}^\infty Y_n$ be a homogeneous free generating set of $H$, where elements of $Y_n$ are of degree $n$ with respect to $X$. We describe the growth of the generating function of $Y$ and prove that $|Y_n|$ grow exponentially.