On invariants of free restricted Lie algebras

We prove that the invariant subalgebra $L^G$ is infinitely generated, where $L=L(X)$ is the free restricted Lie algebra of finite rank $k$ with free generating set $X=\{x_1,\dots,x_k\}$ over an arbitrary field of positive characteristic and $G$ is a non-trivial finite group of homogeneous automorphisms of $L(X)$. We show that the sequence $|Y_n|$, $n\geqslant1$, grows exponentially with base $k$, where $Y=\bigcup_{n=1}^\infty Y_n$ is a free homogeneous generating set of $L^G$ and all the elements of $Y_n$ are of degree $n$ in $X$, $n\geqslant1$. We prove that the radius of convergence of the generating function $\mathcal H(Y,t)=\sum_{n=1}^\infty|Y_n|t^n$ is equal to $1/k$ and find an asymptotic formula for the growth of $\mathcal H(Y,t)$ as $t\to1/k-0$.