Let $X$ be a subshift over a finitely generated group $G$.

Let $K(X)$ denote the smallest cardinality of any alphabet $A$ such that $X$ embeds (continuously and $G$-equivariantly) into $A^G$. For example, if $G$ is surjunctive, and $X$ is the full shift on $d$ letters, then $K(X)=d$.

Let $X^n$ denote the cartesian product of $n$ copies of $X$, equipped with the diagonal $G$-action--this is again a subshift over $G$. For example, if $X$ is the full shift on $d$ letters, then $X^n$ is the full shift on $d^n$ letters.

Because $K(X\times Y)\leq K(X)K(Y)$, the function $n\mapsto \log(K(X^n))$ is manifestly subadditive, so we know that $\frac{\log(K(X^n))}{n}$ converges. Let $K^*(X)=\exp\left(\lim_{n\rightarrow\infty}\frac{\log(K(X^n))}{n}\right)$. For example, if $G=\mathbb{Z}$ and $X$ consists of $6$ points, each having stabilizer $2\mathbb{Z}$, then $K^*(X)=6^{1/2}$, since if $\#A=d$, then $A^\mathbb{Z}$ has $d^2-d$ points of period exactly $2$.

**Questions**

First, is this invariant (i.e., $K^*$) well known?

Second, does $K^*(A^G)=\#A$ for all finite sets $A$ and all finitely generated groups $G$?

Finally, if $X$ is an nontrivial SFT over $G=\mathbb{Z}$, is it always true that $K^*(X)$ is a rational power of a natural number?