What is known about the group of accumulation points of $(x^n)_n$ in a compact semigroup?

Question

Let $S$ be a compact Hausdorff semigroup. Let $x ∈ S$ and let $A$ be the set of accumulation points of $(x^n)_{n>0}$.

Here is a proof that $A$ is a group. I will write $ℕ$ for the set of strictly positive integers. Let $P := \overline{\{x^n\,|\,n∈ℕ\}}$. Define the following continuous functions, where $$ is the Stone-Čech compactification.

1. $f:(ℕ^2)→P^3$ with $f(n,k) = (x^n,x^k,x^{n+k})$.
2. $p:P^3→P^2$ with $p(x,y,z) = (x,z)$.
3. $q:(ℕ^2)→(ℕ)$ with $q(n,k) = k$.

Let $(a,b) ∈ A^2$. We can choose $U ∈ (ℕ^2)$ such that $p(f(U)) = (a,b)$ and such that $q(U)$ is nonprincipal. Indeed, for every neighborhood $V$ of $(a,b)$ and every $K∈ℕ$, there is $(n,k) ∈ ℕ^2$ such that $k≥K$ and $(s^n,s^{n+k}) ∈ V$. We have $f(U) = (a,y,b)$ with $ay = b$ and $y ∈ A$ since $U$ is nonprincipal. Thus, $aA = A$ and symmetrically $Aa = A$. This implies that $A$ is a group.

If $S$ is profinite, then $A$ is a quotient of the group $\widehat{ℤ}$ of profinite integers. What is known in general about the abelian group $A$? Is it the adherence of the positive powers of an element of $A$?