  Every representation of the Cuntz algebra $\mathcal{O}_n$ leads to a unitary
representation of the Higman-Thompson group $V_n$. We consider the family
$\{\pi_x\}_{x\in [0,1[}$ of permutative representations of $\mathcal{O}_n$ that
arise from the interval map $f(x)=nx$ (mod 1) acting on the Hilbert space that
underlies each orbit, and then study the unitary equivalence and the
irreducibility of the corresponding family $\{\rho_x\}_{x\in [0,1[}$ of
representations of Higman-Thompson group $V_n$, showing that that these
representations are indeed irreducible and moreover $\rho_x$ and $\rho_y$ are
equivalent if and only if the orbits of $x$ and $y$ coincide.
