On the entropy and index of the winding endomorphisms of $p$-adic ring $C^*$-algebras

For $p\geq 2$, the $p$-adic ring $C^*$-algebra $\mathcal{Q}_p$ is the universal $C^*$-algebra generated by a unitary $U$ and an isometry $S_p$ such that $S_pU=U^pS_p$ and $\sum_{l=0}^{p-1}U^lS_pS_p^*U^{-l}=1$. For any $k$ coprime with $p$ we define an endomorphism $\chi_k\in{\rm End}(\mathcal{Q}_p)$ by setting $\chi_k(U):=U^k$ and $\chi_k(S_p):=S_p$. We then compute the entropy of $\chi_k$, which turns out to be $\log |k|$. Finally, for selected values of $k$ we also compute the Watatani index of $\chi_k$ showing that the entropy is the natural logarithm of the index.