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

For $pgeq 2$, the $p$-adic ring $C^*$-algebra $CQ_p$ 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 $chi_kin{ m End}(CQ_p)$ 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.