Ill-posedness for the periodic Camassa--Holm type equations in the end-point critical Besov space $B_{\mathrm{\infty },1}^1$

For the real-line case, it is shown that both the Camassa–Holm \cite{Guo} and Novikov equations \cite{Li-arx} are ill-posed in $B_{\mathrm{\infty },1}^1$. In this paper, by presenting a new construction of initial data which leads to the norm inflation phenomena, we prove that both the periodic Camassa–Holm and Novikov equations are also ill-posed in $B_{\mathrm{\infty },1}^1$.