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Peaked periodic waves in the Camassa-Holm equation are revisited. Linearizedevolution equations are derived for perturbations to the peaked periodic wavesand linearized instability is proven both in $H^1$ and $W^{1,\infty}$ norms.Dynamics of perturbations in $H^1$ is related to the existence of two conservedquantities and is bounded in the full nonlinear system due to these conservedquantities. On the other hand, perturbations to the peaked periodic wave growin $W^{1,\infty}$ norm and may blow up in a finite time in the nonlinearevolution of the Camassa-Holm equation.

Growth of perturbations to the peaked periodic waves in the Camassa-Holm equation,