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We show that the decay of approximation numbers of compact compositionoperators on the Dirichlet space $\mathcal{D}$ can be as slow as we wish, whichwas left open in the cited work. We also prove the optimality of a result ofO.~El-Fallah, K.~Kellay, M.~Shabankhah and A.~Youssfi on boundedness on$\mathcal{D}$ of self-maps of the disk all of whose powers are norm-bounded in$\mathcal{D}$.

Two remarks on composition operators on the Dirichlet space