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We study fixed points of contractive convolution operators associated tocontractive quantum measures on locally compact quantum groups. We characterisethe existence of non-zero fixed points respectively on $L^\infty(\mathbb{G})$and on $C_0(\mathbb{G})$, and exploit these results to obtain for example thestructure of the fixed points on the non-commutative $L_p$-spaces. Someconsequences for the fixed points of classical convolution operators andHerz-Schur multipliers are also indicated.

Fixed points and limits of convolution powers of contractive quantum measures