  We study fixed points of contractive convolution operators associated to
contractive quantum measures on locally compact quantum groups. We characterise
the 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 the
structure of the fixed points on the non-commutative $L_p$-spaces. Some
consequences for the fixed points of classical convolution operators and
Herz-Schur multipliers are also indicated.
