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We analyze the ergodic properties of power-bounded operators on a reflexiveBanach space of the form "scalar plus compact-power", and show that they arealmost periodic (all the orbits are conditionally compact). If such an operatoris weakly mixing, then it is stable (its powers converge in the strong operatortopology).Let X-ISP be the separable reflexive indecomposable Banach spaceconstructed by Argyros and Motakis, in which every operator has an invariantsubspace. We conclude that every power-bounded operator on a closed subspace ofX-ISP is almost periodic.

Reflexive Banach spaces with all power-bounded operators almost periodic