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We investigate Sarnak's M\"obius Disjointness Conjecture throughasymptotically periodic functions. It is shown that Sarnak's conjecture forrigid dynamical systems is equivalent to the disjointness of M\"obius fromasymptotically periodic functions. We give sufficient conditions and a partialanswer to the later one. As an application, we show that Sarnak's conjectureholds for a class of rigid dynamical systems, which improves an earlier resultof Kanigowski-Lema{\'{n}}czyk-Radziwi{\l}{\l}.

Disjointness of M\"{o}bius from asymptotically periodic functions