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The Morse boundary of a proper geodesic metric space is designed to encodehypberbolic-like behavior in the space. A key property of this boundary is thata quasi-isometry between two such spaces induces a homeomorphism on their Morseboundaries. In this paper we investigate when the converse holds. We prove thatfor cocompact CAT(0) spaces, a homeomorphism of Morse boundaries is induced bya quasi-isometry if and only if the homeomorphism is quasi-mobius and 2-stable.

A rank-one CAT(0) group is determined by its Morse boundary