Maps on the Morse boundary

Abstract

For a proper geodesic metric space, the Morse boundary focuses on the hyperbolic-like directions in the space. The Morse boundary is a quasi-isometry invariant. That is, a quasi-isometry between two spaces induces a homeomorphism on their Morse boundaries. In this paper, we investigate additional structures on the Morse boundary which determine the space up to a quasi-isometry. We prove that a homeomorphism between the Morse boundaries of two proper, cocompact spaces is induced by a quasi-isometry if and only if both the homeomorphism and its inverse are bihölder, or quasi-symmetric, or strongly quasi-conformal.