Quasi-isometries between visual hyperbolic spaces

We prove that a power quasi-symmetric (or PQ-symmetric) homeomorphism between two complete metric spaces can be extended to a quasi-isometry between their hyperbolic approximations. This result can be used to prove that two visual Gromov hyperbolic spaces are quasi-isometric if and only if there is a PQ-symmetric homeomorphism between their boundaries with bounded visual metrics. Also, in the case of trees, we prove that two geodesically complete trees are quasi-isometric if and only if there is a PQ-symmetric homeomorphism between their boundaries with visual metrics based at infinity. We also give a characterization for a map to be PQ-symmetric based on the relative distortion of subsets.