This preprint was accepted October 1996.
Contact: S. V. Buyalo
ABSTRACT: For every quasi-isometric map $f:X\to Y$ of Hadamard spaces we define its asymptotic limit $s_f$ which sends the boundary at infinity $\bin X$ to the cone $\Cin Y$ over $\bin Y$ and establish its analytical properties. In the case when $X$ and $Y$ are cocompact rank one spaces with respect to the same discrete isometry group $\Ga$ and hence $\Ga$-equivariantly quasi-isometric we give a sufficient condition for $s_f$ to be an equivariant homeomorphism between $\bin X$ and $\bin Y$ w.r.t. the standard topologies and biLipschitz homeomorphism w.r.t. Tits metrics. Apart of this condition there is a large number of equivariantly quasi-isometric cocompact Hadamard spaces whose boundaries at infinity are not equivariantly homeomorphic. This answers a question of M. Gromov.