It is well-known that the Hausdorff dimension is invariant under bi-Lipschitz mappings. I would be interested in a specific converse of this invariance. Let $X$ and $Y$ be two CAT(0) spaces having the same Hausdorff dimension. Can we find two open balls $B_1 subset X$ and $B_2subset Y$ and a bi-Lipschitz surjective mapping $fcolon B_1to B_2$?
Does it help if both spaces are geodesically complete?