Quasisymmetric mappings in b-metric spaces

Considering quasisymmetric mappings between b-metric spaces we have found a new estimation for the ratio of diameters of two subsets which are images of two bounded subsets. This result generalizes the well-known Tukia-Vaisala inequality. The condition under which the image of a b-metric space under a quasisymmetric mapping is also a b-metric space is established. Moreover, the latter question is investigated for additive metric spaces.

Fractal Percolation and Quasisymmetric Mappings

We study the conformal dimension of fractal percolation and show that, almost surely, the conformal dimension of a fractal percolation is strictly smaller than its Hausdorff dimension.

LARGE-SCALE SUBLINEARLY LIPSCHITZ GEOMETRY OF HYPERBOLIC SPACES

Large-scale sublinearly Lipschitz maps have been introduced by Yves Cornulier in order to precisely state his theorems about asymptotic cones of Lie groups. In particular, Sublinearly bi-Lipschitz Equivalences (SBE) are a weak variant of quasi-isometries, with the only requirement of still inducing bi-Lipschitz maps at the level of asymptotic cones. We focus here on hyperbolic metric spaces and study properties of boundary extensions of SBEs, reminiscent of quasi-Möbius (or quasisymmetric) mappings. We give a dimensional invariant of the boundary that allows to distinguish hyperbolic symmetric spaces up to SBE, answering a question of Druţu.