Quasi-visual approximations
Mario Bonk, Mikhail Hlushchanka, Daniel Meyer
TL;DR
The paper develops a rigorous foundation for quasi-visual approximations, a framework that encodes the geometry of bounded metric spaces through hierarchical covers with controlled relative scales. It connects these combinatorial structures to quasisymmetries, Gromov hyperbolic geometry via tile graphs, and dynamical systems, notably Julia sets, by showing equivalences between metric/quasisymmetric data and combinatorial encodings. Key contributions include a precise four-condition definition of quasi-visual approximations, a complete treatment of their relation to visual approximations, a correspondence with Gromov boundaries, and a dynamical characterization of Julia sets for semi-hyperbolic maps through dynamical quasi-visual approximations. The results provide a unifying framework linking fractal geometry, geometric group theory, and complex dynamics, with potential applications to conformal dimension, CXC systems, and beyond.
Abstract
We develop the foundations of the theory of quasi-visual approximations of bounded metric spaces. Roughly speaking, these are sequences of covers of a given space for which the diameters of the sets in the covers shrink to zero and for which relative metric quantities (such as ratios of diameters and distances) are uniformly controlled. This framework has applications to questions in quasiconformal geometry. In particular, quasi-visual approximations can be used to detect whether a given homeomorphism between two bounded metric spaces is a quasisymmetry. We also explore the connection to the theory of Gromov hyperbolic spaces via the tile graph associated with a quasi-visual approximation. As an application, we relate these ideas to the dynamics of semi-hyperbolic rational maps. More specifically, we show that the Julia set of a rational map admits a dynamical quasi-visual approximation if and only if the map is semi-hyperbolic.