Quasi-visual approximations

TL;DR

This paper introduces the concept of quasi-visual approximations for bounded metric spaces, providing a new framework to analyze quasiconformal geometry, quasisymmetries, and dynamics of semi-hyperbolic rational maps.

Contribution

It develops the foundational theory of quasi-visual approximations, linking them to Gromov hyperbolic spaces and dynamical systems, and characterizes semi-hyperbolic rational maps via these approximations.

Findings

Quasi-visual approximations can detect quasisymmetries between metric spaces.

The framework connects to Gromov hyperbolic spaces through tile graphs.

Julia sets of semi-hyperbolic rational maps admit dynamical quasi-visual approximations.

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.