# Mapping properties of a scale invariant Cassinian metric and a Gromov hyperbolic metric

## TL;DR

This paper studies the properties of a scale invariant Cassinian metric and a Gromov hyperbolic metric, focusing on their distortion, continuity, and invariance under various maps.

## Contribution

It introduces new distortion and invariance results for these metrics under Möbius and quasiconformal maps.

## Key findings

- Distortion property of the Cassinian metric under Möbius maps
- Modulus of continuity for identity maps between metrics
- Quasi-invariance of metrics under quasiconformal maps

## Abstract

In this paper, we consider a scale invariant Cassinian metric and a Gromov hyperbolic metric. We discuss a distortion property of the scale invariant Cassinian metric under M\"obius maps of a punctured ball onto another punctured ball. We obtain a modulus of continuity of the identity map from a domain equipped with the scale invariant Cassinian metric (or the Gromov hyperbolic metric) onto the same domain equipped with the Euclidean metric. The quasi-invariance properties of both the metrics under quasiconformal maps are also established.