Apollonian metric, uniformity and Gromov hyperbolicity. Issue 2 (1st February 2020)

• Record Type: Journal Article
• Authors: Li, Yaxiang; Vuorinen, Matti; Zhou, Qingshan
• Is Part Of: Complex variables and elliptic equations. Volume 65:Issue 2(2020)
• Page Start: 215
• Page End: 228
• Publication Date: 2020-02-01
• Subjects: Roughly Apollonian bilipschitz mappings -- uniform domains -- Apollonian metric -- Gromov hyperbolicity; Primary: 30C65 -- 30F45 -- Secondary:30C20; Functions of complex variables -- Periodicals; Differential equations, Elliptic -- Periodicals; 515.905
• DOI: 10.1080/17476933.2019.1579203
• Languages: English
• ISSNs: 1747-6933

Abstract

ABSTRACT: The main purpose of this paper is to investigate the properties of a mapping which is required to be roughly bilipschitz with respect to the Apollonian metric (roughly Apollonian bilipschitz) of its domain. We prove that under these mappings the uniformity, ϕ -uniformity and δ -hyperbolicity (in the sense of Gromov with respect to quasihyperbolic metric) of proper domains of R n are invariant. As applications, we give four equivalent conditions for a quasiconformal mapping which is defined on a uniform domain to be roughly Apollonian bilipschitz, and we conclude that ϕ -uniformity is invariant under quasimöbius mappings.