A1 Refereed original research article in a scientific journal
Apollonian metric, uniformity and Gromov hyperbolicity
Authors: Li YX, Vuorinen M, Zhou QS
Publisher: TAYLOR & FRANCIS LTD
Publication year: 2019
Journal: Complex Variables and Elliptic Equations
Journal name in source: COMPLEX VARIABLES AND ELLIPTIC EQUATIONS
Journal acronym: COMPLEX VAR ELLIPTIC
Number of pages: 14
ISSN: 1747-6933
eISSN: 1747-6941
DOI: https://doi.org/10.1080/17476933.2019.1579203
Self-archived copy’s web address: https://research.utu.fi/converis/portal/detail/Publication/40257516
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, phi-uniformity and delta-hyperbolicity (in the sense of Gromov with respect to quasihyperbolic metric) of proper domains of 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 phi-uniformity is invariant under quasimobius mappings.
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, phi-uniformity and delta-hyperbolicity (in the sense of Gromov with respect to quasihyperbolic metric) of proper domains of 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 phi-uniformity is invariant under quasimobius mappings.
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