TY - JOUR

T1 - Metrics ρ, quasimetrics ρ and pseudometrics inf ρs

AU - Storozhuk, K. V.

PY - 2017/1/1

Y1 - 2017/1/1

N2 - Let ρ be a metric on a space X and let s≥1. The function ρs(a, b) = ρ(a, b)s is a quasimetric (it need not satisfy the triangle inequality). The function inf ρss(a, b) defined by the condition inf ρs(a, b) = inf(σn 0ρs(zi, zi+1) z0 = a, zn = b) is a pseudometric (i.e., satisfies the triangle inequality but can be degenerate). We show how this degeneracy can be connected with the Hausdorff dimension of the space (X,ρ). We also give some examples showing how the topology of the space (X, infρs) can change as s changes.

AB - Let ρ be a metric on a space X and let s≥1. The function ρs(a, b) = ρ(a, b)s is a quasimetric (it need not satisfy the triangle inequality). The function inf ρss(a, b) defined by the condition inf ρs(a, b) = inf(σn 0ρs(zi, zi+1) z0 = a, zn = b) is a pseudometric (i.e., satisfies the triangle inequality but can be degenerate). We show how this degeneracy can be connected with the Hausdorff dimension of the space (X,ρ). We also give some examples showing how the topology of the space (X, infρs) can change as s changes.

UR - http://www.scopus.com/inward/record.url?scp=85022000669&partnerID=8YFLogxK

U2 - 10.1090/ecgd/311

DO - 10.1090/ecgd/311

M3 - Article

AN - SCOPUS:85022000669

VL - 21

SP - 264

EP - 272

JO - Conformal Geometry and Dynamics

JF - Conformal Geometry and Dynamics

SN - 1088-4173

IS - 10

ER -